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.github/workflows/ci.ymlnew file+43
since splitsnapshots
unifiedsplit
@@ -0,0 +1,43 @@
1+name: CI
2+
3+# Trigger the workflow on push or pull request, but only for the master branch
4+on:
5+ pull_request:
6+ push:
7+ branches: [master]
8+
9+jobs:
10+ build:
11+ name: ghc ${{ matrix.ghc }}
12+ runs-on: ubuntu-16.04
13+ strategy:
14+ matrix:
15+ cabal: ["3.0"]
16+ ghc:
17+ - "8.6.5"
18+ - "8.8.3"
19+
20+ steps:
21+ - uses: actions/checkout@v2
22+ if: github.event.action == 'opened' || github.event.action == 'synchronize' || github.event.ref == 'refs/heads/master'
23+
24+ - uses: actions/setup-haskell@v1
25+ name: Setup Haskell
26+ with:
27+ ghc-version: ${{ matrix.ghc }}
28+ cabal-version: ${{ matrix.cabal }}
29+
30+ - uses: actions/cache@v1
31+ name: Cache ~/.cabal/store
32+ with:
33+ path: ~/.cabal/store
34+ key: ${{ runner.os }}-${{ matrix.ghc }}-cabal
35+
36+ - name: Build
37+ run: |
38+ cabal v2-update
39+ cabal v2-build --enable-tests --enable-benchmarks
40+
41+ - name: Test
42+ run: |
43+ cabal v2-test --enable-tests
.gitignore+54−2
@@ -1,3 +1,55 @@
1+### Haskell
2+dist
3+dist-*
4+cabal-dev
5+*.o
6+*.hi
7+*.chi
8+*.chs.h
9+*.dyn_o
10+*.dyn_hi
11+*.prof
12+*.aux
13+*.hp
14+*.eventlog
15+.virtualenv
16+.hsenv
17+.hpc
18+.cabal-sandbox/
19+cabal.sandbox.config
20+cabal.config
21+cabal.project.local
22+.ghc.environment.*
23+.HTF/
24+# Stack
1.stack-work/
25.stack-work/
2−laop.cabal
26+stack.yaml.lock
27+
28+### IDE/support
29+# Vim
30+[._]*.s[a-v][a-z]
31+[._]*.sw[a-p]
32+[._]s[a-v][a-z]
33+[._]sw[a-p]
3−*~
34+*~
35+tags
36+
37+# IntellijIDEA
38+.idea/
39+.ideaHaskellLib/
40+*.iml
41+
42+# Atom
43+.haskell-ghc-mod.json
44+
45+# VS
46+.vscode/
47+
48+# Emacs
49+*#
50+.dir-locals.el
51+TAGS
52+
53+# other
54+.DS_Store
55+hackage.sh
.travis.ymlnew file+47
@@ -0,0 +1,47 @@
1+sudo: true
2+language: haskell
3+
4+git:
5+ depth: 5
6+
7+cabal: "3.0"
8+
9+cache:
10+ directories:
11+ - "$HOME/.cabal/store"
12+ - "$HOME/.stack"
13+ - "$TRAVIS_BUILD_DIR/.stack-work"
14+
15+matrix:
16+ include:
17+ - ghc: 8.6.5
18+ - ghc: 8.8.3
19+
20+ - ghc: 8.8.3
21+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack-8.8.3.yaml"
22+
23+ - ghc: 8.6.5
24+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack.yaml"
25+
26+
27+install:
28+ - |
29+ if [ -z "$STACK_YAML" ]; then
30+ cabal v2-update
31+ cabal v2-build --enable-tests --enable-benchmarks
32+ else
33+ curl -sSL https://get.haskellstack.org/ | sh
34+ stack --version
35+ stack build --system-ghc --test --bench --no-run-tests --no-run-benchmarks
36+ fi
37+
38+script:
39+ - |
40+ if [ -z "$STACK_YAML" ]; then
41+ cabal v2-test --enable-tests
42+ else
43+ stack test --system-ghc
44+ fi
45+
46+notifications:
47+ email: false
CHANGELOG.mdnew file+17
@@ -0,0 +1,17 @@
1+# Changelog
2+
3+`laop` uses [PVP Versioning][1].
4+The changelog is available [on GitHub][2].
5+
6+## 0.1.1.1
7+
8+* Bump base version to work with cabal
9+
10+## 0.1.1.0
11+
12+* Package is more organized and now has tests and benchmarks separated
13+* Repository is more organized and clean
14+* Added CI
15+
16+[1]: https://pvp.haskell.org
17+[2]: https://github.com/bolt12/laop2/releases
ChangeLog.mddeleted−3
@@ -1,3 +0,0 @@
1−# Changelog for laop
2−
3−## Unreleased changes
LICENSE+17−26
@@ -1,30 +1,21 @@
1−Copyright Armando Santos (c) 2019
1+MIT License
2
2
3−All rights reserved.
3+Copyright (c) 2020 Armando Santos
4
4
5−Redistribution and use in source and binary forms, with or without
5+Permission is hereby granted, free of charge, to any person obtaining a copy
6+of this software and associated documentation files (the "Software"), to deal
7+in the Software without restriction, including without limitation the rights
8+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
9+copies of the Software, and to permit persons to whom the Software is
6−modification, are permitted provided that the following conditions are met:
10+furnished to do so, subject to the following conditions:
7
11
8− * Redistributions of source code must retain the above copyright
12+The above copyright notice and this permission notice shall be included in all
9− notice, this list of conditions and the following disclaimer.
13+copies or substantial portions of the Software.
10
14
11− * Redistributions in binary form must reproduce the above
15+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
12− copyright notice, this list of conditions and the following
13− disclaimer in the documentation and/or other materials provided
14− with the distribution.
15−
16− * Neither the name of Armando Santos nor the names of other
17− contributors may be used to endorse or promote products derived
18− from this software without specific prior written permission.
19−
20−THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
21−"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
22−LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
16+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
23−A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
17+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
24−OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
18+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
25−SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
26−LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
27−DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
28−THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
19+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
29−(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
20+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
30−OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
21+SOFTWARE.
README.md+107−24
@@ -1,5 +1,11 @@
1# laop - Linear Algebra of Programming library
1# laop - Linear Algebra of Programming library
2
2
3+[![GitHub CI](https://github.com/bolt12/laop/workflows/CI/badge.svg)](https://github.com/bolt12/laop/actions)
4+[![Build status](https://img.shields.io/travis/bolt12/laop.svg?logo=travis)](https://travis-ci.org/bolt12/laop)
5+[![Hackage](https://img.shields.io/hackage/v/laop.svg?logo=haskell)](https://hackage.haskell.org/package/laop)
6+[![Stackage Lts](http://stackage.org/package/laop/badge/lts)](http://stackage.org/lts/package/laop)
7+[![Stackage Nightly](http://stackage.org/package/laop/badge/nightly)](http://stackage.org/nightly/package/laop)
8+
3The LAoP discipline generalises relations and functions treating them as
9The LAoP discipline generalises relations and functions treating them as
4Boolean matrices and in turn consider these as arrows.
10Boolean matrices and in turn consider these as arrows.
5
11
@@ -10,33 +16,38 @@ motivation behind the library, the underlying theory, and implementation details
10This module offers many of the combinators mentioned in the work of
16This module offers many of the combinators mentioned in the work of
11[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).
17[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).
12
18
13−See the candidate package in hackage [here](https://hackage.haskell.org/package/laop-0.1.0.0/candidate)
19+See the package in hackage [here](https://hackage.haskell.org/package/laop-0.1.1.0)
20+
21+A Functional Pearl has been written and can be regarded as the [reference document](https://github.com/bolt12/tymfgg-pearl) for this library.
14
22
15## Features
23## Features
16
24
17−This library offers:
25+This library offers 3 main matrix programming modules:
18−
26+
19−- An internal representation that uses normal datatypes (`Void`, `()`, `Either`) for matrix dimensions and two
27+- One in which matrices are typed with type level natural numbers;
20− type families `FromNat` and `Count` to make it easier to work with these type of matrices.
28+- One in which matrices are typed with arbitrary generic data types;
21−- A module that uses generalised types for the dimensions and is just a newtype wrapper
29+- One in which matrices are regarded as relations, i.e. boolean matrices;
22− around the internal matrix data type.
30+
23−- Other that uses type level naturals for the dimensions and is just a newtype wrapper
31+There's also an experimental module that uses matrices to represent probability
24− around the internal matrix data type.
32+distributions.
25−- A very simple Probabilistic Programming module and several functions and data types to
33+
26− make it easier to deal with sample space.
34+The most interesting feature is that matrices are represented as an inductive data type.
27−
35+Given this formulation, matrix algorithms can be expressed in a much more elegant and
28−Given this, this new matrix formulation compared to other libraries one has much more advantages, such as:
36+calculational style than the traditional vector of vectors representation. This data type
29−
37+guarantees that a matrix will always have valid dimensions. It can also express block matrix
30−- Has an inductive definition enabling writing matrix manipulation functions in a much more elegant and calculational way;
38+computations naturally, which leads to total, efficient and statically typed manipulation
31−- Statically typed dimensions;
39+and transformation functions.
32−- Polymorphic data type dimensions;
40+
33−- Polymorphic matrix content;
41+Like matrix multiplication, other common operations, such as matrix transposition, benefit from a block-oriented structure that leads to a simple and natural divide-and-conquer algorithmic solution. Performance wise, this means that without much effort we can obtain optimal cache-oblivious algorithms.
34−- Awesome dimension type inference;
42+
35−- Fast type natural conversion via `FromNat` type family;
43+Given this, this new matrix formulation compared to other libraries:
36−- Matrix `Junc` and `Split`-ing in O(1);
44+
37−- Matrix composition takes advantage of divide-and-conquer and fusion laws.
45+- Is more compositional and polymorphic and does not have partial matrix manipulation functions (hence less chances for usage errors);
38−
46+- Our implementation of matrices enables simple manipulation of submatrices, making it particularly suitable for formal verification and equation reasoning, using the mathematical framework defined by the linear algebra of programming. Furthermore, the data type constructors ensure that the matrices of this kind are sound, i.e. malformed matrices with incorrect dimensions of the sort, can not be constructed.
47+
48+## Known issues
49+
39−Unfortunately, this approach does not solve the issue of type dimensions being in some way constrainted making it impossible to write Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed.
50+Unfortunately, due to the use of type-level programming features, this approach sometimes requires type dimensions to be constrainted, in some way, impossible to write idiomatic Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed. And not all kinds of programs can be modeled using matrices, namely programs that deal with arbitrary infinite data types such as lists or integers (although there are some workarounds).
40
51
41## Notes
52## Notes
42
53
@@ -118,6 +129,73 @@ grass_wet = row [0,1] . kp1
118rainning :: Matrix Double (Bool, (Bool, Bool)) One
129rainning :: Matrix Double (Bool, (Bool, Bool)) One
119rainning = row [0,1] . kp2 . kp2
130rainning = row [0,1] . kp2 . kp2
120
131
132+-- Alcuin Puzzle
133+
134+data Being = Farmer | Fox | Goose | Beans
135+ deriving (Bounded, Enum, Eq, Show, Generic)
136+
137+data Bank = LeftB | RightB
138+ deriving (Bounded, Enum, Eq, Show, Generic)
139+
140+eats :: Being -> Being -> Bool
141+eats Fox Goose = True
142+eats Goose Beans = True
143+eats _ _ = False
144+
145+eatsR :: R.Relation Being Being
146+eatsR = R.toRel eats
147+
148+cross :: Bank -> Bank
149+cross LeftB = RightB
150+cross RightB = LeftB
151+
152+crossR :: R.Relation Bank Bank
153+crossR = R.fromF' cross
154+
155+-- | Initial state, everyone in the left bank
156+locationLeft :: Being -> Bank
157+locationLeft _ = LeftB
158+
159+locationLeftR :: R.Relation Being Bank
160+locationLeftR = R.fromF' locationLeft
161+
162+-- | Initial state, everyone in the right bank
163+locationRight :: Being -> Bank
164+locationRight _ = RightB
165+
166+locationRightR :: R.Relation Being Bank
167+locationRightR = R.fromF' locationRight
168+
169+-- Properties
170+
171+-- Being at the same bank
172+sameBank :: R.Relation Being Bank -> R.Relation Being Being
173+sameBank = R.ker
174+
175+-- Risk of somebody eating somebody else
176+canEat :: R.Relation Being Bank -> R.Relation Being Being
177+canEat w = sameBank w `R.intersection` eatsR
178+
179+-- "Starvation" property.
180+inv :: R.Relation Being Bank -> Bool
181+inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)
182+ where
183+ farmer :: R.Relation Being Being
184+ farmer = R.fromF' (const Farmer)
185+
186+-- Arbitrary state
187+bankState :: Being -> Bank -> Bool
188+bankState Farmer LeftB = True
189+bankState Fox LeftB = True
190+bankState Goose RightB = True
191+bankState Beans RightB = True
192+bankState _ _ = False
193+
194+bankStateR :: R.Relation Being Bank
195+bankStateR = R.toRel bankState
196+
197+-- Main
198+
121main :: IO ()
199main :: IO ()
122main = do
200main = do
123 putStrLn "Monty Hall Problem solution:"
201 putStrLn "Monty Hall Problem solution:"
@@ -132,6 +210,8 @@ main = do
132 prettyPrint (grass_wet . state)
210 prettyPrint (grass_wet . state)
133 putStrLn "\n Probability of rain:"
211 putStrLn "\n Probability of rain:"
134 prettyPrint (rainning . state)
212 prettyPrint (rainning . state)
213+ putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
214+ print (inv bankStateR)
135```
215```
136
216
137```Shell
217```Shell
@@ -190,4 +270,7 @@ Monty Hall Problem solution:
190┌ ┐
270┌ ┐
191│ 0.2 │
271│ 0.2 │
192└ ┘
272└ ┘
273+
274+Is the arbitrary state a valid state? (Alcuin Puzzle)
275+False
193```
276```
Setup.hsdeleted−2
@@ -1,2 +0,0 @@
1−import Distribution.Simple
2−main = defaultMain
appveyor.ymlnew file+34
@@ -0,0 +1,34 @@
1+clone_folder: "c:\\WORK"
2+clone_depth: 5
3+
4+# Do not build feature branch with open Pull Requests
5+skip_branch_with_pr: true
6+
7+platform:
8+ - x86_64
9+
10+cache:
11+ - "C:\\SR"
12+ - dist-newstyle
13+
14+environment:
15+ global:
16+ CABOPTS: --store-dir=C:\\SR
17+
18+ matrix:
19+ - GHCVER: 8.6.5
20+
21+install:
22+ - choco source add -n mistuke -s https://www.myget.org/F/mistuke/api/v2
23+ - choco install -y cabal --version 2.4.1.0
24+ - choco install -y ghc --version 8.6.3
25+ - refreshenv
26+
27+before_build:
28+ - cabal --version
29+ - ghc --version
30+ - cabal %CABOPTS% v2-update
31+
32+build_script:
33+ - cabal %CABOPTS% v2-build --enable-tests
34+ - cabal %CABOPTS% v2-test --enable-tests
benchmark/Main.hsnew file+175
@@ -0,0 +1,175 @@
1+{-# LANGUAGE KindSignatures #-}
2+{-# LANGUAGE GeneralizedNewtypeDeriving #-}
3+{-# LANGUAGE DerivingVia #-}
4+{-# LANGUAGE DataKinds #-}
5+{-# LANGUAGE BangPatterns #-}
6+{-# LANGUAGE TypeApplications #-}
7+{-# LANGUAGE FlexibleContexts #-}
8+{-# LANGUAGE AllowAmbiguousTypes #-}
9+{-# LANGUAGE ScopedTypeVariables #-}
10+
11+module Main (main) where
12+
13+import LAoP.Dist.Internal
14+import LAoP.Utils
15+import LAoP.Matrix.Type hiding (CountableN, Countable, FLN)
16+import qualified LAoP.Matrix.Type as T (FLN)
17+import Criterion.Main
18+import Control.DeepSeq
19+import Test.QuickCheck
20+import GHC.TypeLits
21+import Data.Proxy
22+import Prelude hiding (id, (.))
23+
24+selectM2 ::
25+ ( Num e,
26+ T.FLN b b,
27+ CountableN b
28+ ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
29+selectM2 m y = join y iden `comp` m
30+
31+selectD2 ::
32+ ( FLN b b,
33+ CountableN b
34+ ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
35+selectD2 (D d) m = D (join m iden `comp` d)
36+
37+-- Composition with iden
38+compId m = comp iden m
39+
40+randomDist :: forall a . (Countable a, FLN () a) => Gen (Dist a)
41+randomDist = do
42+ let size = fromInteger (natVal (Proxy :: Proxy (Count a)))
43+ l <- vectorOf size (arbitrary :: Gen Prob)
44+ let ln = normalize l
45+ lr = map (:[]) ln
46+ m = fromLists lr
47+ return (D m)
48+
49+randomMatrix :: forall a b . (CountableDims a b, FLN a b) => Gen (Matrix Prob a b)
50+randomMatrix = do
51+ let cols = fromInteger (natVal (Proxy :: Proxy (Count a)))
52+ rows = fromInteger (natVal (Proxy :: Proxy (Count b)))
53+ l <- vectorOf (cols * rows) arbitrary
54+ let lr = buildList l cols
55+ m = fromLists lr
56+ return m
57+
58+buildList [] _ = []
59+buildList l r = take r l : buildList (drop r l) r
60+
61+normalize :: [Prob] -> [Prob]
62+normalize l = let l' = map abs l
63+ s = sum l'
64+ in map (/ s) l'
65+
66+-- Probability
67+newtype Probability = P Double
68+ deriving (Num, Show, Fractional, NFData) via Double
69+newtype Dist' a = D' { unD :: [(a,Probability)] }
70+ deriving (Show, NFData)
71+
72+instance Functor Dist' where
73+ fmap f (D' d) = D' [(f x,p) | (x,p) <- d]
74+
75+instance Applicative Dist' where
76+ pure x = D' [(x, 1)]
77+ fm <*> m = D' [(f x,q*p) | (f,p) <- unD fm, (x,q) <- unD m]
78+
79+class Applicative f => Selective f where
80+ select :: f (Either a b) -> f (a -> b) -> f b
81+
82+instance Selective Dist' where
83+ select x y = (\e f -> either f id e) <$> x <*> y -- selectA
84+
85+instance Monad Dist' where
86+ return = pure
87+ (D' d) >>= f = D' [(y,q*p) | (x,p) <- d, (y,q) <- unD (f x)]
88+
89+normalize2 :: forall a . (Enum a, Bounded a) => [Probability] -> [(a, Probability)]
90+normalize2 l = let l' = map abs l
91+ as = [minBound .. maxBound]
92+ s = sum l'
93+ probs = map (/ s) l'
94+ in zip as probs
95+
96+randomDist2 :: forall a . (Countable a, Enum a, Bounded a) => Gen (Dist' a)
97+randomDist2 = do
98+ let size = fromInteger (natVal (Proxy :: Proxy (Count a)))
99+ l <- vectorOf size (arbitrary :: Gen Prob)
100+ let ln = normalize2 @a (map P l)
101+ return (D' ln)
102+
103+normalize3 :: forall a . [a] -> [Probability] -> [(a, Probability)]
104+normalize3 as l = let s = sum l
105+ probs = map (/ s) l
106+ in zip as probs
107+
108+randomDistF :: forall a b . (CountableDims a b, CoArbitrary a, Arbitrary b) => Gen (Dist' (a -> b))
109+randomDistF = do
110+ let a = fromInteger (natVal (Proxy :: Proxy (Count a)))
111+ b = fromInteger (natVal (Proxy :: Proxy (Count b)))
112+ size = a * b
113+ l <- vectorOf size (arbitrary :: Gen Prob)
114+ l2 <- vectorOf size (arbitrary :: Gen (a -> b))
115+ let !ln = normalize3 l2 (map P l)
116+ return (D' ln)
117+
118+instance CoArbitrary (Natural a b)
119+
120+instance forall (a :: Nat) (b :: Nat). (KnownNat a, KnownNat b) => Arbitrary (Natural a b)
121+ where
122+ arbitrary =
123+ let bottom = fromInteger (natVal (Proxy :: Proxy a))
124+ top = fromInteger (natVal (Proxy :: Proxy b))
125+ in do
126+ x <- arbitrary `suchThat` (\n -> n >= bottom && n <= top)
127+ return (nat x)
128+
129+setupEnv = do
130+ m11 <- generate (resize 1 (randomMatrix @(Natural 0 10) @(Natural 0 10) ))
131+ m12 <- generate (resize 1 (randomMatrix @(Natural 0 10) @(Natural 0 10) ))
132+ m21 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
133+ m22 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
134+ m31 <- generate (resize 1 (randomMatrix @(Natural 0 200) @(Natural 0 200) ))
135+ m32 <- generate (resize 1 (randomMatrix @(Natural 0 200) @(Natural 0 200) ))
136+ return (m11, m12, m21, m22, m31, m32)
137+
138+setupEnv2 = do
139+ m21 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
140+ m40 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Either (Natural 0 100) (Natural 0 100) )))
141+ dist <- generate (resize 1 (randomDist @(Either (Natural 0 100) (Natural 0 100)) ))
142+ dist2 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
143+ distList1 <- generate (resize 1 (randomDist2 @(Either (Natural 0 100) (Natural 0 100))))
144+ distList2 <- generate (resize 1 (randomDistF @(Natural 0 100) @(Natural 0 100)))
145+ return (m21, m40, dist, dist2, distList1, distList2)
146+
147+main :: IO ()
148+main = defaultMain [
149+ env setupEnv $ \ ~(m11, m12, m21, m22, m31, m32) -> bgroup "Matrix Composition" [
150+ bgroup "10x10" [
151+ bench "WHNF - 10x10" $ whnf (comp m11) m12
152+ , bench "NF - 10x10" $ nf (comp m11) m12
153+ ],
154+ bgroup "100x100" [
155+ bench "WHNF - 100x100" $ whnf (comp m21) m22
156+ , bench "NF - 100x100" $ nf (comp m21) m22
157+ ],
158+ bgroup "200x200" [
159+ bench "WHNF - 200x200" $ whnf (comp m31) m32
160+ , bench "NF - 200x200" $ nf (comp m31) m32
161+ ],
162+ bgroup "200x200 - RULES" [
163+ bench "No rules - 200x200" $ nf (comp m21) iden
164+ , bench "Rules - 200x200" $ nf compId m21
165+ ] ],
166+ env setupEnv2 $ \ ~(m21, m40, dist, dist2, dl1, dl2) -> bgroup "Matrix vs List - `select`" [
167+ bgroup "Distribution `select` - 100+100 / 100x100" [
168+ bench "List Distribution - Applicative version" $ nf (select dl1) dl2
169+ , bench "Matrix Distribution - Applicative version" $ nf (selectD2 dist) dist2
170+ , bench "Matrix Distribution - Selective version" $ nf (selectD dist) dist2
171+ ],
172+ bgroup "Matrix `select` - 100x(100+100) / 100x100" [
173+ bench "Applicative version" $ nf (selectM2 m40) m21
174+ , bench "Selective version" $ nf (selectM m40) m21
175+ ] ] ]
laop.cabalnew file+89
@@ -0,0 +1,89 @@
1+cabal-version: 2.4
2+name: laop
3+version: 0.1.1.1
4+synopsis: Matrix programming library
5+description: Matrix programming library that uses a type-safe inductive definition
6+ inspired by the Linear Algebra of Programming
7+homepage: https://github.com/bolt12/laop
8+bug-reports: https://github.com/bolt12/laop/issues
9+license: MIT
10+license-file: LICENSE
11+author: Armando Santos
12+maintainer: Armando Santos <armandoifsantos@gmail.com>
13+copyright: 2020 Armando Santos
14+category: Data
15+build-type: Simple
16+extra-doc-files: README.md
17+ CHANGELOG.md
18+tested-with: GHC == 8.6.5
19+ GHC == 8.8.3
20+
21+source-repository head
22+ type: git
23+ location: https://github.com/bolt12/laop.git
24+
25+common common-options
26+ build-depends: base >= 4.12.0.0 && < 5,
27+ deepseq
28+
29+ ghc-options: -Wall
30+ -Wcompat
31+ -Widentities
32+ -Wincomplete-uni-patterns
33+ -Wincomplete-record-updates
34+ if impl(ghc >= 8.0)
35+ ghc-options: -Wredundant-constraints
36+ if impl(ghc >= 8.2)
37+ ghc-options: -fhide-source-paths
38+ if impl(ghc >= 8.4)
39+ ghc-options: -Wmissing-export-lists
40+ -Wpartial-fields
41+ if impl(ghc >= 8.8)
42+ ghc-options: -Wmissing-deriving-strategies
43+
44+ default-language: Haskell2010
45+
46+library
47+ import: common-options
48+ hs-source-dirs: src
49+ exposed-modules: LAoP.Dist
50+ LAoP.Dist.Internal
51+ LAoP.Matrix.Internal
52+ LAoP.Matrix.Nat
53+ LAoP.Matrix.Type
54+ LAoP.Relation
55+ LAoP.Relation.Internal
56+ LAoP.Utils
57+ LAoP.Utils.Internal
58+
59+test-suite laop-test
60+ import: common-options
61+ type: exitcode-stdio-1.0
62+ hs-source-dirs: test
63+ other-modules: Examples.Readme
64+ main-is: Spec.hs
65+ build-depends: laop
66+ ghc-options: -threaded
67+ -rtsopts
68+ -with-rtsopts=-N
69+ -funfolding-use-threshold=16
70+ -fexcess-precision
71+ -optc-O3
72+ -optc-ffast-math
73+
74+benchmark laop-benchmark
75+ import: common-options
76+ type: exitcode-stdio-1.0
77+ hs-source-dirs: benchmark
78+ main-is: Main.hs
79+ build-depends: criterion
80+ , QuickCheck
81+ , deepseq
82+ , laop
83+ ghc-options: -threaded
84+ -rtsopts
85+ -with-rtsopts=-N
86+ -funfolding-use-threshold=16
87+ -fexcess-precision
88+ -optc-O3
89+ -optc-ffast-math
package.yamldeleted−48
@@ -1,48 +0,0 @@
1−name: laop
2−version: 0.1.0.0
3−github: "bolt12/laop"
4−license: BSD3
5−author: "Armando Santos"
6−maintainer: "armandoifsantos@gmail.com"
7−copyright: "MIT"
8−
9−extra-source-files:
10−- README.md
11−- ChangeLog.md
12−
13−# Metadata used when publishing your package
14−# synopsis: Short description of your package
15−# category: Web
16−
17−# To avoid duplicated efforts in documentation and dealing with the
18−# complications of embedding Haddock markup inside cabal files, it is
19−# common to point users to the README.md file.
20−description: Please see the README on GitHub at <https://github.com/bolt12/laop#readme>
21−
22−dependencies:
23−- base >= 4.7 && < 5
24−
25−library:
26− source-dirs: src
27−
28−executables:
29− laop-exe:
30− main: Main.hs
31− source-dirs: app
32− ghc-options:
33− - -threaded
34− - -rtsopts
35− - -with-rtsopts=-N
36− dependencies:
37− - laop
38−
39−tests:
40− laop-test:
41− main: Spec.hs
42− source-dirs: test
43− ghc-options:
44− - -threaded
45− - -rtsopts
46− - -with-rtsopts=-N
47− dependencies:
48− - laop
src/LAoP/Dist.hs+10
@@ -36,6 +36,14 @@ module LAoP.Dist
36 Dist (..),
36 Dist (..),
37 Prob,
37 Prob,
38
38
39+ -- * Constraint type synonyms
40+ Countable,
41+ CountableN,
42+ CountableDimsN,
43+ FLN,
44+ Liftable,
45+ TrivialP,
46+
39 -- * Functor instance equivalent functions
47 -- * Functor instance equivalent functions
40 fmapD,
48 fmapD,
41
49
@@ -45,6 +53,8 @@ module LAoP.Dist
45
53
46 -- * Selective equivalent functions
54 -- * Selective equivalent functions
47 selectD,
55 selectD,
56+ branchD,
57+ ifD,
48
58
49 -- * Monad equivalent functions
59 -- * Monad equivalent functions
50 returnD,
60 returnD,
src/LAoP/Dist/Internal.hs+87−36
@@ -1,3 +1,5 @@
1+{-# LANGUAGE DerivingVia #-}
2+{-# LANGUAGE ConstraintKinds #-}
1{-# LANGUAGE GADTs #-}
3{-# LANGUAGE GADTs #-}
2{-# LANGUAGE AllowAmbiguousTypes #-}
4{-# LANGUAGE AllowAmbiguousTypes #-}
3{-# LANGUAGE TypeApplications #-}
5{-# LANGUAGE TypeApplications #-}
@@ -8,18 +10,27 @@
8{-# LANGUAGE RankNTypes #-}
10{-# LANGUAGE RankNTypes #-}
9
11
10module LAoP.Dist.Internal
12module LAoP.Dist.Internal
11− (
13+ (
12 Dist(..),
14 Dist(..),
13 Prob,
15 Prob,
14
16
17+ Countable,
18+ CountableN,
19+ CountableDimsN,
20+ FLN,
21+ Liftable,
22+ TrivialP,
23+
15 fmapD,
24 fmapD,
16 unitD,
25 unitD,
17 multD,
26 multD,
18 selectD,
27 selectD,
28+ branchD,
29+ ifD,
19 returnD,
30 returnD,
20 bindD,
31 bindD,
21 (??),
32 (??),
22−
33+
23 choose,
34 choose,
24 shape,
35 shape,
25 linear,
36 linear,
@@ -32,11 +43,15 @@ module LAoP.Dist.Internal
32 )
43 )
33 where
44 where
34
45
35−import LAoP.Matrix.Type
46+import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDims, CountableN, CountableDimsN, Liftable, FLN)
47+import Prelude hiding (id, (.))
48+import qualified LAoP.Matrix.Internal as I
36import LAoP.Utils
49import LAoP.Utils
37import GHC.TypeLits
50import GHC.TypeLits
38import Data.Proxy
51import Data.Proxy
39import Data.List (sortBy)
52import Data.List (sortBy)
53+import Control.DeepSeq
54+import Data.Bool
40
55
41-- | Type synonym for probability value
56-- | Type synonym for probability value
42type Prob = Double
57type Prob = Double
@@ -44,17 +59,21 @@ type Prob = Double
44-- | Type synonym for column vector matrices. This represents a probability
59-- | Type synonym for column vector matrices. This represents a probability
45-- distribution.
60-- distribution.
46newtype Dist a = D (Matrix Prob () a)
61newtype Dist a = D (Matrix Prob () a)
62+ deriving (Show, Num, Eq, Ord, NFData) via (Matrix Prob () a)
63+
64+-- | Constraint type synonyms to keep the type signatures less convoluted
65+type Countable a = KnownNat (I.Count a)
66+type CountableN a = KnownNat (I.Count (I.Normalize a))
67+type CountableDimsN a b = (CountableN a, CountableN b)
68+type FLN a b = I.FL (I.Normalize a) (I.Normalize b)
69+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Prob, Ord Prob)
70+type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)
47
71
48-- | Functor instance
72-- | Functor instance
49fmapD ::
73fmapD ::
50− ( Bounded a,
74+ ( Liftable a b,
51− Bounded b,
75+ CountableDimsN a b,
52− Enum a,
76+ FLN b a
53− Enum b,
54− Eq b,
55− KnownNat (Count (Normalize a)),
56− KnownNat (Count (Normalize b)),
57− FromLists Prob (Normalize b) (Normalize a)
58 )
77 )
59 =>
78 =>
60 (a -> b) -> Dist a -> Dist b
79 (a -> b) -> Dist a -> Dist b
@@ -66,26 +85,58 @@ unitD = D (one 1)
66
85
67-- | Applicative/Monoidal instance 'mult' function
86-- | Applicative/Monoidal instance 'mult' function
68multD ::
87multD ::
69− ( KnownNat (Count (Normalize a)),
88+ ( CountableDimsN a b,
70− KnownNat (Count (Normalize b)),
89+ CountableN (a, b),
71− KnownNat (Count (Normalize (a, b))),
90+ FLN (a, b) a,
72− FromLists Prob (Normalize (a, b)) (Normalize a),
91+ FLN (a, b) b,
73− FromLists Prob (Normalize (a, b)) (Normalize b),
92+ TrivialP a b
74− Normalize (a, b) ~ Normalize (Normalize a, Normalize b)
75 ) => Dist a -> Dist b -> Dist (a, b)
93 ) => Dist a -> Dist b -> Dist (a, b)
76−multD (D a) (D b) = D (khatri a b)
94+multD (D a) (D b) = D (kr a b)
77
95
78-- | Selective instance function
96-- | Selective instance function
79selectD ::
97selectD ::
80− ( Normalize (Either a b) ~ Either (Normalize a) (Normalize b),
98+ ( FLN b b,
81− FromLists Prob (Normalize b) (Normalize b),
99+ CountableN b
82− KnownNat (Count (Normalize b))
83 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
100 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
84−selectD (D d) m = D (junc m identity `comp` d)
101+selectD (D d) m = D (selectM d m)
85−
102+
103+-- | Chooses which of the two given effectful
104+-- functions to apply to a given argument;
105+branchD ::
106+ ( Num e,
107+ CountableDimsN a b,
108+ CountableDimsN c (Either b c),
109+ FLN c b,
110+ FLN a b,
111+ FLN a a,
112+ FLN b b,
113+ FLN c c,
114+ FLN b a,
115+ FLN b c,
116+ FLN (Either b c) b,
117+ FLN (Either b c) c
118+ )
119+ => Dist (Either a b) -> Matrix Prob a c -> Matrix Prob b c -> Dist c
120+branchD x l r = f x `selectD` g l `selectD` r
121+ where
122+ f (D m) = D (fork (tr i1) (i1 `comp` tr i2) `comp` m)
123+ g m = i2 `comp` m
124+
125+-- | Branch on a Boolean value, skipping unnecessary computations.
126+ifD ::
127+ ( CountableDimsN a (Either () a),
128+ FLN a a,
129+ FLN a (),
130+ FLN () a,
131+ FLN (Either () a) a
132+ )
133+ => Dist Bool -> Dist a -> Dist a -> Dist a
134+ifD x (D t) (D e) = branchD x' t e
135+ where
136+ x' = bool (Right ()) (Left ()) `fmapD` x
86
137
87-- | Monad instance 'return' function
138-- | Monad instance 'return' function
88−returnD :: forall a . (Enum a, FromLists Prob () (Normalize a), KnownNat (Count a)) => a -> Dist a
139+returnD :: forall a . (Enum a, FLN () a, Countable a) => a -> Dist a
89returnD a = D (col l)
140returnD a = D (col l)
90 where
141 where
91 i = fromInteger $ natVal (Proxy :: Proxy (Count a))
142 i = fromInteger $ natVal (Proxy :: Proxy (Count a))
@@ -99,8 +150,8 @@ bindD (D d) m = D (m `comp` d)
99-- | Extract probabilities given an Event.
150-- | Extract probabilities given an Event.
100(??) ::
151(??) ::
101 ( Enum a,
152 ( Enum a,
102− KnownNat (Count a),
153+ Countable a,
103− FromLists Prob () (Normalize a)
154+ FLN () a
104 ) => (a -> Bool) -> Dist a -> Prob
155 ) => (a -> Bool) -> Dist a -> Prob
105(??) p d =
156(??) p d =
106 let l = toValues d
157 let l = toValues d
@@ -110,11 +161,11 @@ bindD (D d) m = D (m `comp` d)
110-- Distribution Construction
161-- Distribution Construction
111
162
112-- | Constructs a Bernoulli distribution
163-- | Constructs a Bernoulli distribution
113−choose :: (FromLists Prob () (Normalize a)) => Prob -> Dist a
164+choose :: (FLN () a) => Prob -> Dist a
114choose prob = D (col [prob, 1 - prob])
165choose prob = D (col [prob, 1 - prob])
115
166
116-- | Creates a distribution given a shape function
167-- | Creates a distribution given a shape function
117−shape :: (FromLists Prob () (Normalize a)) => (Prob -> Prob) -> [a] -> Dist a
168+shape :: (FLN () a) => (Prob -> Prob) -> [a] -> Dist a
118shape _ [] = error "Probability.shape: empty list"
169shape _ [] = error "Probability.shape: empty list"
119shape f xs =
170shape f xs =
120 let incr = 1 / fromIntegral (length xs - 1)
171 let incr = 1 / fromIntegral (length xs - 1)
@@ -122,23 +173,23 @@ shape f xs =
122 in fromFreqs (zip xs ps)
173 in fromFreqs (zip xs ps)
123
174
124-- | Constructs a Linear distribution
175-- | Constructs a Linear distribution
125−linear :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
176+linear :: (FLN () a) => [a] -> Dist a
126linear = shape id
177linear = shape id
127
178
128-- | Constructs an Uniform distribution
179-- | Constructs an Uniform distribution
129−uniform :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
180+uniform :: (FLN () a) => [a] -> Dist a
130uniform = shape (const 1)
181uniform = shape (const 1)
131
182
132-- | Constructs an Negative Exponential distribution
183-- | Constructs an Negative Exponential distribution
133−negExp :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
184+negExp :: (FLN () a) => [a] -> Dist a
134negExp = shape (\x -> exp (-x))
185negExp = shape (\x -> exp (-x))
135
186
136-- | Constructs an Normal distribution
187-- | Constructs an Normal distribution
137−normal :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
188+normal :: (FLN () a) => [a] -> Dist a
138normal = shape (normalCurve 0.5 0.5)
189normal = shape (normalCurve 0.5 0.5)
139
190
140-- | Transforms a 'Dist' into a list of pairs.
191-- | Transforms a 'Dist' into a list of pairs.
141−toValues :: forall a . (Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> [(a, Prob)]
192+toValues :: forall a . (Enum a, Countable a, FLN () a) => Dist a -> [(a, Prob)]
142toValues (D d) =
193toValues (D d) =
143 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))
194 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))
144 probs = toList d
195 probs = toList d
@@ -146,7 +197,7 @@ toValues (D d) =
146 in res
197 in res
147
198
148-- | Pretty a distribution
199-- | Pretty a distribution
149−prettyDist :: forall a. (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> String
200+prettyDist :: forall a. (Show a, Enum a, Countable a, FLN () a) => Dist a -> String
150prettyDist d =
201prettyDist d =
151 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)
202 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)
152 w = maximum (map (length . show . fst) values)
203 w = maximum (map (length . show . fst) values)
@@ -158,12 +209,12 @@ prettyDist d =
158 showR n x = show x ++ " "
209 showR n x = show x ++ " "
159
210
160-- | Pretty Print a distribution
211-- | Pretty Print a distribution
161−prettyPrintDist :: forall a . (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> IO ()
212+prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FLN () a) => Dist a -> IO ()
162prettyPrintDist = putStrLn . prettyDist @a
213prettyPrintDist = putStrLn . prettyDist @a
163
214
164-- Auxiliary functions
215-- Auxiliary functions
165
216
166−fromFreqs :: (FromLists Prob () (Normalize a)) => [(a,Prob)] -> Dist a
217+fromFreqs :: (FLN () a) => [(a,Prob)] -> Dist a
167fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))
218fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))
168 where q = sum $ map snd xs
219 where q = sum $ map snd xs
169
220
src/LAoP/Matrix/Internal.hs+445−436
@@ -1,4 +1,6 @@
1+{-# LANGUAGE StrictData #-}
1{-# LANGUAGE TypeFamilyDependencies #-}
2{-# LANGUAGE TypeFamilyDependencies #-}
3+{-# LANGUAGE ConstraintKinds #-}
2{-# LANGUAGE AllowAmbiguousTypes #-}
4{-# LANGUAGE AllowAmbiguousTypes #-}
3{-# LANGUAGE DataKinds #-}
5{-# LANGUAGE DataKinds #-}
4{-# LANGUAGE FlexibleContexts #-}
6{-# LANGUAGE FlexibleContexts #-}
@@ -28,7 +30,7 @@
28-- motivation behind the library, the underlying theory, and implementation details.
30-- motivation behind the library, the underlying theory, and implementation details.
29--
31--
30-- This module offers many of the combinators mentioned in the work of
32-- This module offers many of the combinators mentioned in the work of
31−-- Macedo (2012) and Oliveira (2012).
33+-- Macedo (2012) and Oliveira (2012).
32--
34--
33-- This is an Internal module and it is no supposed to be imported.
35-- This is an Internal module and it is no supposed to be imported.
34--
36--
@@ -41,17 +43,26 @@ module LAoP.Matrix.Internal
41 --
43 --
42 -- There exists two type families that make it easier to write
44 -- There exists two type families that make it easier to write
43 -- matrix dimensions: 'FromNat' and 'Count'. This approach
45 -- matrix dimensions: 'FromNat' and 'Count'. This approach
44− -- leads to a very straightforward implementation
46+ -- leads to a very straightforward implementation
45− -- of LAoP combinators.
47+ -- of LAoP combinators.
46
48
47 -- * Type safe matrix representation
49 -- * Type safe matrix representation
48 Matrix (..),
50 Matrix (..),
49
51
52+ -- * Constraint type aliases
53+ Countable,
54+ CountableDims,
55+ CountableN,
56+ CountableDimsN,
57+ FL,
58+ FLN,
59+ Liftable,
60+ Trivial,
61+
50 -- * Primitives
62 -- * Primitives
51− empty,
52 one,
63 one,
53− junc,
64+ join,
54− split,
65+ fork,
55
66
56 -- * Auxiliary type families
67 -- * Auxiliary type families
57 FromNat,
68 FromNat,
@@ -64,6 +75,7 @@ module LAoP.Matrix.Internal
64 toLists,
75 toLists,
65 toList,
76 toList,
66 matrixBuilder,
77 matrixBuilder,
78+ matrixBuilder',
67 row,
79 row,
68 col,
80 col,
69 zeros,
81 zeros,
@@ -74,28 +86,38 @@ module LAoP.Matrix.Internal
74 -- * Misc
86 -- * Misc
75 -- ** Get dimensions
87 -- ** Get dimensions
76 columns,
88 columns,
89+ columns',
77 rows,
90 rows,
91+ rows',
78
92
79 -- ** Matrix Transposition
93 -- ** Matrix Transposition
80 tr,
94 tr,
81
95
96+ -- ** Scalar multiplication/division of matrices
97+ (.|),
98+ (./),
99+
82 -- ** Selective operator
100 -- ** Selective operator
83− select,
101+ select,
102+ branch,
84
103
85 -- ** McCarthy's Conditional
104 -- ** McCarthy's Conditional
86 cond,
105 cond,
87
106
88 -- ** Matrix "abiding"
107 -- ** Matrix "abiding"
89− abideJS,
108+ abideJF,
90− abideSJ,
109+ abideFJ,
110+
111+ -- ** Zip matrices
112+ zipWithM,
91
113
92 -- * Biproduct approach
114 -- * Biproduct approach
93− -- ** Split
115+ -- ** Fork
94 (===),
116 (===),
95 -- *** Projections
117 -- *** Projections
96 p1,
118 p1,
97 p2,
119 p2,
98− -- ** Junc
120+ -- ** Join
99 (|||),
121 (|||),
100 -- *** Injections
122 -- *** Injections
101 i1,
123 i1,
@@ -108,12 +130,13 @@ module LAoP.Matrix.Internal
108
130
109 -- | Note that given the restrictions imposed it is not possible to
131 -- | Note that given the restrictions imposed it is not possible to
110 -- implement the standard type classes present in standard Haskell.
132 -- implement the standard type classes present in standard Haskell.
133+
111 -- *** Matrix pairing projections
134 -- *** Matrix pairing projections
112− kp1,
135+ fstM,
113− kp2,
136+ sndM,
114
137
115 -- *** Matrix pairing
138 -- *** Matrix pairing
116− khatri,
139+ kr,
117
140
118 -- * Matrix composition and lifting
141 -- * Matrix composition and lifting
119
142
@@ -121,10 +144,10 @@ module LAoP.Matrix.Internal
121
144
122 -- | Note that given the restrictions imposed it is not possible to
145 -- | Note that given the restrictions imposed it is not possible to
123 -- implement the standard type classes present in standard Haskell.
146 -- implement the standard type classes present in standard Haskell.
124− identity,
147+ iden,
125 comp,
148 comp,
126− fromF,
127 fromF',
149 fromF',
150+ fromF,
128
151
129 -- * Matrix printing
152 -- * Matrix printing
130 pretty,
153 pretty,
@@ -151,21 +174,22 @@ import LAoP.Utils.Internal
151import Data.Bool
174import Data.Bool
152import Data.Kind
175import Data.Kind
153import Data.List
176import Data.List
177+import Data.Maybe
154import Data.Proxy
178import Data.Proxy
179+import Data.Typeable
155import Data.Void
180import Data.Void
156import GHC.TypeLits
181import GHC.TypeLits
157import Data.Type.Equality
182import Data.Type.Equality
158import GHC.Generics
183import GHC.Generics
159import Control.DeepSeq
184import Control.DeepSeq
160−import Control.Category
185+import Prelude hiding (id, (.))
161−import Prelude hiding ((.))
186+import qualified Prelude (id, (.))
162
187
163-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
164data Matrix e cols rows where
189data Matrix e cols rows where
165− Empty :: Matrix e Void Void
166 One :: e -> Matrix e () ()
190 One :: e -> Matrix e () ()
167− Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
191+ Join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
168− Split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
192+ Fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
169
193
170deriving instance (Show e) => Show (Matrix e cols rows)
194deriving instance (Show e) => Show (Matrix e cols rows)
171
195
@@ -175,221 +199,214 @@ deriving instance (Show e) => Show (Matrix e cols rows)
175-- 'Generic' instance.
199-- 'Generic' instance.
176type family Count (d :: Type) :: Nat where
200type family Count (d :: Type) :: Nat where
177 Count (Natural n m) = (m - n) + 1
201 Count (Natural n m) = (m - n) + 1
202+ Count (List a) = (^) 2 (Count a)
178− Count (Either a b) = (+) (Count a) (Count b)
203+ Count (Either a b) = (+) (Count a) (Count b)
179− Count (a, b) = (*) (Count a) (Count b)
204+ Count (a, b) = (*) (Count a) (Count b)
180− Count (a -> b) = (^) (Count b) (Count a)
205+ Count (a -> b) = (^) (Count b) (Count a)
181 -- Generics
206 -- Generics
182− Count (M1 _ _ f p) = Count (f p)
207+ Count (M1 _ _ f p) = Count (f p)
183− Count (K1 _ _ _) = 1
208+ Count (K1 _ _ _) = 1
184− Count (V1 _) = 0
209+ Count (V1 _) = 0
185− Count (U1 _) = 1
210+ Count (U1 _) = 1
186 Count ((:*:) a b p) = Count (a p) * Count (b p)
211 Count ((:*:) a b p) = Count (a p) * Count (b p)
187 Count ((:+:) a b p) = Count (a p) + Count (b p)
212 Count ((:+:) a b p) = Count (a p) + Count (b p)
188− Count d = Count (Rep d R)
213+ Count d = Count (Rep d R)
189
214
190-- | Type family that computes of a given type dimension from a given natural
215-- | Type family that computes of a given type dimension from a given natural
191--
216--
192-- Thanks to Li-Yao Xia this type family is super fast.
217-- Thanks to Li-Yao Xia this type family is super fast.
193type family FromNat (n :: Nat) :: Type where
218type family FromNat (n :: Nat) :: Type where
194− FromNat 0 = Void
195 FromNat 1 = ()
219 FromNat 1 = ()
196 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
197
221
198type family FromNat' (b :: Bool) (m :: Type) :: Type where
222type family FromNat' (b :: Bool) (m :: Type) :: Type where
199− FromNat' 'True m = Either m m
223+ FromNat' 'True m = Either m m
200 FromNat' 'False m = Either () (Either m m)
224 FromNat' 'False m = Either () (Either m m)
201
225
202-- | Type family that normalizes the representation of a given data
226-- | Type family that normalizes the representation of a given data
203-- structure
227-- structure
204type family Normalize (d :: Type) :: Type where
228type family Normalize (d :: Type) :: Type where
229+ Normalize (Either a b) = Either (Normalize a) (Normalize b)
230+ Normalize d = FromNat (Count d)
231+
232+-- | Constraint type synonyms to keep the type signatures less convoluted
233+type Countable a = KnownNat (Count a)
234+type CountableN a = KnownNat (Count (Normalize a))
235+type CountableDims a b = (Countable a, Countable b)
236+type CountableDimsN a b = (CountableN a, CountableN b)
237+type FL a b = FromLists a b
238+type FLN a b = FromLists (Normalize a) (Normalize b)
239+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
205− Normalize d = FromNat (Count d)
240+type Trivial a = FromNat (Count a) ~ a
206−
241+
207−-- | It isn't possible to implement the 'id' function so it's
242+-- | It is possible to implement a constrained version of the category type
208−-- implementation is 'undefined'. However 'comp' can be and this partial
243+-- class.
209−-- class implementation exists just to make the code more readable.
210−--
211−-- Please use 'identity' instead.
212instance (Num e) => Category (Matrix e) where
244instance (Num e) => Category (Matrix e) where
213− id = undefined
245+ type Object (Matrix e) a = (FL a a, Countable a)
246+ id = iden
214− (.) = comp
247+ (.) = comp
215
248
216instance NFData e => NFData (Matrix e cols rows) where
249instance NFData e => NFData (Matrix e cols rows) where
217− rnf Empty = ()
218− rnf (One e) = rnf e
250+ rnf (One e) = rnf e
219− rnf (Junc a b) = rnf a `seq` rnf b
251+ rnf (Join a b) = rnf a `seq` rnf b
220− rnf (Split a b) = rnf a `seq` rnf b
252+ rnf (Fork a b) = rnf a `seq` rnf b
221
253
222instance Eq e => Eq (Matrix e cols rows) where
254instance Eq e => Eq (Matrix e cols rows) where
223− Empty == Empty = True
224− (One a) == (One b) = a == b
255+ (One a) == (One b) = a == b
225− (Junc a b) == (Junc c d) = a == c && b == d
256+ (Join a b) == (Join c d) = a == c && b == d
226− (Split a b) == (Split c d) = a == c && b == d
257+ (Fork a b) == (Fork c d) = a == c && b == d
227− x@(Split _ _) == y@(Junc _ _) = x == abideJS y
258+ x@(Fork _ _) == y@(Join _ _) = x == abideJF y
228− x@(Junc _ _) == y@(Split _ _) = abideJS x == y
259+ x@(Join _ _) == y@(Fork _ _) = abideJF x == y
229
260
230instance Num e => Num (Matrix e cols rows) where
261instance Num e => Num (Matrix e cols rows) where
262+ a + b = zipWithM (+) a b
263+
264+ a - b = zipWithM (-) a b
265+
266+ a * b = zipWithM (*) a b
267+
268+ abs (One a) = One (abs a)
269+ abs (Join a b) = Join (abs a) (abs b)
270+ abs (Fork a b) = Fork (abs a) (abs b)
231
271
232− Empty + Empty = Empty
233− (One a) + (One b) = One (a + b)
234− (Junc a b) + (Junc c d) = Junc (a + c) (b + d)
235− (Split a b) + (Split c d) = Split (a + c) (b + d)
236− x@(Split _ _) + y@(Junc _ _) = x + abideJS y
237− x@(Junc _ _) + y@(Split _ _) = abideJS x + y
238−
239− Empty - Empty = Empty
240− (One a) - (One b) = One (a - b)
241− (Junc a b) - (Junc c d) = Junc (a - c) (b - d)
242− (Split a b) - (Split c d) = Split (a - c) (b - d)
243− x@(Split _ _) - y@(Junc _ _) = x - abideJS y
244− x@(Junc _ _) - y@(Split _ _) = abideJS x - y
245−
246− Empty * Empty = Empty
247− (One a) * (One b) = One (a * b)
248− (Junc a b) * (Junc c d) = Junc (a * c) (b * d)
249− (Split a b) * (Split c d) = Split (a * c) (b * d)
250− x@(Split _ _) * y@(Junc _ _) = x * abideJS y
251− x@(Junc _ _) * y@(Split _ _) = abideJS x * y
252−
253− abs Empty = Empty
254− abs (One a) = One (abs a)
255− abs (Junc a b) = Junc (abs a) (abs b)
256− abs (Split a b) = Split (abs a) (abs b)
257−
258− signum Empty = Empty
259− signum (One a) = One (signum a)
272+ signum (One a) = One (signum a)
260− signum (Junc a b) = Junc (signum a) (signum b)
273+ signum (Join a b) = Join (signum a) (signum b)
261− signum (Split a b) = Split (signum a) (signum b)
274+ signum (Fork a b) = Fork (signum a) (signum b)
262
275
263instance Ord e => Ord (Matrix e cols rows) where
276instance Ord e => Ord (Matrix e cols rows) where
264− Empty <= Empty = True
265− (One a) <= (One b) = a <= b
277+ (One a) <= (One b) = a <= b
266− (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)
278+ (Join a b) <= (Join c d) = (a <= c) && (b <= d)
267− (Split a b) <= (Split c d) = (a <= c) && (b <= d)
279+ (Fork a b) <= (Fork c d) = (a <= c) && (b <= d)
268− x@(Split _ _) <= y@(Junc _ _) = x <= abideJS y
280+ x@(Fork _ _) <= y@(Join _ _) = x <= abideJF y
269− x@(Junc _ _) <= y@(Split _ _) = abideJS x <= y
281+ x@(Join _ _) <= y@(Fork _ _) = abideJF x <= y
270
282
271-- Primitives
283-- Primitives
272
284
273−-- | Empty matrix constructor
274−empty :: Matrix e Void Void
275−empty = Empty
276−
277-- | Unit matrix constructor
285-- | Unit matrix constructor
278one :: e -> Matrix e () ()
286one :: e -> Matrix e () ()
279one = One
287one = One
280
288
281−-- | Matrix 'Junc' constructor
289+-- | Matrix 'Join' constructor
282−junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
290+join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
283−junc = Junc
291+join = Join
284
292
285infixl 3 |||
293infixl 3 |||
286
294
287−-- | Matrix 'Junc' constructor
295+-- | Matrix 'Join' constructor
288(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
289−(|||) = Junc
297+(|||) = Join
290
298
291−-- | Matrix 'Split' constructor
299+-- | Matrix 'Fork' constructor
292−split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
300+fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
293−split = Split
301+fork = Fork
294
302
295infixl 2 ===
303infixl 2 ===
296
304
297−-- | Matrix 'Split' constructor
305+-- | Matrix 'Fork' constructor
298(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
299−(===) = Split
307+(===) = Fork
300
308
301-- Construction
309-- Construction
302
310
303-- | Type class for defining the 'fromList' conversion function.
311-- | Type class for defining the 'fromList' conversion function.
304--
312--
305-- Given that it is not possible to branch on types at the term level type
313-- Given that it is not possible to branch on types at the term level type
306−-- classes are needed bery much like an inductive definition but on types.
314+-- classes are needed very much like an inductive definition but on types.
307−class FromLists e cols rows where
315+class FromLists cols rows where
308 -- | Build a matrix out of a list of list of elements. Throws a runtime
316 -- | Build a matrix out of a list of list of elements. Throws a runtime
309 -- error if the dimensions do not match.
317 -- error if the dimensions do not match.
310 fromLists :: [[e]] -> Matrix e cols rows
318 fromLists :: [[e]] -> Matrix e cols rows
311
319
312−instance FromLists e Void Void where
320+instance FromLists () () where
313− fromLists [] = Empty
314− fromLists _ = error "Wrong dimensions"
315−
316−instance {-# OVERLAPPING #-} FromLists e () () where
317 fromLists [[e]] = One e
321 fromLists [[e]] = One e
318 fromLists _ = error "Wrong dimensions"
322 fromLists _ = error "Wrong dimensions"
319
323
320−instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where
324+instance (FromLists cols ()) => FromLists (Either () cols) () where
321− fromLists [h : t] = Junc (One h) (fromLists [t])
325+ fromLists [h : t] = Join (One h) (fromLists [t])
322 fromLists _ = error "Wrong dimensions"
326 fromLists _ = error "Wrong dimensions"
323
327
328+instance {-# OVERLAPPABLE #-}
324−instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), KnownNat (Count a)) => FromLists e (Either a b) () where
329+ (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where
325− fromLists [l] =
330+ fromLists [l] =
326 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
327− in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])
332+ in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])
328 fromLists _ = error "Wrong dimensions"
333 fromLists _ = error "Wrong dimensions"
329
334
330−instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where
335+instance (FromLists () rows) => FromLists () (Either () rows) where
331− fromLists ([h] : t) = Split (One h) (fromLists t)
336+ fromLists ([h] : t) = Fork (One h) (fromLists t)
332 fromLists _ = error "Wrong dimensions"
337 fromLists _ = error "Wrong dimensions"
333
338
339+instance {-# OVERLAPPABLE #-}
334−instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, KnownNat (Count a)) => FromLists e () (Either a b) where
340+ (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where
335− fromLists l@([_] : _) =
341+ fromLists l@([_] : _) =
336 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
337− in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))
343+ in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))
338 fromLists _ = error "Wrong dimensions"
344 fromLists _ = error "Wrong dimensions"
339
345
340−instance {-# OVERLAPPABLE #-} (FromLists e (Either a b) c, FromLists e (Either a b) d, KnownNat (Count c)) => FromLists e (Either a b) (Either c d) where
346+instance (FromLists (Either a b) c, FromLists (Either a b) d, Countable c)
347+ => FromLists (Either a b) (Either c d) where
341 fromLists l@(h : t) =
348 fromLists l@(h : t) =
342 let lh = length h
349 let lh = length h
343 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
344− condition = all (== lh) (map length t)
351+ condition = all ((== lh) . length) t
345 in if lh > 0 && condition
352 in if lh > 0 && condition
346− then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))
353+ then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))
347 else error "Not all rows have the same length"
354 else error "Not all rows have the same length"
348
355
349-- | Matrix builder function. Constructs a matrix provided with
356-- | Matrix builder function. Constructs a matrix provided with
350−-- a construction function.
357+-- a construction function that operates with indices.
351−matrixBuilder ::
358+matrixBuilder' ::
352 forall e cols rows.
359 forall e cols rows.
353− ( FromLists e cols rows,
360+ ( FL cols rows,
354− KnownNat (Count cols),
361+ CountableDims cols rows
355− KnownNat (Count rows)
356− ) =>
357− ((Int, Int) -> e) ->
358− Matrix e cols rows
362+ ) => ((Int, Int) -> e) -> Matrix e cols rows
359−matrixBuilder f =
363+matrixBuilder' f =
360 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
361 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
362 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
363 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
364
368
369+-- | Matrix builder function. Constructs a matrix provided with
370+-- a construction function that operates with arbitrary types.
371+matrixBuilder ::
372+ forall e a b.
373+ ( FLN a b,
374+ Enum a,
375+ Enum b,
376+ Bounded a,
377+ Bounded b,
378+ Eq a,
379+ Countable b
380+ ) => ((a, b) -> e) -> Matrix e (Normalize a) (Normalize b)
381+matrixBuilder f =
382+ let r = fromInteger $ natVal (Proxy :: Proxy (Count b))
383+ positions = [(a, b) | a <- [minBound .. maxBound], b <- [minBound .. maxBound]]
384+ in fromLists . map (map f) . transpose . buildList r $ positions
385+ where
386+ buildList _ [] = []
387+ buildList r l = take r l : buildList r (drop r l)
388+
365-- | Constructs a column vector matrix
389-- | Constructs a column vector matrix
366−col :: (FromLists e () rows) => [e] -> Matrix e () rows
390+col :: (FL () rows) => [e] -> Matrix e () rows
367col = fromLists . map (: [])
391col = fromLists . map (: [])
368
392
369-- | Constructs a row vector matrix
393-- | Constructs a row vector matrix
370−row :: (FromLists e cols ()) => [e] -> Matrix e cols ()
394+row :: (FL cols ()) => [e] -> Matrix e cols ()
371row = fromLists . (: [])
395row = fromLists . (: [])
372
396
373-- | Lifts functions to matrices with arbitrary dimensions.
397-- | Lifts functions to matrices with arbitrary dimensions.
374--
398--
375-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
376-- types @a@ or @b@ allows.
400-- types @a@ or @b@ allows.
377−fromF ::
401+fromF' ::
378 forall a b cols rows e.
402 forall a b cols rows e.
379− ( Bounded a,
403+ ( Liftable e a b,
380− Bounded b,
404+ CountableDims cols rows,
381− Enum a,
382− Enum b,
383− Eq b,
384− Num e,
385− Ord e,
386− KnownNat (Count cols),
387− KnownNat (Count rows),
388− FromLists e rows cols
405+ FL rows cols
389 ) =>
406 ) =>
390 (a -> b) ->
407 (a -> b) ->
391 Matrix e cols rows
408 Matrix e cols rows
392−fromF f =
409+fromF' f =
393 let minA = minBound @a
410 let minA = minBound @a
394 maxA = maxBound @a
411 maxA = maxBound @a
395 minB = minBound @b
412 minB = minBound @b
@@ -399,8 +416,8 @@ fromF f =
399 elementsA = take ccols [minA .. maxA]
416 elementsA = take ccols [minA .. maxA]
400 elementsB = take rrows [minB .. maxB]
417 elementsB = take rrows [minB .. maxB]
401 combinations = (,) <$> elementsA <*> elementsB
418 combinations = (,) <$> elementsA <*> elementsB
402− combAp = map snd . sort . map (\(a, b) -> if f a == b
419+ combAp = map snd . sort . map (\(a, b) -> if f a == b
403− then ((fromEnum a, fromEnum b), 1)
420+ then ((fromEnum a, fromEnum b), 1)
404 else ((fromEnum a, fromEnum b), 0)) $ combinations
421 else ((fromEnum a, fromEnum b), 0)) $ combinations
405 mList = buildList combAp rrows
422 mList = buildList combAp rrows
406 in tr $ fromLists mList
423 in tr $ fromLists mList
@@ -410,48 +427,23 @@ fromF f =
410
427
411-- | Lifts functions to matrices with dimensions matching @a@ and @b@
428-- | Lifts functions to matrices with dimensions matching @a@ and @b@
412-- cardinality's.
429-- cardinality's.
413−fromF' ::
430+fromF ::
414 forall a b e.
431 forall a b e.
415− ( Bounded a,
432+ ( Liftable e a b,
416− Bounded b,
433+ CountableDimsN a b,
417− Enum a,
434+ FLN b a
418− Enum b,
419− Eq b,
420− Num e,
421− Ord e,
422− KnownNat (Count (Normalize a)),
423− KnownNat (Count (Normalize b)),
424− FromLists e (Normalize b) (Normalize a)
425 ) =>
435 ) =>
426 (a -> b) ->
436 (a -> b) ->
427 Matrix e (Normalize a) (Normalize b)
437 Matrix e (Normalize a) (Normalize b)
428−fromF' f =
438+fromF = fromF'
429− let minA = minBound @a
430− maxA = maxBound @a
431− minB = minBound @b
432− maxB = maxBound @b
433− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
434− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
435− elementsA = take ccols [minA .. maxA]
436− elementsB = take rrows [minB .. maxB]
437− combinations = (,) <$> elementsA <*> elementsB
438− combAp = map snd . sort . map (\(a, b) -> if f a == b
439− then ((fromEnum a, fromEnum b), 1)
440− else ((fromEnum a, fromEnum b), 0)) $ combinations
441− mList = buildList combAp rrows
442− in tr $ fromLists mList
443− where
444− buildList [] _ = []
445− buildList l r = take r l : buildList (drop r l) r
446
439
447-- Conversion
440-- Conversion
448
441
449-- | Converts a matrix to a list of lists of elements.
442-- | Converts a matrix to a list of lists of elements.
450toLists :: Matrix e cols rows -> [[e]]
443toLists :: Matrix e cols rows -> [[e]]
451−toLists Empty = []
452toLists (One e) = [[e]]
444toLists (One e) = [[e]]
453−toLists (Split l r) = toLists l ++ toLists r
445+toLists (Fork l r) = toLists l ++ toLists r
454−toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)
446+toLists (Join l r) = zipWith (++) (toLists l) (toLists r)
455
447
456-- | Converts a matrix to a list of elements.
448-- | Converts a matrix to a list of elements.
457toList :: Matrix e cols rows -> [e]
449toList :: Matrix e cols rows -> [e]
@@ -460,37 +452,38 @@ toList = concat . toLists
460-- Zeros Matrix
452-- Zeros Matrix
461
453
462-- | The zero matrix. A matrix wholly filled with zeros.
454-- | The zero matrix. A matrix wholly filled with zeros.
463−zeros :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows
455+zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
464−zeros = matrixBuilder (const 0)
456+zeros = matrixBuilder' (const 0)
465
457
466-- Ones Matrix
458-- Ones Matrix
467
459
468-- | The ones matrix. A matrix wholly filled with ones.
460-- | The ones matrix. A matrix wholly filled with ones.
469--
461--
470-- Also known as T (Top) matrix.
462-- Also known as T (Top) matrix.
471−ones :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows
463+ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
472−ones = matrixBuilder (const 1)
464+ones = matrixBuilder' (const 1)
473
465
474-- Const Matrix
466-- Const Matrix
475
467
476-- | The constant matrix constructor. A matrix wholly filled with a given
468-- | The constant matrix constructor. A matrix wholly filled with a given
477-- value.
469-- value.
478−constant :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => e -> Matrix e cols rows
470+constant :: (Num e, FL cols rows, CountableDims cols rows) => e -> Matrix e cols rows
479−constant e = matrixBuilder (const e)
471+constant e = matrixBuilder' (const e)
480
472
481-- Bang Matrix
473-- Bang Matrix
482
474
483-- | The T (Top) row vector matrix.
475-- | The T (Top) row vector matrix.
484−bang :: forall e cols. (Num e, Enum e, FromLists e cols (), KnownNat (Count cols)) => Matrix e cols ()
476+bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()
485bang =
477bang =
486 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
487 in fromLists [take c [1, 1 ..]]
479 in fromLists [take c [1, 1 ..]]
488
480
489−-- Identity Matrix
481+-- iden Matrix
490
482
491−-- | Identity matrix.
483+-- | iden matrix.
492−identity :: (Num e, FromLists e cols cols, KnownNat (Count cols)) => Matrix e cols cols
484+iden :: forall e cols . (Num e, FL cols cols, Countable cols) => Matrix e cols cols
493−identity = matrixBuilder (bool 0 1 . uncurry (==))
485+iden = matrixBuilder' (bool 0 1 . uncurry (==))
486+{-# NOINLINE iden #-}
494
487
495-- Matrix composition (MMM)
488-- Matrix composition (MMM)
496
489
@@ -499,60 +492,90 @@ identity = matrixBuilder (bool 0 1 . uncurry (==))
499-- This definition takes advantage of divide-and-conquer and fusion laws
492-- This definition takes advantage of divide-and-conquer and fusion laws
500-- from LAoP.
493-- from LAoP.
501comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
502−comp Empty Empty = Empty
503−comp (One a) (One b) = One (a * b)
495+comp (One a) (One b) = One (a * b)
504−comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law
496+comp (Join a b) (Fork c d) = comp a c + comp b d -- Divide-and-conquer law
505−comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law
497+comp (Fork a b) c = Fork (comp a c) (comp b c) -- Fork fusion law
506−comp c (Junc a b) = Junc (comp c a) (comp c b) -- Junc fusion law
498+comp c (Join a b) = Join (comp c a) (comp c b) -- Join fusion law
499+{-# NOINLINE comp #-}
500+{-# RULES
501+ "comp/iden1" forall m. comp m iden = m ;
502+ "comp/iden2" forall m. comp iden m = m
503+#-}
504+
505+-- Scalar multiplication of matrices
506+
507+infixl 7 .|
508+-- | Scalar multiplication of matrices.
509+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows
510+(.|) e (One a) = One (e * a)
511+(.|) e (Join a b) = Join (e .| a) (e .| b)
512+(.|) e (Fork a b) = Fork (e .| a) (e .| b)
513+
514+-- Scalar division of matrices
515+
516+infixl 7 ./
517+-- | Scalar multiplication of matrices.
518+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows
519+(./) (One a) e = One (a / e)
520+(./) (Join a b) e = Join (a ./ e) (b ./ e)
521+(./) (Fork a b) e = Fork (a ./ e) (b ./ e)
507
522
508-- Projections
523-- Projections
509
524
510-- | Biproduct first component projection
525-- | Biproduct first component projection
511−p1 :: forall e m n. (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e (Either m n) m
526+p1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e (Either m n) m
512−p1 =
527+p1 = join id zeros
513− let iden = identity :: Matrix e m m
514− zero = zeros :: Matrix e n m
515− in junc iden zero
516
528
517-- | Biproduct second component projection
529-- | Biproduct second component projection
518−p2 :: forall e m n. (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e (Either m n) n
530+p2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e (Either m n) n
519−p2 =
531+p2 = join zeros id
520− let iden = identity :: Matrix e n n
521− zero = zeros :: Matrix e m n
522− in junc zero iden
523
532
524-- Injections
533-- Injections
525
534
526-- | Biproduct first component injection
535-- | Biproduct first component injection
527−i1 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e m (Either m n)
536+i1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e m (Either m n)
528i1 = tr p1
537i1 = tr p1
529
538
530-- | Biproduct second component injection
539-- | Biproduct second component injection
531−i2 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e n (Either m n)
540+i2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e n (Either m n)
532i2 = tr p2
541i2 = tr p2
533
542
534-- Dimensions
543-- Dimensions
535
544
536-- | Obtain the number of rows.
545-- | Obtain the number of rows.
537--
546--
538−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
547+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
539−-- dimensions in constant time.
548+-- dimensions in constant time. For a version that doesn't require the
540−--
549+-- constraint see 'rows''.
541−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
542−-- exchange for performance.
543−rows :: forall e cols rows. (KnownNat (Count rows)) => Matrix e cols rows -> Int
550+rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int
544rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
551rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
545
552
553+-- | Obtain the number of rows in an inefficient manner, but without any
554+-- constraints.
555+--
556+-- For a more efficient version see 'rows'.
557+rows' :: Matrix e cols rows -> Int
558+rows' (One _) = 1
559+rows' (Join lhs _) = rows' lhs
560+rows' (Fork top bottom) = rows' top + rows' bottom
561+
546-- | Obtain the number of columns.
562-- | Obtain the number of columns.
547−--
548−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
549−-- dimensions in constant time.
550--
563--
551−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
564+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
552−-- exchange for performance.
565+-- dimensions in constant time. For a version that doesn't require the
566+-- constraint see 'columns''.
553−columns :: forall e cols rows. (KnownNat (Count cols)) => Matrix e cols rows -> Int
567+columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int
554columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
555
569
570+-- | Obtain the number of columns in an inefficient manner, but without any
571+-- constraints.
572+--
573+-- For a more efficient version see 'columns'.
574+columns' :: Matrix e cols rows -> Int
575+columns' (One _) = 1
576+columns' (Join lhs rhs) = columns' lhs + columns' rhs
577+columns' (Fork top _) = columns' top
578+
556-- Coproduct Bifunctor
579-- Coproduct Bifunctor
557
580
558infixl 5 -|-
581infixl 5 -|-
@@ -561,157 +584,172 @@ infixl 5 -|-
561(-|-) ::
584(-|-) ::
562 forall e n k m j.
585 forall e n k m j.
563 ( Num e,
586 ( Num e,
564− KnownNat (Count j),
587+ CountableDims j k,
565− KnownNat (Count k),
588+ FL k k,
566− FromLists e k k,
589+ FL j k,
567− FromLists e j k,
590+ FL k j,
568− FromLists e k j,
591+ FL j j
569− FromLists e j j
570 ) =>
592 ) =>
571 Matrix e n k ->
593 Matrix e n k ->
572 Matrix e m j ->
594 Matrix e m j ->
573 Matrix e (Either n m) (Either k j)
595 Matrix e (Either n m) (Either k j)
574−(-|-) a b = Junc (i1 . a) (i2 . b)
596+(-|-) a b = Join (i1 . a) (i2 . b)
575
597
576-- Khatri Rao Product and projections
598-- Khatri Rao Product and projections
577
599
578-- | Khatri Rao product first component projection matrix.
600-- | Khatri Rao product first component projection matrix.
579−kp1 ::
601+fstM ::
580 forall e m k .
602 forall e m k .
581 ( Num e,
603 ( Num e,
582− KnownNat (Count k),
604+ CountableDims k m,
583− FromLists e (FromNat (Count m * Count k)) m,
584− KnownNat (Count m),
585− KnownNat (Count (Normalize (m, k)))
605+ FL (Normalize (m, k)) m,
606+ CountableN (m, k)
586 ) => Matrix e (Normalize (m, k)) m
607 ) => Matrix e (Normalize (m, k)) m
587−kp1 = matrixBuilder f
608+fstM = matrixBuilder' f
588 where
609 where
589 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
590 f (x, y)
611 f (x, y)
591 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
592− | otherwise = 0
613+ | otherwise = 0
593
614
594-- | Khatri Rao product second component projection matrix.
615-- | Khatri Rao product second component projection matrix.
595−kp2 ::
616+sndM ::
596 forall e m k .
617 forall e m k .
597 ( Num e,
618 ( Num e,
598− KnownNat (Count k),
619+ CountableDims k m,
599− FromLists e (FromNat (Count m * Count k)) k,
600− KnownNat (Count m),
601− KnownNat (Count (Normalize (m, k)))
620+ FL (Normalize (m, k)) k,
621+ CountableN (m, k)
602 ) => Matrix e (Normalize (m, k)) k
622 ) => Matrix e (Normalize (m, k)) k
603−kp2 = matrixBuilder f
623+sndM = matrixBuilder' f
604 where
624 where
605 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
606 f (x, y)
626 f (x, y)
607 | x == y || mod (y - x) offset == 0 = 1
627 | x == y || mod (y - x) offset == 0 = 1
608− | otherwise = 0
628+ | otherwise = 0
609
629
610-- | Khatri Rao Matrix product also known as matrix pairing.
630-- | Khatri Rao Matrix product also known as matrix pairing.
611--
631--
612-- NOTE: That this is not a true categorical product, see for instance:
632-- NOTE: That this is not a true categorical product, see for instance:
613−--
633+--
614-- @
634-- @
615−-- | kp1 . khatri a b == a
635+-- | fstM . kr a b == a
616−-- khatri a b ==> |
636+-- kr a b ==> |
617−-- | kp2 . khatri a b == b
637+-- | sndM . kr a b == b
618-- @
638-- @
619--
639--
620-- __Emphasis__ on the implication symbol.
640-- __Emphasis__ on the implication symbol.
621−khatri ::
641+kr ::
622− forall e cols a b.
642+ forall e cols a b.
623 ( Num e,
643 ( Num e,
624− KnownNat (Count a),
644+ CountableDims a b,
625− KnownNat (Count b),
645+ CountableN (a, b),
626− KnownNat (Count (Normalize (a, b))),
627− FromLists e (Normalize (a, b)) a,
646+ FL (Normalize (a, b)) a,
628− FromLists e (Normalize (a, b)) b
647+ FL (Normalize (a, b)) b
629 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
630−khatri a b =
649+kr a b =
631− let kp1' = kp1 @e @a @b
650+ let fstM' = fstM @e @a @b
632− kp2' = kp2 @e @a @b
651+ sndM' = sndM @e @a @b
633− in (tr kp1') . a * (tr kp2') . b
652+ in (tr fstM' . a) * (tr sndM' . b)
634
653
635-- Product Bifunctor (Kronecker)
654-- Product Bifunctor (Kronecker)
636
655
637infixl 4 ><
656infixl 4 ><
638
657
639-- | Matrix product functor also known as kronecker product
658-- | Matrix product functor also known as kronecker product
640−(><) ::
659+(><) ::
641− forall e m p n q.
660+ forall e m p n q.
642 ( Num e,
661 ( Num e,
643− KnownNat (Count m),
662+ CountableDims m n,
644− KnownNat (Count n),
663+ CountableDims p q,
645− KnownNat (Count p),
664+ CountableDimsN (m, n) (p, q),
646− KnownNat (Count q),
647− KnownNat (Count (Normalize (m, n))),
648− FromLists e (Normalize (m, n)) m,
665+ FL (Normalize (m, n)) m,
649− FromLists e (Normalize (m, n)) n,
666+ FL (Normalize (m, n)) n,
650− KnownNat (Count (Normalize (p, q))),
651− FromLists e (Normalize (p, q)) p,
667+ FL (Normalize (p, q)) p,
652− FromLists e (Normalize (p, q)) q
668+ FL (Normalize (p, q)) q
653− )
669+ )
654 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
655(><) a b =
671(><) a b =
656− let kp1' = kp1 @e @m @n
672+ let fstM' = fstM @e @m @n
657− kp2' = kp2 @e @m @n
673+ sndM' = sndM @e @m @n
658− in khatri (a . kp1') (b . kp2')
674+ in kr (a . fstM') (b . sndM')
659
675
660−-- Matrix abide Junc Split
676+-- Matrix abide Join Fork
661
677
662−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
678+-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.
663−--
679+--
664-- Law:
680-- Law:
665--
681--
666-- @
682-- @
667−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)
683+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)
668-- @
684-- @
669−abideJS :: Matrix e cols rows -> Matrix e cols rows
685+abideJF :: Matrix e cols rows -> Matrix e cols rows
670−abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law
686+abideJF (Join (Fork a c) (Fork b d)) = Fork (Join (abideJF a) (abideJF b)) (Join (abideJF c) (abideJF d)) -- Join-Fork abide law
671−abideJS Empty = Empty
672−abideJS (One e) = One e
687+abideJF (One e) = One e
673−abideJS (Junc a b) = Junc (abideJS a) (abideJS b)
688+abideJF (Join a b) = Join (abideJF a) (abideJF b)
674−abideJS (Split a b) = Split (abideJS a) (abideJS b)
689+abideJF (Fork a b) = Fork (abideJF a) (abideJF b)
675
690
676−-- Matrix abide Split Junc
691+-- Matrix abide Fork Join
677
692
678−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
693+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
679−--
694+--
680-- @
695-- @
681−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)
696+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)
682-- @
697-- @
683−abideSJ :: Matrix e cols rows -> Matrix e cols rows
698+abideFJ :: Matrix e cols rows -> Matrix e cols rows
684−abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law
699+abideFJ (Fork (Join a b) (Join c d)) = Join (Fork (abideFJ a) (abideFJ c)) (Fork (abideFJ b) (abideFJ d)) -- Fork-Join abide law
685−abideSJ Empty = Empty
686−abideSJ (One e) = One e
700+abideFJ (One e) = One e
687−abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)
701+abideFJ (Join a b) = Join (abideFJ a) (abideFJ b)
688−abideSJ (Split a b) = Split (abideSJ a) (abideSJ b)
702+abideFJ (Fork a b) = Fork (abideFJ a) (abideFJ b)
689
703
690-- Matrix transposition
704-- Matrix transposition
691
705
692-- | Matrix transposition.
706-- | Matrix transposition.
693tr :: Matrix e cols rows -> Matrix e rows cols
707tr :: Matrix e cols rows -> Matrix e rows cols
694−tr Empty = Empty
695−tr (One e) = One e
708+tr (One e) = One e
696−tr (Junc a b) = Split (tr a) (tr b)
709+tr (Join a b) = Fork (tr a) (tr b)
697−tr (Split a b) = Junc (tr a) (tr b)
710+tr (Fork a b) = Join (tr a) (tr b)
698
711
699-- Selective 'select' operator
712-- Selective 'select' operator
700
713
701-- | Selective functors 'select' operator equivalent inspired by the
714-- | Selective functors 'select' operator equivalent inspired by the
702-- ArrowMonad solution presented in the paper.
715-- ArrowMonad solution presented in the paper.
716+select :: (Num e, FL b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
717+select (Fork a b) y = y . a + b -- Divide-and-conquer law
718+select (Join (Fork a c) (Fork b d)) y = join (y . a + c) (y . b + d) -- Pattern matching + DnC law
719+select m y = join y id . m
720+
721+branch ::
722+ ( Num e,
723+ CountableDims a b,
724+ CountableDims c (Either b c),
725+ FL c b,
726+ FL a b,
727+ FL a a,
728+ FL b b,
729+ FL c c,
730+ FL b a,
731+ FL b c,
732+ FL (Either b c) b,
733+ FL (Either b c) c
734+ )
703−select :: (Num e, FromLists e b b, KnownNat (Count b)) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
735+ => Matrix e cols (Either a b) -> Matrix e a c -> Matrix e b c -> Matrix e cols c
704−select m y = junc y identity . m
736+branch x l r = f x `select` g l `select` r
737+ where
738+ f :: (Num e, Countable a, CountableDims b c, FL a b, FL c b, FL b b, FL b a, FL a a)
739+ => Matrix e cols (Either a b) -> Matrix e cols (Either a (Either b c))
740+ f m = fork (tr i1) (i1 . tr i2) . m
741+ g :: (Num e, CountableDims b c, FL b c, FL c c) => Matrix e a c -> Matrix e a (Either b c)
742+ g m = i2 . m
705
743
706-- McCarthy's Conditional
744-- McCarthy's Conditional
707
745
708-- | McCarthy's Conditional expresses probabilistic choice.
746-- | McCarthy's Conditional expresses probabilistic choice.
709cond ::
747cond ::
710− ( cols ~ FromNat (Count cols),
748+ ( Trivial cols,
711− KnownNat (Count cols),
749+ Countable cols,
712− FromLists e () cols,
750+ FL () cols,
713− FromLists e cols (),
751+ FL cols (),
714− FromLists e cols cols,
752+ FL cols cols,
715 Bounded a,
753 Bounded a,
716 Enum a,
754 Enum a,
717 Num e,
755 Num e,
@@ -719,14 +757,14 @@ cond ::
719 )
757 )
720 =>
758 =>
721 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
722−cond p f g = junc f g . grd p
760+cond p f g = join f g . grd p
723−
761+
724−grd ::
762+grd ::
725− ( q ~ FromNat (Count q),
763+ ( Trivial q,
726− KnownNat (Count q),
764+ Countable q,
727− FromLists e () q,
765+ FL () q,
728− FromLists e q (),
766+ FL q (),
729− FromLists e q q,
767+ FL q q,
730 Bounded a,
768 Bounded a,
731 Enum a,
769 Enum a,
732 Num e,
770 Num e,
@@ -734,63 +772,86 @@ grd ::
734 )
772 )
735 =>
773 =>
736 (a -> Bool) -> Matrix e q (Either q q)
774 (a -> Bool) -> Matrix e q (Either q q)
737−grd f = split (corr f) (corr (not . f))
775+grd f = fork (corr f) (corr (not . f))
738−
776+
739−corr ::
777+corr ::
740− forall e a q .
778+ forall e a q .
741− ( q ~ FromNat (Count q),
779+ ( Trivial q,
742− KnownNat (Count q),
780+ Countable q,
743− FromLists e () q,
781+ FL () q,
744− FromLists e q (),
782+ FL q (),
745− FromLists e q q,
783+ FL q q,
746− Bounded a,
784+ Liftable e a Bool
747− Enum a,
748− Num e,
749− Ord e
750− )
785+ )
751 => (a -> Bool) -> Matrix e q q
786 => (a -> Bool) -> Matrix e q q
752−corr p = let f = fromF p :: Matrix e q ()
787+corr p = let f = fromF' p :: Matrix e q ()
753− in khatri f (identity :: Matrix e q q)
788+ in kr f (id :: Matrix e q q)
754
789
755-- Pretty print
790-- Pretty print
756
791
757prettyAux :: Show e => [[e]] -> [[e]] -> String
792prettyAux :: Show e => [[e]] -> [[e]] -> String
758prettyAux [] _ = ""
793prettyAux [] _ = ""
759−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
794+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
760 where
795 where
761 v = fmap show m
796 v = fmap show m
762 widest = maximum $ fmap length v
797 widest = maximum $ fmap length v
763 fill str = replicate (widest - length str - 2) ' ' ++ str
798 fill str = replicate (widest - length str - 2) ' ' ++ str
764−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
799+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
765 where
800 where
766− v = fmap show m
801+ v = fmap show m
767− widest = maximum $ fmap length v
802+ widest = maximum $ fmap length v
768 fill str = replicate (widest - length str - 2) ' ' ++ str
803 fill str = replicate (widest - length str - 2) ' ' ++ str
769−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
804+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
770 prettyAux t l
805 prettyAux t l
771 where
806 where
772− v = fmap show l
807+ v = fmap show l
773− widest = maximum $ fmap length v
808+ widest = maximum $ fmap length v
774 fill str = replicate (widest - length str - 2) ' ' ++ str
809 fill str = replicate (widest - length str - 2) ' ' ++ str
775
810
776-- | Matrix pretty printer
811-- | Matrix pretty printer
777−pretty :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> String
812+pretty :: (CountableDims cols rows, Show e) => Matrix e cols rows -> String
813+pretty m = concat
778−pretty m = "┌ " ++ unwords (replicate (columns m) blank) ++ " ┐\n" ++
814+ [ "┌ ", unwords (replicate (columns m) blank), " ┐\n"
779− prettyAux (toLists m) (toLists m) ++
815+ , unlines
816+ [ "│ " ++ unwords (fmap (\j -> fill $ show $ getElem i j m) [1..columns m]) ++ " │" | i <- [1..rows m] ]
780− "└ " ++ unwords (replicate (columns m) blank) ++ " ┘"
817+ , "└ ", unwords (replicate (columns m) blank), " ┘"
818+ ]
781− where
819+ where
782− v = fmap show (toList m)
820+ strings = map show (toList m)
783− widest = maximum $ fmap length v
821+ widest = maximum $ map length strings
784 fill str = replicate (widest - length str) ' ' ++ str
822 fill str = replicate (widest - length str) ' ' ++ str
785− blank = fill ""
823+ blank = fill ""
824+ safeGet i j m
825+ | i > rows m || j > columns m || i < 1 || j < 1 = Nothing
826+ | otherwise = Just $ unsafeGet i j m (toList m)
827+ unsafeGet i j m l = l !! encode (columns m) (i,j)
828+ encode m (i,j) = (i-1)*m + j - 1
829+ getElem i j m =
830+ fromMaybe
831+ (error $
832+ "getElem: Trying to get the "
833+ ++ show (i, j)
834+ ++ " element from a "
835+ ++ show (rows m) ++ "x" ++ show (columns m)
836+ ++ " matrix."
837+ )
838+ (safeGet i j m)
786
839
787-- | Matrix pretty printer
840-- | Matrix pretty printer
788−prettyPrint :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> IO ()
841+prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()
789prettyPrint = putStrLn . pretty
842prettyPrint = putStrLn . pretty
790
843
844+-- | Zip two matrices with a given binary function
845+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows
846+zipWithM f (One a) (One b) = One (f a b)
847+zipWithM f (Join a b) (Join c d) = Join (zipWithM f a c) (zipWithM f b d)
848+zipWithM f (Fork a b) (Fork c d) = Fork (zipWithM f a c) (zipWithM f b d)
849+zipWithM f x@(Fork _ _) y@(Join _ _) = zipWithM f x (abideJF y)
850+zipWithM f x@(Join _ _) y@(Fork _ _) = zipWithM f (abideJF x) y
851+
791-- Relational operators functions
852-- Relational operators functions
792
853
793−type Boolean = Natural 0 1
854+type Boolean = Natural 0 1
794type Relation a b = Matrix Boolean a b
855type Relation a b = Matrix Boolean a b
795
856
796-- | Helper conversion function
857-- | Helper conversion function
@@ -806,54 +867,47 @@ fromBool False = nat 0
806
867
807-- | Relational negation
868-- | Relational negation
808negateM :: Relation cols rows -> Relation cols rows
869negateM :: Relation cols rows -> Relation cols rows
809−negateM Empty = Empty
810−negateM (One (Nat 0)) = One (Nat 1)
870+negateM (One (Nat p)) = One (Nat (negate p))
811−negateM (One (Nat 1)) = One (Nat 0)
812−negateM (Junc a b) = Junc (negateM a) (negateM b)
871+negateM (Join a b) = Join (negateM a) (negateM b)
813−negateM (Split a b) = Split (negateM a) (negateM b)
872+negateM (Fork a b) = Fork (negateM a) (negateM b)
814
873
815-- | Relational addition
874-- | Relational addition
816orM :: Relation cols rows -> Relation cols rows -> Relation cols rows
875orM :: Relation cols rows -> Relation cols rows -> Relation cols rows
817−orM Empty Empty = Empty
818−orM (One a) (One b) = One (fromBool (toBool a || toBool b))
876+orM (One a) (One b) = One (fromBool (toBool a || toBool b))
819−orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)
877+orM (Join a b) (Join c d) = Join (orM a c) (orM b d)
820−orM (Split a b) (Split c d) = Split (orM a c) (orM b d)
878+orM (Fork a b) (Fork c d) = Fork (orM a c) (orM b d)
821−orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)
879+orM x@(Fork _ _) y@(Join _ _) = orM x (abideJF y)
822−orM x@(Junc _ _) y@(Split _ _) = orM (abideJS x) y
880+orM x@(Join _ _) y@(Fork _ _) = orM (abideJF x) y
823
881
824-- | Relational multiplication
882-- | Relational multiplication
825andM :: Relation cols rows -> Relation cols rows -> Relation cols rows
883andM :: Relation cols rows -> Relation cols rows -> Relation cols rows
826−andM Empty Empty = Empty
827−andM (One a) (One b) = One (fromBool (toBool a && toBool b))
884+andM (One a) (One b) = One (fromBool (toBool a && toBool b))
828−andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)
885+andM (Join a b) (Join c d) = Join (andM a c) (andM b d)
829−andM (Split a b) (Split c d) = Split (andM a c) (andM b d)
886+andM (Fork a b) (Fork c d) = Fork (andM a c) (andM b d)
830−andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)
887+andM x@(Fork _ _) y@(Join _ _) = andM x (abideJF y)
831−andM x@(Junc _ _) y@(Split _ _) = andM (abideJS x) y
888+andM x@(Join _ _) y@(Fork _ _) = andM (abideJF x) y
832
889
833-- | Relational subtraction
890-- | Relational subtraction
834subM :: Relation cols rows -> Relation cols rows -> Relation cols rows
891subM :: Relation cols rows -> Relation cols rows -> Relation cols rows
835−subM Empty Empty = Empty
836−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
892+subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
837−subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)
893+subM (Join a b) (Join c d) = Join (subM a c) (subM b d)
838−subM (Split a b) (Split c d) = Split (subM a c) (subM b d)
894+subM (Fork a b) (Fork c d) = Fork (subM a c) (subM b d)
839−subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)
895+subM x@(Fork _ _) y@(Join _ _) = subM x (abideJF y)
840−subM x@(Junc _ _) y@(Split _ _) = subM (abideJS x) y
896+subM x@(Join _ _) y@(Fork _ _) = subM (abideJF x) y
841
897
842-- | Matrix relational composition.
898-- | Matrix relational composition.
843compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows
899compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows
844−compRel Empty Empty = Empty
845−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
900+compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
846−compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
901+compRel (Join a b) (Fork c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
847−compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law
902+compRel (Fork a b) c = Fork (compRel a c) (compRel b c) -- Fork fusion law
848−compRel c (Junc a b) = Junc (compRel c a) (compRel c b) -- Junc fusion law
903+compRel c (Join a b) = Join (compRel c a) (compRel c b) -- Join fusion law
849
904
850-- | Matrix relational right division
905-- | Matrix relational right division
851divR :: Relation b c -> Relation b a -> Relation a c
906divR :: Relation b c -> Relation b a -> Relation a c
852−divR Empty Empty = Empty
853divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a
907divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a
854−divR (Junc a b) (Junc c d) = andM (divR a c) (divR b d)
908+divR (Join a b) (Join c d) = andM (divR a c) (divR b d)
855−divR (Split a b) c = Split (divR a c) (divR b c)
909+divR (Fork a b) c = Fork (divR a c) (divR b c)
856−divR c (Split a b) = Junc (divR c a) (divR c b)
910+divR c (Fork a b) = Join (divR c a) (divR c b)
857
911
858-- | Matrix relational left division
912-- | Matrix relational left division
859divL :: Relation c b -> Relation a b -> Relation a c
913divL :: Relation c b -> Relation a b -> Relation a c
@@ -869,97 +923,52 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)
869--
923--
870-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
871-- types @a@ or @b@ allows.
925-- types @a@ or @b@ allows.
872−fromFRel ::
926+fromFRel' ::
873 forall a b cols rows.
927 forall a b cols rows.
874− ( Bounded a,
928+ ( Liftable Boolean a b,
875− Bounded b,
929+ CountableDims cols rows,
876− Enum a,
877− Enum b,
878− Eq b,
879− KnownNat (Count cols),
880− KnownNat (Count rows),
881− FromLists (Natural 0 1) rows cols
930+ FL rows cols
882 ) =>
931 ) =>
883 (a -> b) ->
932 (a -> b) ->
884 Relation cols rows
933 Relation cols rows
885−fromFRel f =
934+fromFRel' = fromF'
886− let minA = minBound @a
887− maxA = maxBound @a
888− minB = minBound @b
889− maxB = maxBound @b
890− ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))
891− rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))
892− elementsA = take ccols [minA .. maxA]
893− elementsB = take rrows [minB .. maxB]
894− combinations = (,) <$> elementsA <*> elementsB
895− combAp = map snd . sort . map (\(a, b) -> if f a == b
896− then ((fromEnum a, fromEnum b), nat 1)
897− else ((fromEnum a, fromEnum b), nat 0)) $ combinations
898− mList = buildList combAp rrows
899− in tr $ fromLists mList
900− where
901− buildList [] _ = []
902− buildList l r = take r l : buildList (drop r l) r
903
935
904-- | Lifts functions to relations with dimensions matching @a@ and @b@
936-- | Lifts functions to relations with dimensions matching @a@ and @b@
905-- cardinality's.
937-- cardinality's.
906−fromFRel' ::
938+fromFRel ::
907 forall a b.
939 forall a b.
908− ( Bounded a,
940+ ( Liftable Boolean a b,
909− Bounded b,
941+ CountableDimsN a b,
910− Enum a,
942+ FLN b a
911− Enum b,
912− Eq b,
913− KnownNat (Count (Normalize a)),
914− KnownNat (Count (Normalize b)),
915− FromLists (Natural 0 1) (Normalize b) (Normalize a)
916 ) =>
943 ) =>
917 (a -> b) ->
944 (a -> b) ->
918 Relation (Normalize a) (Normalize b)
945 Relation (Normalize a) (Normalize b)
919−fromFRel' f =
946+fromFRel = fromFRel'
920− let minA = minBound @a
921− maxA = maxBound @a
922− minB = minBound @b
923− maxB = maxBound @b
924− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
925− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
926− elementsA = take ccols [minA .. maxA]
927− elementsB = take rrows [minB .. maxB]
928− combinations = (,) <$> elementsA <*> elementsB
929− combAp = map snd . sort . map (\(a, b) -> if f a == b
930− then ((fromEnum a, fromEnum b), nat 1)
931− else ((fromEnum a, fromEnum b), nat 0)) $ combinations
932− mList = buildList combAp rrows
933− in tr $ fromLists mList
934− where
935− buildList [] _ = []
936− buildList l r = take r l : buildList (drop r l) r
937
947
938-- | Lifts a relation function to a Boolean Matrix
948-- | Lifts a relation function to a Boolean Matrix
939−toRel ::
949+toRel ::
940 forall a b.
950 forall a b.
941 ( Bounded a,
951 ( Bounded a,
942 Bounded b,
952 Bounded b,
943 Enum a,
953 Enum a,
944 Enum b,
954 Enum b,
945 Eq b,
955 Eq b,
946− KnownNat (Count (Normalize a)),
956+ CountableDims a b,
947− KnownNat (Count (Normalize b)),
957+ FLN b a
948− FromLists (Natural 0 1) (Normalize b) (Normalize a)
949− )
958+ )
950 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)
959 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)
951toRel f =
960toRel f =
952 let minA = minBound @a
961 let minA = minBound @a
953 maxA = maxBound @a
962 maxA = maxBound @a
954 minB = minBound @b
963 minB = minBound @b
955 maxB = maxBound @b
964 maxB = maxBound @b
956− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
965+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))
957− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
966+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))
958 elementsA = take ccols [minA .. maxA]
967 elementsA = take ccols [minA .. maxA]
959 elementsB = take rrows [minB .. maxB]
968 elementsB = take rrows [minB .. maxB]
960 combinations = (,) <$> elementsA <*> elementsB
969 combinations = (,) <$> elementsA <*> elementsB
961− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
970+ combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
962− then ((fromEnum a, fromEnum b), nat 1)
971+ then ((fromEnum a, fromEnum b), nat 1)
963 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
964 mList = buildList combAp rrows
973 mList = buildList combAp rrows
965 in tr $ fromLists mList
974 in tr $ fromLists mList
src/LAoP/Matrix/Nat.hs+194−187
@@ -1,3 +1,4 @@
1+{-# LANGUAGE ConstraintKinds #-}
1{-# LANGUAGE AllowAmbiguousTypes #-}
2{-# LANGUAGE AllowAmbiguousTypes #-}
2{-# LANGUAGE DataKinds #-}
3{-# LANGUAGE DataKinds #-}
3{-# LANGUAGE DerivingVia #-}
4{-# LANGUAGE DerivingVia #-}
@@ -53,11 +54,22 @@ module LAoP.Matrix.Nat
53 -- * Type safe matrix representation
54 -- * Type safe matrix representation
54 Matrix (..),
55 Matrix (..),
55
56
57+ -- * Constraint type synonyms
58+ Countable,
59+ CountableDims,
60+ CountableN,
61+ CountableNz,
62+ CountableDimsN,
63+ FLN,
64+ FLNz,
65+ Liftable,
66+ TrivialE,
67+ TrivialP,
68+
56 -- * Primitives
69 -- * Primitives
57− empty,
58 one,
70 one,
59− junc,
71+ join,
60− split,
72+ fork,
61
73
62 -- * Auxiliary type families
74 -- * Auxiliary type families
63 I.FromNat,
75 I.FromNat,
@@ -65,11 +77,11 @@ module LAoP.Matrix.Nat
65 I.Normalize,
77 I.Normalize,
66
78
67 -- * Matrix construction and conversion
79 -- * Matrix construction and conversion
68− I.FromLists,
80+ I.FL,
69 fromLists,
81 fromLists,
70 toLists,
82 toLists,
71 toList,
83 toList,
72− matrixBuilder,
84+ matrixBuilder',
73 row,
85 row,
74 col,
86 col,
75 zeros,
87 zeros,
@@ -85,6 +97,10 @@ module LAoP.Matrix.Nat
85 -- ** Matrix Transposition
97 -- ** Matrix Transposition
86 tr,
98 tr,
87
99
100+ -- ** Scalar multiplication/division of matrices
101+ (.|),
102+ (./),
103+
88 -- ** Selective operator
104 -- ** Selective operator
89 select,
105 select,
90
106
@@ -92,16 +108,19 @@ module LAoP.Matrix.Nat
92 cond,
108 cond,
93
109
94 -- ** Matrix "abiding"
110 -- ** Matrix "abiding"
95− abideJS,
111+ abideJF,
96− abideSJ,
112+ abideFJ,
113+
114+ -- ** Zip Matrices
115+ zipWithM,
97
116
98 -- * Biproduct approach
117 -- * Biproduct approach
99− -- ** Split
118+ -- ** Fork
100 (===),
119 (===),
101 -- *** Projections
120 -- *** Projections
102 p1,
121 p1,
103 p2,
122 p2,
104− -- ** Junc
123+ -- ** Join
105 (|||),
124 (|||),
106 -- *** Injections
125 -- *** Injections
107 i1,
126 i1,
@@ -115,11 +134,11 @@ module LAoP.Matrix.Nat
115 -- | Note that given the restrictions imposed it is not possible to
134 -- | Note that given the restrictions imposed it is not possible to
116 -- implement the standard type classes present in standard Haskell.
135 -- implement the standard type classes present in standard Haskell.
117 -- *** Matrix pairing projections
136 -- *** Matrix pairing projections
118− kp1,
137+ fstM,
119− kp2,
138+ sndM,
120
139
121 -- *** Matrix pairing
140 -- *** Matrix pairing
122− khatri,
141+ kr,
123
142
124 -- * Matrix composition and lifting
143 -- * Matrix composition and lifting
125
144
@@ -127,10 +146,10 @@ module LAoP.Matrix.Nat
127
146
128 -- | Note that given the restrictions imposed it is not possible to
147 -- | Note that given the restrictions imposed it is not possible to
129 -- implement the standard type classes present in standard Haskell.
148 -- implement the standard type classes present in standard Haskell.
130− identity,
149+ iden,
131 comp,
150 comp,
132− fromF,
133 fromF',
151 fromF',
152+ fromF,
134
153
135 -- * Matrix printing
154 -- * Matrix printing
136 pretty,
155 pretty,
@@ -141,107 +160,95 @@ where
141import Data.Proxy
160import Data.Proxy
142import GHC.TypeLits
161import GHC.TypeLits
143import Control.DeepSeq
162import Control.DeepSeq
144−import qualified Control.Category as C
145import qualified LAoP.Matrix.Internal as I
163import qualified LAoP.Matrix.Internal as I
146
164
147newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))
165newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))
148 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))
166 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))
149
167
150−-- | It isn't possible to implement the 'id' function so it's
168+-- | Constraint type synonyms to keep the type signatures less convoluted
151−-- implementation is 'undefined'. However 'comp' can be and this partial
169+type Countable a = KnownNat (I.Count a)
152−-- class implementation exists just to make the code more readable.
170+type CountableDims a b = (Countable a, Countable b)
153−--
171+type CountableN a = KnownNat (I.Count (I.FromNat a))
154−-- Please use 'identity' instead.
172+type CountableNz a = KnownNat (I.Count (I.Normalize a))
155−instance (Num e) => C.Category (Matrix e) where
173+type CountableDimsN a b = (CountableN a, CountableN b)
156− id = undefined
174+type FLN e a b = I.FL (I.FromNat a) (I.FromNat b)
157− (.) = comp
175+type FLNz e a b = I.FL (I.Normalize a) (I.Normalize b)
176+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
177+type TrivialE a b = I.FromNat (a + b) ~ Either (I.FromNat a) (I.FromNat b)
178+type TrivialP a b = I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))
158
179
159-- Primitives
180-- Primitives
160
181
161−empty :: Matrix e 0 0
162−empty = M I.Empty
163−
164one :: e -> Matrix e 1 1
182one :: e -> Matrix e 1 1
165one = M . I.One
183one = M . I.One
166
184
167−junc ::
185+join ::
168− (I.FromNat cols3 ~ Either (I.FromNat cols1) (I.FromNat cols2)) =>
186+ (TrivialE a b) =>
169− Matrix e cols1 rows ->
187+ Matrix e a rows ->
170− Matrix e cols2 rows ->
188+ Matrix e b rows ->
171− Matrix e cols3 rows
189+ Matrix e (a + b) rows
172−junc (M a) (M b) = M (I.Junc a b)
190+join (M a) (M b) = M (I.Join a b)
173
191
174infixl 3 |||
192infixl 3 |||
175(|||) ::
193(|||) ::
176− (I.FromNat cols3 ~ Either (I.FromNat cols1) (I.FromNat cols2)) =>
194+ (TrivialE a b) =>
177− Matrix e cols1 rows ->
195+ Matrix e a rows ->
178− Matrix e cols2 rows ->
196+ Matrix e b rows ->
179− Matrix e cols3 rows
197+ Matrix e (a + b) rows
180−(|||) = junc
198+(|||) = join
181−
199+
182−split ::
200+fork ::
183− (I.FromNat rows3 ~ Either (I.FromNat rows1) (I.FromNat rows2)) =>
201+ (TrivialE a b) =>
184− Matrix e cols rows1 ->
202+ Matrix e cols a ->
185− Matrix e cols rows2 ->
203+ Matrix e cols b ->
186− Matrix e cols rows3
204+ Matrix e cols (a + b)
187−split (M a) (M b) = M (I.Split a b)
205+fork (M a) (M b) = M (I.Fork a b)
188
206
189infixl 2 ===
207infixl 2 ===
190(===) ::
208(===) ::
191− (I.FromNat rows3 ~ Either (I.FromNat rows1) (I.FromNat rows2)) =>
209+ (TrivialE a b) =>
192− Matrix e cols rows1 ->
210+ Matrix e cols a ->
193− Matrix e cols rows2 ->
211+ Matrix e cols b ->
194− Matrix e cols rows3
212+ Matrix e cols (a + b)
195−(===) = split
213+(===) = fork
196
214
197-- Construction
215-- Construction
198
216
199−fromLists :: (I.FromLists e (I.FromNat cols) (I.FromNat rows)) => [[e]] -> Matrix e cols rows
217+fromLists :: (FLN e cols rows) => [[e]] -> Matrix e cols rows
200fromLists = M . I.fromLists
218fromLists = M . I.fromLists
201
219
220+-- | Matrix builder function. Constructs a matrix provided with
221+-- a construction function that operates with indices.
202−matrixBuilder ::
222+matrixBuilder' ::
203− (I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
223+ (FLN e cols rows, CountableN cols, CountableN rows)
204− ((Int, Int) -> e) ->
205− Matrix e cols rows
224+ => ((Int, Int) -> e) -> Matrix e cols rows
206−matrixBuilder = M . I.matrixBuilder
225+matrixBuilder' = M . I.matrixBuilder'
207
226
208−col :: (I.FromLists e () (I.FromNat rows)) => [e] -> Matrix e 1 rows
227+col :: (I.FL () (I.FromNat rows)) => [e] -> Matrix e 1 rows
209col = M . I.col
228col = M . I.col
210
229
211−row :: (I.FromLists e (I.FromNat cols) ()) => [e] -> Matrix e cols 1
230+row :: (I.FL (I.FromNat cols) ()) => [e] -> Matrix e cols 1
212row = M . I.row
231row = M . I.row
213
232
214−fromF ::
233+fromF' ::
215− ( Bounded a,
234+ ( Liftable e a b,
216− Bounded b,
235+ CountableN cols,
217− Enum a,
236+ CountableN rows,
218− Enum b,
237+ FLN e rows cols
219− Eq b,
220− Num e,
221− Ord e,
222− KnownNat (I.Count (I.FromNat cols)),
223− KnownNat (I.Count (I.FromNat rows)),
224− I.FromLists e (I.FromNat rows) (I.FromNat cols)
225 ) =>
238 ) =>
226 (a -> b) ->
239 (a -> b) ->
227 Matrix e cols rows
240 Matrix e cols rows
228−fromF = M . I.fromF
241+fromF' = M . I.fromF'
229
242
230−fromF' ::
243+fromF ::
231− ( Bounded a,
244+ ( Liftable e a b,
232− Bounded b,
245+ CountableNz a,
233− Enum a,
246+ CountableNz b,
234− Enum b,
247+ FLNz e b a
235− Eq b,
236− Num e,
237− Ord e,
238− KnownNat (I.Count (I.Normalize a)),
239− KnownNat (I.Count (I.Normalize b)),
240− I.FromLists e (I.Normalize b) (I.Normalize a)
241 ) =>
248 ) =>
242 (a -> b) ->
249 (a -> b) ->
243 Matrix e (I.Count a) (I.Count b)
250 Matrix e (I.Count a) (I.Count b)
244−fromF' = M . I.fromF'
251+fromF = undefined -- M . I.fromF
245
252
246-- Conversion
253-- Conversion
247
254
@@ -254,21 +261,21 @@ toList (M m) = I.toList m
254-- Zeros Matrix
261-- Zeros Matrix
255
262
256zeros ::
263zeros ::
257− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
264+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
258 Matrix e cols rows
265 Matrix e cols rows
259zeros = M I.zeros
266zeros = M I.zeros
260
267
261-- Ones Matrix
268-- Ones Matrix
262
269
263ones ::
270ones ::
264− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
271+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
265 Matrix e cols rows
272 Matrix e cols rows
266ones = M I.ones
273ones = M I.ones
267
274
268-- Const Matrix
275-- Const Matrix
269
276
270constant ::
277constant ::
271− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
278+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
272 e ->
279 e ->
273 Matrix e cols rows
280 Matrix e cols rows
274constant = M . I.constant
281constant = M . I.constant
@@ -277,40 +284,52 @@ constant = M . I.constant
277
284
278bang ::
285bang ::
279 forall e cols.
286 forall e cols.
280− (Num e, Enum e, I.FromLists e (I.FromNat cols) (), KnownNat (I.Count (I.FromNat cols))) =>
287+ (Num e, Enum e, I.FL (I.FromNat cols) (), CountableN cols) =>
281 Matrix e cols 1
288 Matrix e cols 1
282bang = M I.bang
289bang = M I.bang
283
290
284−-- Identity Matrix
291+-- iden Matrix
285
292
286−identity ::
293+iden ::
287− (Num e, I.FromLists e (I.FromNat cols) (I.FromNat cols), KnownNat (I.Count (I.FromNat cols))) =>
294+ (Num e, FLN e cols cols, CountableN cols) =>
288 Matrix e cols cols
295 Matrix e cols cols
289−identity = M I.identity
296+iden = M I.iden
290
297
291-- Matrix composition (MMM)
298-- Matrix composition (MMM)
292
299
293comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
300comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
294comp (M a) (M b) = M (I.comp a b)
301comp (M a) (M b) = M (I.comp a b)
295
302
303+-- Scalar multiplication of matrices
304+
305+infixl 7 .|
306+-- | Scalar multiplication of matrices.
307+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows
308+(.|) e (M m) = M (e I..| m)
309+
310+-- Scalar division of matrices
311+
312+infixl 7 ./
313+-- | Scalar multiplication of matrices.
314+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows
315+(./) (M m) e = M (m I../ e)
316+
296p1 ::
317p1 ::
297 ( Num e,
318 ( Num e,
298− KnownNat (I.Count (I.FromNat n)),
319+ CountableDimsN n m,
299− KnownNat (I.Count (I.FromNat m)),
320+ FLN e n m,
300− I.FromLists e (I.FromNat n) (I.FromNat m),
321+ FLN e m m,
301− I.FromLists e (I.FromNat m) (I.FromNat m),
322+ TrivialE m n
302− I.FromNat (m + n) ~ Either (I.FromNat m) (I.FromNat n)
303 ) =>
323 ) =>
304 Matrix e (m + n) m
324 Matrix e (m + n) m
305p1 = M I.p1
325p1 = M I.p1
306
326
307p2 ::
327p2 ::
308 ( Num e,
328 ( Num e,
309− KnownNat (I.Count (I.FromNat n)),
329+ CountableDimsN n m,
310− KnownNat (I.Count (I.FromNat m)),
330+ FLN e m n,
311− I.FromLists e (I.FromNat m) (I.FromNat n),
331+ FLN e n n,
312− I.FromLists e (I.FromNat n) (I.FromNat n),
332+ TrivialE m n
313− I.FromNat (m + n) ~ Either (I.FromNat m) (I.FromNat n)
314 ) =>
333 ) =>
315 Matrix e (m + n) n
334 Matrix e (m + n) n
316p2 = M I.p2
335p2 = M I.p2
@@ -319,138 +338,125 @@ p2 = M I.p2
319
338
320i1 ::
339i1 ::
321 ( Num e,
340 ( Num e,
322− KnownNat (I.Count (I.FromNat n)),
341+ CountableDimsN n rows,
323− KnownNat (I.Count (I.FromNat rows)),
342+ FLN e n rows,
324− I.FromLists e (I.FromNat n) (I.FromNat rows),
343+ FLN e rows rows,
325− I.FromLists e (I.FromNat rows) (I.FromNat rows),
344+ TrivialE rows n
326− I.FromNat (rows + n) ~ Either (I.FromNat rows) (I.FromNat n)
327 ) =>
345 ) =>
328 Matrix e rows (rows + n)
346 Matrix e rows (rows + n)
329i1 = tr p1
347i1 = tr p1
330
348
331i2 ::
349i2 ::
332 ( Num e,
350 ( Num e,
333− KnownNat (I.Count (I.FromNat rows)),
351+ CountableDimsN rows m,
334− KnownNat (I.Count (I.FromNat m)),
352+ FLN e m rows,
335− I.FromLists e (I.FromNat m) (I.FromNat rows),
353+ FLN e rows rows,
336− I.FromLists e (I.FromNat rows) (I.FromNat rows),
354+ TrivialE m rows
337− I.FromNat (m + rows) ~ Either (I.FromNat m) (I.FromNat rows)
338 ) =>
355 ) =>
339 Matrix e rows (m + rows)
356 Matrix e rows (m + rows)
340i2 = tr p2
357i2 = tr p2
341
358
342-- Dimensions
359-- Dimensions
343
360
344−rows :: (KnownNat (I.Count (I.FromNat rows))) => Matrix e cols rows -> Int
361+rows :: (CountableN rows) => Matrix e cols rows -> Int
345rows (M m) = I.rows m
362rows (M m) = I.rows m
346
363
347−columns :: (KnownNat (I.Count (I.FromNat cols))) => Matrix e cols rows -> Int
364+columns :: (CountableN cols) => Matrix e cols rows -> Int
348columns (M m) = I.columns m
365columns (M m) = I.columns m
349
366
350−-- Coproduct Bifunctor
351−
352infixl 5 -|-
367infixl 5 -|-
353
368
369+-- | Coproduct Bifunctor (Direct sum)
354(-|-) ::
370(-|-) ::
355 ( Num e,
371 ( Num e,
356− KnownNat (I.Count (I.FromNat j)),
372+ CountableDimsN j k,
357− KnownNat (I.Count (I.FromNat k)),
373+ FLN e k k,
358− I.FromLists e (I.FromNat k) (I.FromNat k),
374+ FLN e j k,
359− I.FromLists e (I.FromNat j) (I.FromNat k),
375+ FLN e k j,
360− I.FromLists e (I.FromNat k) (I.FromNat j),
376+ FLN e j j,
361− I.FromLists e (I.FromNat j) (I.FromNat j),
377+ TrivialE n m,
362− I.FromNat (n + m) ~ Either (I.FromNat n) (I.FromNat m),
378+ TrivialE k j
363− I.FromNat (k + j) ~ Either (I.FromNat k) (I.FromNat j)
364 ) =>
379 ) =>
365 Matrix e n k ->
380 Matrix e n k ->
366 Matrix e m j ->
381 Matrix e m j ->
367 Matrix e (n + m) (k + j)
382 Matrix e (n + m) (k + j)
368(-|-) (M a) (M b) = M ((I.-|-) a b)
383(-|-) (M a) (M b) = M ((I.-|-) a b)
369
384
370−-- Khatri Rao Product and projections
385+-- | Khatri Rao Product first projection
371−
386+fstM ::
372−kp1 ::
373 forall e m k .
387 forall e m k .
374 ( Num e,
388 ( Num e,
375− KnownNat (I.Count (I.FromNat m)),
389+ CountableDimsN m k,
376− KnownNat (I.Count (I.FromNat k)),
390+ CountableN (m * k),
377− KnownNat (I.Count (I.FromNat (m * k))),
391+ FLN e (m * k) m,
378− I.FromLists e (I.FromNat (m * k)) (I.FromNat m),
392+ TrivialP m k
379− I.FromNat (m * k) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat k))
380 ) => Matrix e (m * k) m
393 ) => Matrix e (m * k) m
381−kp1 = M (I.kp1 @e @(I.FromNat m) @(I.FromNat k))
394+fstM = M (I.fstM @e @(I.FromNat m) @(I.FromNat k))
382
395
383−kp2 ::
396+-- | Khatri Rao Product second projection
397+sndM ::
384 forall e m k.
398 forall e m k.
385 ( Num e,
399 ( Num e,
386− KnownNat (I.Count (I.FromNat k)),
400+ CountableDimsN k m,
387− I.FromLists e (I.FromNat (m * k)) (I.FromNat k),
401+ FLN e (m * k) k,
388− KnownNat (I.Count (I.FromNat m)),
402+ CountableN (m * k),
389− KnownNat (I.Count (I.FromNat (m * k))),
403+ TrivialP m k
390− I.FromNat (m * k) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat k))
391 ) => Matrix e (m * k) k
404 ) => Matrix e (m * k) k
392−kp2 = M (I.kp2 @e @(I.FromNat m) @(I.FromNat k))
405+sndM = M (I.sndM @e @(I.FromNat m) @(I.FromNat k))
393
406
394−khatri ::
407+-- | Khatri Rao Product
408+kr ::
395 forall e cols a b.
409 forall e cols a b.
396 ( Num e,
410 ( Num e,
397− KnownNat (I.Count (I.FromNat a)),
411+ CountableDimsN a b,
398− KnownNat (I.Count (I.FromNat b)),
412+ CountableN (a * b),
399− KnownNat (I.Count (I.FromNat (a * b))),
413+ FLN e (a * b) a,
400− I.FromLists e (I.FromNat (a * b)) (I.FromNat a),
414+ FLN e (a * b) b,
401− I.FromLists e (I.FromNat (a * b)) (I.FromNat b),
415+ TrivialP a b
402− I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))
403 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)
416 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)
404−khatri a b =
417+kr a b =
405− let kp1' = kp1 @e @a @b
418+ let fstM' = fstM @e @a @b
406− kp2' = kp2 @e @a @b
419+ sndM' = sndM @e @a @b
407− in comp (tr kp1') a * comp (tr kp2') b
420+ in comp (tr fstM') a * comp (tr sndM') b
408−
409−-- Product Bifunctor (Kronecker)
410
421
422+-- | Product Bifunctor (Kronecker)
411infixl 4 ><
423infixl 4 ><
412
424
413(><) ::
425(><) ::
414 forall e m p n q.
426 forall e m p n q.
415 ( Num e,
427 ( Num e,
416− KnownNat (I.Count (I.FromNat m)),
428+ CountableDimsN m n,
417− KnownNat (I.Count (I.FromNat n)),
429+ CountableDimsN p q,
418− KnownNat (I.Count (I.FromNat p)),
430+ CountableDimsN (m * n) (p * q),
419− KnownNat (I.Count (I.FromNat q)),
431+ FLN e (m * n) m,
420− KnownNat (I.Count (I.FromNat (m * n))),
432+ FLN e (m * n) n,
421− KnownNat (I.Count (I.FromNat (p * q))),
433+ FLN e (p * q) p,
422− I.FromLists e (I.FromNat (m * n)) (I.FromNat m),
434+ FLN e (p * q) q,
423− I.FromLists e (I.FromNat (m * n)) (I.FromNat n),
435+ TrivialP m n,
424− I.FromLists e (I.FromNat (p * q)) (I.FromNat p),
436+ TrivialP p q
425− I.FromLists e (I.FromNat (p * q)) (I.FromNat q),
426− I.FromNat (m * n) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat n)),
427− I.FromNat (p * q) ~ I.FromNat (I.Count (I.FromNat p) * I.Count (I.FromNat q))
428 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)
437 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)
429(><) a b =
438(><) a b =
430− let kp1' = kp1 @e @m @n
439+ let fstM' = fstM @e @m @n
431− kp2' = kp2 @e @m @n
440+ sndM' = sndM @e @m @n
432− in khatri (comp a kp1') (comp b kp2')
441+ in kr (comp a fstM') (comp b sndM')
433−
434−-- Matrix abide Junc Split
435
442
443+-- | Matrix abide Join Fork
436−abideJS :: Matrix e cols rows -> Matrix e cols rows
444+abideJF :: Matrix e cols rows -> Matrix e cols rows
437−abideJS (M m) = M (I.abideJS m)
445+abideJF (M m) = M (I.abideJF m)
438
446
439−-- Matrix abide Split Junc
447+-- | Matrix abide Fork Join
440−
441−abideSJ :: Matrix e cols rows -> Matrix e cols rows
448+abideFJ :: Matrix e cols rows -> Matrix e cols rows
442−abideSJ (M m) = M (I.abideSJ m)
449+abideFJ (M m) = M (I.abideFJ m)
443−
444−-- Matrix transposition
445
450
451+-- | Matrix transposition
446tr :: Matrix e cols rows -> Matrix e rows cols
452tr :: Matrix e cols rows -> Matrix e rows cols
447tr (M m) = M (I.tr m)
453tr (M m) = M (I.tr m)
448
454
449-- Selective 'select' operator
455-- Selective 'select' operator
450select ::
456select ::
451 ( Num e,
457 ( Num e,
452− I.FromLists e (I.FromNat rows1) (I.FromNat rows1),
458+ FLN e rows1 rows1,
453− KnownNat (I.Count (I.FromNat rows1)),
459+ CountableN rows1,
454 I.FromNat rows2 ~ I.FromNat rows1,
460 I.FromNat rows2 ~ I.FromNat rows1,
455 I.FromNat cols1 ~ I.FromNat cols2,
461 I.FromNat cols1 ~ I.FromNat cols2,
456 I.FromNat rows3 ~ Either (I.FromNat cols3) (I.FromNat rows1)
462 I.FromNat rows3 ~ Either (I.FromNat cols3) (I.FromNat rows1)
@@ -461,14 +467,11 @@ select (M m) (M y) = M (I.select m y)
461
467
462cond ::
468cond ::
463 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,
469 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,
464− KnownNat (I.Count (I.FromNat cols)),
470+ CountableN cols,
465− I.FromLists e () (I.FromNat cols),
471+ I.FL () (I.FromNat cols),
466− I.FromLists e (I.FromNat cols) (),
472+ I.FL (I.FromNat cols) (),
467− I.FromLists e (I.FromNat cols) (I.FromNat cols),
473+ FLN e cols cols,
468− Bounded a,
474+ Liftable e a Bool
469− Enum a,
470− Num e,
471− Ord e
472 )
475 )
473 =>
476 =>
474 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
477 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
@@ -476,8 +479,12 @@ cond p (M a) (M b) = M (I.cond p a b)
476
479
477-- Pretty print
480-- Pretty print
478
481
479−pretty :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> String
482+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String
480pretty (M m) = I.pretty m
483pretty (M m) = I.pretty m
481
484
482−prettyPrint :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> IO ()
485+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()
483prettyPrint (M m) = I.prettyPrint m
486prettyPrint (M m) = I.prettyPrint m
487+
488+-- | Zip two matrices with a given binary function
489+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows
490+zipWithM f (M a) (M b) = M (I.zipWithM f a b)
src/LAoP/Matrix/Type.hs+283−240
@@ -1,10 +1,11 @@
1+{-# LANGUAGE ConstraintKinds #-}
2+{-# LANGUAGE TypeFamilies #-}
1{-# LANGUAGE AllowAmbiguousTypes #-}
3{-# LANGUAGE AllowAmbiguousTypes #-}
2{-# LANGUAGE DataKinds #-}
4{-# LANGUAGE DataKinds #-}
3{-# LANGUAGE DerivingVia #-}
5{-# LANGUAGE DerivingVia #-}
4{-# LANGUAGE FlexibleContexts #-}
6{-# LANGUAGE FlexibleContexts #-}
5{-# LANGUAGE FlexibleInstances #-}
7{-# LANGUAGE FlexibleInstances #-}
6{-# LANGUAGE GADTs #-}
8{-# LANGUAGE GADTs #-}
7−{-# LANGUAGE KindSignatures #-}
8{-# LANGUAGE MultiParamTypeClasses #-}
9{-# LANGUAGE MultiParamTypeClasses #-}
9{-# LANGUAGE NoStarIsType #-}
10{-# LANGUAGE NoStarIsType #-}
10{-# LANGUAGE RankNTypes #-}
11{-# LANGUAGE RankNTypes #-}
@@ -55,15 +56,24 @@ module LAoP.Matrix.Type
55 -- * Type safe matrix representation
56 -- * Type safe matrix representation
56 Matrix (..),
57 Matrix (..),
57
58
59+ -- * Constraint type aliases
60+ Countable,
61+ CountableDims,
62+ CountableN,
63+ CountableDimsN,
64+ FLN,
65+ Liftable,
66+ Trivial,
67+ TrivialP,
68+
58 -- * Type aliases
69 -- * Type aliases
59 Zero,
70 Zero,
60 One,
71 One,
61
72
62 -- * Primitives
73 -- * Primitives
63− empty,
64 one,
74 one,
65− junc,
75+ join,
66− split,
76+ fork,
67
77
68 -- * Auxiliary type families
78 -- * Auxiliary type families
69 I.FromNat,
79 I.FromNat,
@@ -75,24 +85,27 @@ module LAoP.Matrix.Type
75 fromLists,
85 fromLists,
76 toLists,
86 toLists,
77 toList,
87 toList,
88+ matrixBuilder',
78 matrixBuilder,
89 matrixBuilder,
79 row,
90 row,
80 col,
91 col,
81 zeros,
92 zeros,
82 ones,
93 ones,
83 bang,
94 bang,
95+ point,
84 constant,
96 constant,
85
97
86 -- * Functor instance equivalent function
98 -- * Functor instance equivalent function
87 fmapM,
99 fmapM,
100+ bimapM,
88
101
89 -- * Applicative/Monoidal instance equivalent functions
102 -- * Applicative/Monoidal instance equivalent functions
90 unitM,
103 unitM,
91 multM,
104 multM,
92
105
93 -- * Selective equivalent instance function
106 -- * Selective equivalent instance function
94− selectM,
107+ selectM,
95−
108+
96 -- * Monad equivalent instance function
109 -- * Monad equivalent instance function
97 returnM,
110 returnM,
98 bindM,
111 bindM,
@@ -100,25 +113,34 @@ module LAoP.Matrix.Type
100 -- * Misc
113 -- * Misc
101 -- ** Get dimensions
114 -- ** Get dimensions
102 columns,
115 columns,
116+ columns',
103 rows,
117 rows,
118+ rows',
104
119
105 -- ** Matrix Transposition
120 -- ** Matrix Transposition
106 tr,
121 tr,
107
122
123+ -- ** Scalar multiplication/division of matrices
124+ (.|),
125+ (./),
126+
108 -- ** McCarthy's Conditional
127 -- ** McCarthy's Conditional
109 cond,
128 cond,
110
129
111 -- ** Matrix "abiding"
130 -- ** Matrix "abiding"
112− abideJS,
131+ abideJF,
113− abideSJ,
132+ abideFJ,
133+
134+ -- ** Zip Matrices
135+ zipWithM,
114
136
115 -- * Biproduct approach
137 -- * Biproduct approach
116− -- ** Split
138+ -- ** Fork
117 (===),
139 (===),
118 -- *** Projections
140 -- *** Projections
119 p1,
141 p1,
120 p2,
142 p2,
121− -- ** Junc
143+ -- ** Join
122 (|||),
144 (|||),
123 -- *** Injections
145 -- *** Injections
124 i1,
146 i1,
@@ -132,11 +154,11 @@ module LAoP.Matrix.Type
132 -- | Note that given the restrictions imposed it is not possible to
154 -- | Note that given the restrictions imposed it is not possible to
133 -- implement the standard type classes present in standard Haskell.
155 -- implement the standard type classes present in standard Haskell.
134 -- *** Matrix pairing projections
156 -- *** Matrix pairing projections
135− kp1,
157+ fstM,
136− kp2,
158+ sndM,
137
159
138 -- *** Matrix pairing
160 -- *** Matrix pairing
139− khatri,
161+ kr,
140
162
141 -- * Matrix composition and lifting
163 -- * Matrix composition and lifting
142
164
@@ -144,10 +166,10 @@ module LAoP.Matrix.Type
144
166
145 -- | Note that given the restrictions imposed it is not possible to
167 -- | Note that given the restrictions imposed it is not possible to
146 -- implement the standard type classes present in standard Haskell.
168 -- implement the standard type classes present in standard Haskell.
147− identity,
169+ iden,
148 comp,
170 comp,
149− fromF,
150 fromF',
171 fromF',
172+ fromF,
151
173
152 -- * Relation
174 -- * Relation
153 toRel,
175 toRel,
@@ -161,23 +183,44 @@ where
161import Data.Void
183import Data.Void
162import Data.Proxy
184import Data.Proxy
163import Data.Kind
185import Data.Kind
164−import GHC.TypeLits
186+import GHC.TypeLits
165import Control.DeepSeq
187import Control.DeepSeq
166import LAoP.Utils
188import LAoP.Utils
167−import qualified Control.Category as C
168import qualified LAoP.Matrix.Internal as I
189import qualified LAoP.Matrix.Internal as I
190+import Prelude hiding (id, (.))
169
191
170newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))
192newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))
171 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))
193 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))
172
194
195+-- | Constraint type synonyms to keep the type signatures less convoluted
196+type Countable a = KnownNat (I.Count a)
197+type CountableDims a b = (Countable a, Countable b)
198+type CountableN a = KnownNat (I.Count (I.Normalize a))
199+type CountableDimsN a b = (CountableN a, CountableN b)
200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)
201+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
202+type Trivial a = I.Normalize (I.Normalize a) ~ I.Normalize (I.Normalize (I.Normalize a))
203+type Trivial2 a = I.Normalize a ~ I.Normalize (I.Normalize a)
204+type Trivial3 a = I.FromNat (I.Count (I.Normalize (I.Normalize a))) ~ I.Normalize (I.Normalize a)
205+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
206+
173−-- | It isn't possible to implement the 'id' function so it's
207+-- | It is possible to implement a constrained version of the category type
174−-- implementation is 'undefined'. However 'comp' can be and this partial
208+-- class.
175−-- class implementation exists just to make the code more readable.
176−--
177−-- Please use 'identity' instead.
178−instance (Num e) => C.Category (Matrix e) where
209+instance (Num e) => Category (Matrix e) where
179− id = undefined
210+ type Object (Matrix e) a = (FLN a a, CountableN a)
211+ id = iden
180− (.) = comp
212+ (.) = comp
213+
214+-- | Bifunctor equivalent function
215+bimapM ::
216+ ( Liftable e a b,
217+ Liftable e c d,
218+ CountableDimsN a c,
219+ CountableDimsN b d,
220+ FLN d c,
221+ FLN b a
222+ ) => (a -> b) -> (c -> d) -> Matrix e a c -> Matrix e b d
223+bimapM f g m = fromF g . m . tr (fromF f)
181
224
182-- | Zero type alias
225-- | Zero type alias
183type Zero = Void
226type Zero = Void
@@ -187,92 +230,75 @@ type One = ()
187
230
188-- Primitives
231-- Primitives
189
232
190−-- | Empty matrix constructor
191−empty :: Matrix e Zero Zero
192−empty = M I.Empty
193−
194-- | Unit matrix constructor
233-- | Unit matrix constructor
195one :: e -> Matrix e One One
234one :: e -> Matrix e One One
196one = M . I.One
235one = M . I.One
197
236
198−-- | Matrix 'Junc' constructor
237+-- | Matrix 'Join' constructor
199−junc ::
238+join ::
200− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
201 Matrix e a rows ->
239 Matrix e a rows ->
202 Matrix e b rows ->
240 Matrix e b rows ->
203 Matrix e (Either a b) rows
241 Matrix e (Either a b) rows
204−junc (M a) (M b) = M (I.Junc a b)
242+join (M a) (M b) = M (I.Join a b)
205
243
206infixl 3 |||
244infixl 3 |||
207−-- | Matrix 'Junc' constructor
245+-- | Matrix 'Join' constructor
208(|||) ::
246(|||) ::
209− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
210 Matrix e a rows ->
247 Matrix e a rows ->
211 Matrix e b rows ->
248 Matrix e b rows ->
212 Matrix e (Either a b) rows
249 Matrix e (Either a b) rows
213−(|||) = junc
250+(|||) = join
214
251
215−-- | Matrix 'Split' constructor
252+-- | Matrix 'Fork' constructor
216−split ::
253+fork ::
217− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
218 Matrix e cols a ->
254 Matrix e cols a ->
219 Matrix e cols b ->
255 Matrix e cols b ->
220 Matrix e cols (Either a b)
256 Matrix e cols (Either a b)
221−split (M a) (M b) = M (I.Split a b)
257+fork (M a) (M b) = M (I.Fork a b)
222
258
223infixl 2 ===
259infixl 2 ===
224−-- | Matrix 'Split' constructor
260+-- | Matrix 'Fork' constructor
225(===) ::
261(===) ::
226− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
227 Matrix e cols a ->
262 Matrix e cols a ->
228 Matrix e cols b ->
263 Matrix e cols b ->
229 Matrix e cols (Either a b)
264 Matrix e cols (Either a b)
230−(===) = split
265+(===) = fork
231
266
232-- Functor hierarchy
267-- Functor hierarchy
233
268
234-- | Functor instance equivalent function
269-- | Functor instance equivalent function
235−fmapM ::
270+fmapM ::
236− ( Bounded a,
271+ ( Liftable e a b,
237− Bounded b,
272+ CountableDimsN a b,
238− Enum a,
273+ FLN b a
239− Enum b,
240− Eq b,
241− Num e,
242− Ord e,
243− KnownNat (I.Count (I.Normalize a)),
244− KnownNat (I.Count (I.Normalize b)),
245− I.FromLists e (I.Normalize b) (I.Normalize a)
246 )
274 )
247 =>
275 =>
248 (a -> b) -> Matrix e c a -> Matrix e c b
276 (a -> b) -> Matrix e c a -> Matrix e c b
249−fmapM f m = fromF' f `comp` m
277+fmapM f m = fromF f . m
250
278
251-- | Applicative instance equivalent 'unit' function,
279-- | Applicative instance equivalent 'unit' function,
252unitM :: (Num e) => Matrix e () ()
280unitM :: (Num e) => Matrix e () ()
253unitM = one 1
281unitM = one 1
254
282
255-- | Applicative instance equivalent 'unit' function,
283-- | Applicative instance equivalent 'unit' function,
256−multM ::
284+multM ::
257− ( KnownNat (I.Count (I.Normalize a)),
285+ ( CountableDimsN a b,
258− KnownNat (I.Count (I.Normalize b)),
286+ CountableN (a, b),
259− KnownNat (I.Count (I.Normalize (a, b))),
260 Num e,
287 Num e,
261− I.FromLists e (I.Normalize (a, b)) (I.Normalize a),
288+ FLN (a, b) a,
262− I.FromLists e (I.Normalize (a, b)) (I.Normalize b),
289+ FLN (a, b) b,
263− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
290+ TrivialP a b
264 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)
291 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)
265−multM a b = khatri a b
292+multM = kr
266−
267
293
268-- | Monad instance equivalent 'return' function,
294-- | Monad instance equivalent 'return' function,
269−returnM ::
295+returnM ::
270− forall e a .
296+ forall e a .
271− ( Num e,
297+ ( Num e,
272− Enum e,
298+ Enum e,
273− Enum a,
299+ Enum a,
274− I.FromLists e () (I.Normalize a),
300+ FLN () a,
275− KnownNat (I.Count a)
301+ Countable a
276 ) => a -> Matrix e One a
302 ) => a -> Matrix e One a
277returnM a = col l
303returnM a = col l
278 where
304 where
@@ -288,73 +314,64 @@ bindM = flip comp
288
314
289-- | Build a matrix out of a list of list of elements. Throws a runtime
315-- | Build a matrix out of a list of list of elements. Throws a runtime
290-- error if the dimensions do not match.
316-- error if the dimensions do not match.
291−fromLists :: (I.FromLists e (I.Normalize cols) (I.Normalize rows)) => [[e]] -> Matrix e cols rows
317+fromLists :: (FLN cols rows) => [[e]] -> Matrix e cols rows
292fromLists = M . I.fromLists
318fromLists = M . I.fromLists
293
319
294-- | Matrix builder function. Constructs a matrix provided with
320-- | Matrix builder function. Constructs a matrix provided with
295−-- a construction function.
321+-- a construction function that operates with indices.
322+matrixBuilder' ::
323+ (FLN cols rows, CountableDimsN cols rows )
324+ => ((Int, Int) -> e) -> Matrix e cols rows
325+matrixBuilder' = M . I.matrixBuilder'
326+
327+-- | Matrix builder function. Constructs a matrix provided with
328+-- a construction function that operates with arbitrary types.
296matrixBuilder ::
329matrixBuilder ::
297− (I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
330+ ( FLN a b,
298− ((Int, Int) -> e) ->
331+ Enum a,
332+ Enum b,
333+ Bounded a,
334+ Bounded b,
335+ Eq a,
336+ Countable b
299− Matrix e cols rows
337+ ) => ((a, b) -> e) -> Matrix e a b
300−matrixBuilder = M . I.matrixBuilder
338+matrixBuilder f = M (I.matrixBuilder f)
301
339
302-- | Constructs a column vector matrix
340-- | Constructs a column vector matrix
303−col :: (I.FromLists e () (I.Normalize rows)) => [e] -> Matrix e One rows
341+col :: (FLN () rows) => [e] -> Matrix e One rows
304col = M . I.col
342col = M . I.col
305
343
306-- | Constructs a row vector matrix
344-- | Constructs a row vector matrix
307−row :: (I.FromLists e (I.Normalize cols) ()) => [e] -> Matrix e cols One
345+row :: (FLN cols ()) => [e] -> Matrix e cols One
308row = M . I.row
346row = M . I.row
309
347
310-- | Lifts functions to matrices with arbitrary dimensions.
348-- | Lifts functions to matrices with arbitrary dimensions.
311--
349--
312-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
350-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
313-- types @a@ or @b@ allows.
351-- types @a@ or @b@ allows.
314−fromF ::
352+fromF' ::
315− ( Bounded a,
353+ ( Liftable e a b,
316− Bounded b,
354+ CountableDimsN cols rows,
317− Enum a,
355+ FLN rows cols
318− Enum b,
319− Eq b,
320− Num e,
321− Ord e,
322− KnownNat (I.Count (I.Normalize cols)),
323− KnownNat (I.Count (I.Normalize rows)),
324− I.FromLists e (I.Normalize rows) (I.Normalize cols)
325 ) =>
356 ) =>
326− (a -> b) ->
327− Matrix e cols rows
357+ (a -> b) -> Matrix e cols rows
328−fromF = M . I.fromF
358+fromF' = M . I.fromF'
329
359
330-- | Lifts functions to matrices with dimensions matching @a@ and @b@
360-- | Lifts functions to matrices with dimensions matching @a@ and @b@
331-- cardinality's.
361-- cardinality's.
332−fromF' ::
362+fromF ::
333− ( Bounded a,
363+ ( Liftable e a b,
334− Bounded b,
364+ CountableDimsN a b,
335− Enum a,
365+ FLN b a
336− Enum b,
337− Eq b,
338− Num e,
339− Ord e,
340− KnownNat (I.Count (I.Normalize a)),
341− KnownNat (I.Count (I.Normalize b)),
342− I.FromLists e (I.Normalize b) (I.Normalize a)
343 ) =>
366 ) =>
344− (a -> b) ->
345− Matrix e a b
367+ (a -> b) -> Matrix e a b
346−fromF' = M . I.fromF'
368+fromF = M . I.fromF
347
369
348-- | Lifts relation functions to Boolean Matrix
370-- | Lifts relation functions to Boolean Matrix
349toRel ::
371toRel ::
350− ( Bounded a,
372+ ( Liftable (Natural 0 1) a b,
351− Bounded b,
373+ CountableDims a b,
352− Enum a,
374+ FLN b a
353− Enum b,
354− Eq b,
355− KnownNat (I.Count (I.Normalize a)),
356− KnownNat (I.Count (I.Normalize b)),
357− I.FromLists (Natural 0 1) (I.Normalize b) (I.Normalize a)
358 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b
375 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b
359toRel = M . I.toRel
376toRel = M . I.toRel
360
377
@@ -372,8 +389,8 @@ toList (M m) = I.toList m
372
389
373-- | The zero matrix. A matrix wholly filled with zeros.
390-- | The zero matrix. A matrix wholly filled with zeros.
374zeros ::
391zeros ::
375− (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
392+ (Num e, FLN cols rows, CountableDimsN cols rows)
376− Matrix e cols rows
393+ => Matrix e cols rows
377zeros = M I.zeros
394zeros = M I.zeros
378
395
379-- Ones Matrix
396-- Ones Matrix
@@ -382,8 +399,8 @@ zeros = M I.zeros
382--
399--
383-- Also known as T (Top) matrix.
400-- Also known as T (Top) matrix.
384ones ::
401ones ::
385− (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
402+ (Num e, FLN cols rows, CountableDimsN cols rows)
386− Matrix e cols rows
403+ => Matrix e cols rows
387ones = M I.ones
404ones = M I.ones
388
405
389-- Const Matrix
406-- Const Matrix
@@ -391,9 +408,8 @@ ones = M I.ones
391-- | The constant matrix constructor. A matrix wholly filled with a given
408-- | The constant matrix constructor. A matrix wholly filled with a given
392-- value.
409-- value.
393constant ::
410constant ::
394− (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
411+ (Num e, FLN cols rows, CountableDimsN cols rows)
395− e ->
396− Matrix e cols rows
412+ => e -> Matrix e cols rows
397constant = M . I.constant
413constant = M . I.constant
398
414
399-- Bang Matrix
415-- Bang Matrix
@@ -401,17 +417,30 @@ constant = M . I.constant
401-- | The T (Top) row vector matrix.
417-- | The T (Top) row vector matrix.
402bang ::
418bang ::
403 forall e cols.
419 forall e cols.
404− (Num e, Enum e, I.FromLists e (I.Normalize cols) (), KnownNat (I.Count (I.Normalize cols))) =>
420+ (Num e, Enum e, FLN cols (), CountableN cols) =>
405 Matrix e cols One
421 Matrix e cols One
406bang = M I.bang
422bang = M I.bang
407
423
424+-- | Point constant relation
425+point ::
426+ ( Bounded a,
427+ Enum a,
428+ Eq a,
429+ Num e,
430+ Ord e,
431+ CountableN a,
432+ FLN a One
433+ ) => a -> Matrix e One a
434+point = fromF . const
435+
408−-- Identity Matrix
436+-- iden Matrix
409
437
410−-- | Identity matrix
438+-- | iden matrix
411−identity ::
439+iden ::
412− (Num e, I.FromLists e (I.Normalize a) (I.Normalize a), KnownNat (I.Count (I.Normalize a))) =>
440+ (Num e, FLN a a, CountableN a) =>
413 Matrix e a a
441 Matrix e a a
414−identity = M I.identity
442+iden = M I.iden
443+{-# NOINLINE iden #-}
415
444
416-- Matrix composition (MMM)
445-- Matrix composition (MMM)
417
446
@@ -421,15 +450,32 @@ identity = M I.identity
421-- from LAoP.
450-- from LAoP.
422comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
451comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
423comp (M a) (M b) = M (I.comp a b)
452comp (M a) (M b) = M (I.comp a b)
453+{-# NOINLINE comp #-}
454+{-# RULES
455+ "comp/iden1" forall m. comp m iden = m ;
456+ "comp/iden2" forall m. comp iden m = m
457+#-}
458+
459+-- Scalar multiplication of matrices
460+
461+infixl 7 .|
462+-- | Scalar multiplication of matrices.
463+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows
464+(.|) e (M m) = M (e I..| m)
465+
466+-- Scalar division of matrices
467+
468+infixl 7 ./
469+-- | Scalar multiplication of matrices.
470+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows
471+(./) (M m) e = M (m I../ e)
424
472
425-- | Biproduct first component projection
473-- | Biproduct first component projection
426p1 ::
474p1 ::
427 ( Num e,
475 ( Num e,
428− KnownNat (I.Count (I.Normalize n)),
476+ CountableDimsN n m,
429− KnownNat (I.Count (I.Normalize m)),
477+ FLN n m,
430− I.FromLists e (I.Normalize n) (I.Normalize m),
478+ FLN m m
431− I.FromLists e (I.Normalize m) (I.Normalize m),
432− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
433 ) =>
479 ) =>
434 Matrix e (Either m n) m
480 Matrix e (Either m n) m
435p1 = M I.p1
481p1 = M I.p1
@@ -437,11 +483,9 @@ p1 = M I.p1
437-- | Biproduct second component projection
483-- | Biproduct second component projection
438p2 ::
484p2 ::
439 ( Num e,
485 ( Num e,
440− KnownNat (I.Count (I.Normalize n)),
486+ CountableDimsN n m,
441− KnownNat (I.Count (I.Normalize m)),
487+ FLN m n,
442− I.FromLists e (I.Normalize m) (I.Normalize n),
488+ FLN n n
443− I.FromLists e (I.Normalize n) (I.Normalize n),
444− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
445 ) =>
489 ) =>
446 Matrix e (Either m n) n
490 Matrix e (Either m n) n
447p2 = M I.p2
491p2 = M I.p2
@@ -451,11 +495,9 @@ p2 = M I.p2
451-- | Biproduct first component injection
495-- | Biproduct first component injection
452i1 ::
496i1 ::
453 ( Num e,
497 ( Num e,
454− KnownNat (I.Count (I.Normalize n)),
498+ CountableDimsN n m,
455− KnownNat (I.Count (I.Normalize m)),
499+ FLN n m,
456− I.FromLists e (I.Normalize n) (I.Normalize m),
500+ FLN m m
457− I.FromLists e (I.Normalize m) (I.Normalize m),
458− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
459 ) =>
501 ) =>
460 Matrix e m (Either m n)
502 Matrix e m (Either m n)
461i1 = tr p1
503i1 = tr p1
@@ -463,11 +505,9 @@ i1 = tr p1
463-- | Biproduct second component injection
505-- | Biproduct second component injection
464i2 ::
506i2 ::
465 ( Num e,
507 ( Num e,
466− KnownNat (I.Count (I.Normalize n)),
508+ CountableDimsN n m,
467− KnownNat (I.Count (I.Normalize m)),
509+ FLN m n,
468− I.FromLists e (I.Normalize m) (I.Normalize n),
510+ FLN n n
469− I.FromLists e (I.Normalize n) (I.Normalize n),
470− I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
471 ) =>
511 ) =>
472 Matrix e n (Either m n)
512 Matrix e n (Either m n)
473i2 = tr p2
513i2 = tr p2
@@ -476,24 +516,34 @@ i2 = tr p2
476
516
477-- | Obtain the number of rows.
517-- | Obtain the number of rows.
478--
518--
479−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
519+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
480−-- dimensions in constant time.
520+-- dimensions in constant time. For a version that doesn't require the
481−--
521+-- constraint see 'rows''.
482−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
483−-- exchange for performance.
484−rows :: (KnownNat (I.Count (I.Normalize rows))) => Matrix e cols rows -> Int
522+rows :: (CountableN rows) => Matrix e cols rows -> Int
485rows (M m) = I.rows m
523rows (M m) = I.rows m
486
524
525+-- | Obtain the number of rows in an inefficient manner, but without any
526+-- constraints.
527+--
528+-- For a more efficient version see 'rows'.
529+rows' :: Matrix e cols rows -> Int
530+rows' (M m) = I.rows' m
531+
487-- | Obtain the number of columns.
532-- | Obtain the number of columns.
488−--
489−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
490−-- dimensions in constant time.
491--
533--
492−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
534+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
493−-- exchange for performance.
535+-- dimensions in constant time. For a version that doesn't require the
536+-- constraint see 'columns''.
494−columns :: (KnownNat (I.Count (I.Normalize cols))) => Matrix e cols rows -> Int
537+columns :: (CountableN cols) => Matrix e cols rows -> Int
495columns (M m) = I.columns m
538columns (M m) = I.columns m
496
539
540+-- | Obtain the number of columns in an inefficient manner, but without any
541+-- constraints.
542+--
543+-- For a more efficient version see 'columns'.
544+columns' :: Matrix e cols rows -> Int
545+columns' (M m) = I.columns' m
546+
497-- Coproduct Bifunctor
547-- Coproduct Bifunctor
498
548
499infixl 5 -|-
549infixl 5 -|-
@@ -501,14 +551,11 @@ infixl 5 -|-
501-- | Matrix coproduct functor also known as matrix direct sum.
551-- | Matrix coproduct functor also known as matrix direct sum.
502(-|-) ::
552(-|-) ::
503 ( Num e,
553 ( Num e,
504− KnownNat (I.Count (I.Normalize j)),
554+ CountableDimsN j k,
505− KnownNat (I.Count (I.Normalize k)),
555+ FLN k k,
506− I.FromLists e (I.Normalize k) (I.Normalize k),
556+ FLN j k,
507− I.FromLists e (I.Normalize j) (I.Normalize k),
557+ FLN k j,
508− I.FromLists e (I.Normalize k) (I.Normalize j),
558+ FLN j j
509− I.FromLists e (I.Normalize j) (I.Normalize j),
510− I.Normalize (Either n m) ~ Either (I.Normalize n) (I.Normalize m),
511− I.Normalize (Either k j) ~ Either (I.Normalize k) (I.Normalize j)
512 ) =>
559 ) =>
513 Matrix e n k ->
560 Matrix e n k ->
514 Matrix e m j ->
561 Matrix e m j ->
@@ -518,104 +565,98 @@ infixl 5 -|-
518-- Khatri Rao Product and projections
565-- Khatri Rao Product and projections
519
566
520-- | Khatri Rao product first component projection matrix.
567-- | Khatri Rao product first component projection matrix.
521−kp1 ::
568+fstM ::
522 forall e m k .
569 forall e m k .
523 ( Num e,
570 ( Num e,
524− KnownNat (I.Count (I.Normalize m)),
571+ CountableDimsN m k,
525− KnownNat (I.Count (I.Normalize k)),
572+ CountableN (m, k),
526− KnownNat (I.Count (I.Normalize (m, k))),
573+ FLN (m, k) m,
527− I.FromLists e (I.Normalize (m, k)) (I.Normalize m),
574+ TrivialP m k
528− I.Normalize (m, k) ~ I.Normalize (I.Normalize m, I.Normalize k)
529 ) => Matrix e (m, k) m
575 ) => Matrix e (m, k) m
530−kp1 = M (I.kp1 @e @(I.Normalize m) @(I.Normalize k))
576+fstM = M (I.fstM @e @(I.Normalize m) @(I.Normalize k))
531
577
532-- | Khatri Rao product second component projection matrix.
578-- | Khatri Rao product second component projection matrix.
533−kp2 ::
579+sndM ::
534 forall e m k.
580 forall e m k.
535 ( Num e,
581 ( Num e,
536− KnownNat (I.Count (I.Normalize k)),
582+ CountableDimsN k m,
537− KnownNat (I.Count (I.Normalize m)),
583+ CountableN (m, k),
538− KnownNat (I.Count (I.Normalize (m, k))),
584+ FLN (m, k) k,
539− I.FromLists e (I.Normalize (m, k)) (I.Normalize k),
585+ TrivialP m k
540− I.Normalize (m, k) ~ I.Normalize (I.Normalize m, I.Normalize k)
541 ) => Matrix e (m, k) k
586 ) => Matrix e (m, k) k
542−kp2 = M (I.kp2 @e @(I.Normalize m) @(I.Normalize k))
587+sndM = M (I.sndM @e @(I.Normalize m) @(I.Normalize k))
543
588
544-- | Khatri Rao Matrix product also known as matrix pairing.
589-- | Khatri Rao Matrix product also known as matrix pairing.
545--
590--
546-- NOTE: That this is not a true categorical product, see for instance:
591-- NOTE: That this is not a true categorical product, see for instance:
547−--
592+--
548-- @
593-- @
549−-- | kp1 `comp` khatri a b == a
594+-- | fstM . kr a b == a
550−-- khatri a b ==> |
595+-- kr a b ==> |
551−-- | kp2 `comp` khatri a b == b
596+-- | sndM . kr a b == b
552-- @
597-- @
553--
598--
554-- __Emphasis__ on the implication symbol.
599-- __Emphasis__ on the implication symbol.
555−khatri ::
600+kr ::
556 forall e cols a b.
601 forall e cols a b.
557 ( Num e,
602 ( Num e,
558− KnownNat (I.Count (I.Normalize a)),
603+ CountableDimsN a b,
559− KnownNat (I.Count (I.Normalize b)),
604+ CountableN (a, b),
560− KnownNat (I.Count (I.Normalize (a, b))),
605+ FLN (a, b) a,
561− I.FromLists e (I.Normalize (a, b)) (I.Normalize a),
606+ FLN (a, b) b,
562− I.FromLists e (I.Normalize (a, b)) (I.Normalize b),
607+ TrivialP a b
563− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
564 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)
608 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)
565−khatri a b =
609+kr a b =
566− let kp1' = kp1 @e @a @b
610+ let fstM' = fstM @e @a @b
567− kp2' = kp2 @e @a @b
611+ sndM' = sndM @e @a @b
568− in comp (tr kp1') a * comp (tr kp2') b
612+ in comp (tr fstM') a * comp (tr sndM') b
569
613
570-- Product Bifunctor (Kronecker)
614-- Product Bifunctor (Kronecker)
571
615
572infixl 4 ><
616infixl 4 ><
573
617
574−-- | Matrix product functor also known as kronecker product
618+-- | Matrix product functor also known as Kronecker product
575(><) ::
619(><) ::
576 forall e m p n q.
620 forall e m p n q.
577 ( Num e,
621 ( Num e,
578− KnownNat (I.Count (I.Normalize m)),
622+ CountableDimsN m n,
579− KnownNat (I.Count (I.Normalize n)),
623+ CountableDimsN p q,
580− KnownNat (I.Count (I.Normalize p)),
624+ CountableDimsN (m, n) (p, q),
581− KnownNat (I.Count (I.Normalize q)),
625+ FLN (m, n) m,
582− KnownNat (I.Count (I.Normalize (m, n))),
626+ FLN (m, n) n,
583− KnownNat (I.Count (I.Normalize (p, q))),
627+ FLN (p, q) p,
584− I.FromLists e (I.Normalize (m, n)) (I.Normalize m),
628+ FLN (p, q) q,
585− I.FromLists e (I.Normalize (m, n)) (I.Normalize n),
629+ TrivialP m n,
586− I.FromLists e (I.Normalize (p, q)) (I.Normalize p),
630+ TrivialP p q
587− I.FromLists e (I.Normalize (p, q)) (I.Normalize q),
588− I.Normalize (m, n) ~ I.Normalize (I.Normalize m, I.Normalize n),
589− I.Normalize (p, q) ~ I.Normalize (I.Normalize p, I.Normalize q)
590 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)
631 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)
591(><) a b =
632(><) a b =
592− let kp1' = kp1 @e @m @n
633+ let fstM' = fstM @e @m @n
593− kp2' = kp2 @e @m @n
634+ sndM' = sndM @e @m @n
594− in khatri (comp a kp1') (comp b kp2')
635+ in kr (comp a fstM') (comp b sndM')
595
636
596−-- Matrix abide Junc Split
637+-- Matrix abide Join Fork
597
638
598−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
639+-- | Matrix "abiding" followin the 'Join'-'Fork' abide law.
599−--
640+--
600-- Law:
641-- Law:
601--
642--
602-- @
643-- @
603−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)
644+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)
604-- @
645-- @
605−abideJS :: Matrix e cols rows -> Matrix e cols rows
646+abideJF :: Matrix e cols rows -> Matrix e cols rows
606−abideJS (M m) = M (I.abideJS m)
647+abideJF (M m) = M (I.abideJF m)
607
648
608−-- Matrix abide Split Junc
649+-- Matrix abide Fork Join
609
650
610−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
651+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
611−--
652+--
612-- Law:
653-- Law:
613--
654--
614-- @
655-- @
615−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)
656+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)
616-- @
657-- @
617−abideSJ :: Matrix e cols rows -> Matrix e cols rows
658+abideFJ :: Matrix e cols rows -> Matrix e cols rows
618−abideSJ (M m) = M (I.abideSJ m)
659+abideFJ (M m) = M (I.abideFJ m)
619
660
620-- Matrix transposition
661-- Matrix transposition
621
662
@@ -627,11 +668,10 @@ tr (M m) = M (I.tr m)
627
668
628-- | Selective functors 'select' operator equivalent inspired by the
669-- | Selective functors 'select' operator equivalent inspired by the
629-- ArrowMonad solution presented in the paper.
670-- ArrowMonad solution presented in the paper.
630−selectM ::
671+selectM ::
631 ( Num e,
672 ( Num e,
632− I.FromLists e (I.Normalize b) (I.Normalize b),
673+ FLN b b,
633− KnownNat (I.Count (I.Normalize b)),
674+ CountableN b
634− I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)
635 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
675 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
636selectM (M m) (M y) = M (I.select m y)
676selectM (M m) (M y) = M (I.select m y)
637
677
@@ -639,15 +679,14 @@ selectM (M m) (M y) = M (I.select m y)
639
679
640-- | McCarthy's Conditional expresses probabilistic choice.
680-- | McCarthy's Conditional expresses probabilistic choice.
641cond ::
681cond ::
642− ( I.Normalize (I.Normalize a) ~ I.Normalize a,
682+ ( Trivial a,
643− KnownNat (I.Count (I.Normalize a)),
683+ Trivial2 a,
644− I.FromLists e () (I.Normalize a),
684+ Trivial3 a,
645− I.FromLists e (I.Normalize a) (),
685+ CountableN a,
646− I.FromLists e (I.Normalize a) (I.Normalize a),
686+ FLN () a,
647− Bounded a,
687+ FLN a (),
648− Enum a,
688+ FLN a a,
649− Num e,
689+ Liftable e a Bool
650− Ord e
651 )
690 )
652 =>
691 =>
653 (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b
692 (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b
@@ -656,9 +695,13 @@ cond p (M a) (M b) = M (I.cond p a b)
656-- Pretty print
695-- Pretty print
657
696
658-- | Matrix pretty printer
697-- | Matrix pretty printer
659−pretty :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> String
698+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String
660pretty (M m) = I.pretty m
699pretty (M m) = I.pretty m
661
700
662-- | Matrix pretty printer
701-- | Matrix pretty printer
663−prettyPrint :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> IO ()
702+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()
664prettyPrint (M m) = I.prettyPrint m
703prettyPrint (M m) = I.prettyPrint m
704+
705+-- | Zip two matrices with a given binary function
706+zipWithM :: (e -> f -> g) -> Matrix e a b -> Matrix f a b -> Matrix g a b
707+zipWithM f (M a) (M b) = M (I.zipWithM f a b)
src/LAoP/Relation.hs+34−7
@@ -30,12 +30,21 @@ module LAoP.Relation
30 Relation (..),
30 Relation (..),
31 Boolean,
31 Boolean,
32
32
33+ -- * Constraint type synonyms
34+ Countable,
35+ CountableDims,
36+ CountableN,
37+ CountableDimsN,
38+ FLN,
39+ Liftable,
40+ Trivial,
41+ TrivialP,
42+
33 -- * Primitives
43 -- * Primitives
34− empty,
35 one,
44 one,
36− junc,
45+ join,
37 (|||),
46 (|||),
38− split,
47+ fork,
39 (===),
48 (===),
40
49
41 -- * Auxiliary type families
50 -- * Auxiliary type families
@@ -53,6 +62,7 @@ module LAoP.Relation
53 toList,
62 toList,
54 toBool,
63 toBool,
55 pt,
64 pt,
65+ belongs,
56 relationBuilder,
66 relationBuilder,
57 zeros,
67 zeros,
58 ones,
68 ones,
@@ -80,15 +90,25 @@ module LAoP.Relation
80 injection,
90 injection,
81 surjection,
91 surjection,
82 bijection,
92 bijection,
93+ domain,
94+ range,
83
95
84 -- * Function division
96 -- * Function division
85 divisionF,
97 divisionF,
86
98
99+ -- * Relation division
100+ divR,
101+ divL,
102+ divS,
103+ shrunkBy,
104+ overriddenBy,
105+
87 -- * Relational pairing
106 -- * Relational pairing
88 splitR,
107 splitR,
108+
89 -- ** Projections
109 -- ** Projections
90− p1,
110+ fstR,
91− p2,
111+ sndR,
92 -- ** Bifunctor
112 -- ** Bifunctor
93 (><),
113 (><),
94 -- * Relational coproduct
114 -- * Relational coproduct
@@ -116,18 +136,25 @@ module LAoP.Relation
116 linearOrder,
136 linearOrder,
117 equivalence,
137 equivalence,
118 partialEquivalence,
138 partialEquivalence,
139+ difunctional,
140+
141+ -- * Conditionals
142+ equalizer,
119
143
120 -- ** McCarthy's Conditional
144 -- ** McCarthy's Conditional
145+ predR,
146+ guard,
121− -- cond,
147+ cond,
122
148
123 -- * Relational composition and lifting
149 -- * Relational composition and lifting
124− identity,
150+ iden,
125 comp,
151 comp,
126 fromF,
152 fromF,
127 fromF',
153 fromF',
128
154
129 -- ** Relational application
155 -- ** Relational application
130 pointAp,
156 pointAp,
157+ pointApBool,
131
158
132 -- * Matrix printing
159 -- * Matrix printing
133 pretty,
160 pretty,
src/LAoP/Relation/Internal.hs+540−276
@@ -1,3 +1,5 @@
1+{-# LANGUAGE TypeFamilies #-}
2+{-# LANGUAGE ConstraintKinds #-}
1{-# LANGUAGE GADTs #-}
3{-# LANGUAGE GADTs #-}
2{-# LANGUAGE UndecidableInstances #-}
4{-# LANGUAGE UndecidableInstances #-}
3{-# LANGUAGE FlexibleContexts #-}
5{-# LANGUAGE FlexibleContexts #-}
@@ -39,12 +41,21 @@ module LAoP.Relation.Internal
39 Relation (..),
41 Relation (..),
40 Boolean,
42 Boolean,
41
43
44+ -- * Constraint type synonyms
45+ Countable,
46+ CountableDims,
47+ CountableN,
48+ CountableDimsN,
49+ FLN,
50+ Liftable,
51+ Trivial,
52+ TrivialP,
53+
42 -- * Primitives
54 -- * Primitives
43− empty,
44 one,
55 one,
45− junc,
56+ join,
46 (|||),
57 (|||),
47− split,
58+ fork,
48 (===),
59 (===),
49
60
50 -- * Auxiliary type families
61 -- * Auxiliary type families
@@ -55,13 +66,15 @@ module LAoP.Relation.Internal
55 -- * Matrix construction and conversion
66 -- * Matrix construction and conversion
56 I.FromLists,
67 I.FromLists,
57 fromLists,
68 fromLists,
58− fromF,
59 fromF',
69 fromF',
70+ fromF,
60 toRel,
71 toRel,
61 toLists,
72 toLists,
62 toList,
73 toList,
63 toBool,
74 toBool,
64 pt,
75 pt,
76+ belongs,
77+ relationBuilder',
65 relationBuilder,
78 relationBuilder,
66 zeros,
79 zeros,
67 ones,
80 ones,
@@ -89,6 +102,8 @@ module LAoP.Relation.Internal
89 injection,
102 injection,
90 surjection,
103 surjection,
91 bijection,
104 bijection,
105+ domain,
106+ range,
92
107
93 -- * Function division
108 -- * Function division
94 divisionF,
109 divisionF,
@@ -97,13 +112,15 @@ module LAoP.Relation.Internal
97 divR,
112 divR,
98 divL,
113 divL,
99 divS,
114 divS,
115+ shrunkBy,
116+ overriddenBy,
100
117
101 -- * Relational pairing
118 -- * Relational pairing
102 splitR,
119 splitR,
103
120
104 -- ** Projections
121 -- ** Projections
105− p1,
122+ fstR,
106− p2,
123+ sndR,
107 -- ** Bifunctor
124 -- ** Bifunctor
108 (><),
125 (><),
109 -- * Relational coproduct
126 -- * Relational coproduct
@@ -131,18 +148,25 @@ module LAoP.Relation.Internal
131 linearOrder,
148 linearOrder,
132 equivalence,
149 equivalence,
133 partialEquivalence,
150 partialEquivalence,
151+ difunctional,
152+
153+ -- * Conditionals
154+ equalizer,
134
155
135 -- ** McCarthy's Conditional
156 -- ** McCarthy's Conditional
157+ predR,
158+ guard,
136− -- cond,
159+ cond,
137
160
138 -- * Relational composition and lifting
161 -- * Relational composition and lifting
139− identity,
162+ iden,
140 comp,
163 comp,
141− fromF,
142 fromF',
164 fromF',
165+ fromF,
143
166
144 -- ** Relational application
167 -- ** Relational application
145 pointAp,
168 pointAp,
169+ pointApBool,
146
170
147 -- * Matrix printing
171 -- * Matrix printing
148 pretty,
172 pretty,
@@ -154,12 +178,13 @@ import Data.Void
154import qualified LAoP.Matrix.Internal as I
178import qualified LAoP.Matrix.Internal as I
155import LAoP.Utils.Internal
179import LAoP.Utils.Internal
156import Control.DeepSeq
180import Control.DeepSeq
157−import qualified Control.Category as C
158import Data.Bool
181import Data.Bool
159import GHC.TypeLits
182import GHC.TypeLits
183+import Prelude hiding (id, (.))
160
184
161-- | Boolean type synonym for working with boolean matrices
185-- | Boolean type synonym for working with boolean matrices
162type Boolean = Natural 0 1
186type Boolean = Natural 0 1
187+type Powerset a = List (List a)
163
188
164-- | Relation data type.
189-- | Relation data type.
165newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))
190newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))
@@ -167,25 +192,34 @@ newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))
167
192
168deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)
193deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)
169
194
195+-- | Constraint type synonyms to keep the type signatures less convoluted
196+type Countable a = KnownNat (I.Count a)
197+type CountableDims a b = (Countable a, Countable b)
198+type CountableN a = KnownNat (I.Count (I.Normalize a))
199+type CountableDimsN a b = (CountableN a, CountableN b)
200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)
201+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Boolean, Ord Boolean)
202+type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)
203+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
204+
170−-- | It isn't possible to implement the 'id' function so it's
205+-- | It is possible to implement a constrained version of the category type
171−-- implementation is 'undefined'. However 'comp' can be and this partial
206+-- class.
172−-- class implementation exists just to make the code more readable.
173−--
174−-- Please use 'identity' instead.
175−instance C.Category Relation where
207+instance Category Relation where
176− id = undefined
208+ type Object Relation a = (FLN a a, CountableN a)
209+ id = iden
177− (.) = comp
210+ (.) = comp
178
211
179instance Num (Relation a b) where
212instance Num (Relation a b) where
180− -- | Matrix addition becomes Boolean matrix disjunction
213+ -- | Matrix addition becomes Boolean matrix disjointion
181 (R a) + (R b) = R (I.orM a b)
214 (R a) + (R b) = R (I.orM a b)
182
215
183 -- | Matrix subtraction becomes Relational subtraction
216 -- | Matrix subtraction becomes Relational subtraction
184 (R a) - (R b) = R (I.subM a b)
217 (R a) - (R b) = R (I.subM a b)
185
218
186− -- | Matrix multiplication becomes Boolean matrix conjunction
219+ -- | Matrix multiplication becomes Boolean matrix conjointion
187 (R a) * (R b) = R (I.andM a b)
220 (R a) * (R b) = R (I.andM a b)
188
221
222+ -- | Matrix negation becomes Boolean matrix negation
189 negate (R a) = R (I.negateM a)
223 negate (R a) = R (I.negateM a)
190
224
191-- Type alias
225-- Type alias
@@ -194,105 +228,107 @@ type One = ()
194
228
195-- Primitives
229-- Primitives
196
230
197−-- | Empty matrix constructor
198−empty :: Relation Zero Zero
199−empty = R I.Empty
200−
201-- | Unit matrix constructor
231-- | Unit matrix constructor
202one :: Boolean -> Relation One One
232one :: Boolean -> Relation One One
203one = R . I.One
233one = R . I.One
204
234
205−-- | Boolean Matrix 'Junc' constructor, also known as relational coproduct.
235+-- | Boolean Matrix 'Join' constructor, also known as relational coproduct.
206--
236--
207-- See 'eitherR'.
237-- See 'eitherR'.
208−junc :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))
209− => Relation a c -> Relation b c -> Relation (Either a b) c
238+join :: Relation a c -> Relation b c -> Relation (Either a b) c
210−junc (R a) (R b) = R (I.Junc a b)
239+join (R a) (R b) = R (I.Join a b)
211
240
212infixl 3 |||
241infixl 3 |||
213−-- | Boolean Matrix 'Junc' constructor
242+-- | Boolean Matrix 'Join' constructor
243+--
244+-- See 'eitherR'.
214(|||) ::
245(|||) ::
215− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
216 Relation a c ->
246 Relation a c ->
217 Relation b c ->
247 Relation b c ->
218 Relation (Either a b) c
248 Relation (Either a b) c
219−(|||) = junc
249+(|||) = join
220
250
221−-- | Boolean Matrix 'Split' constructor, also known as relational product.
251+-- | Boolean Matrix 'Fork' constructor, also known as relational product.
222−split :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))
223− => Relation c a -> Relation c b -> Relation c (Either a b)
252+fork :: Relation c a -> Relation c b -> Relation c (Either a b)
224−split (R a) (R b) = R (I.Split a b)
253+fork (R a) (R b) = R (I.Fork a b)
225
254
226infixl 2 ===
255infixl 2 ===
227−-- | Boolean Matrix 'Split' constructor
256+-- | Boolean Matrix 'Fork' constructor
228(===) ::
257(===) ::
229− (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
230 Relation c a ->
258 Relation c a ->
231 Relation c b ->
259 Relation c b ->
232 Relation c (Either a b)
260 Relation c (Either a b)
233−(===) = split
261+(===) = fork
234
262
235-- Construction
263-- Construction
236
264
237-- | Build a matrix out of a list of list of elements. Throws a runtime
265-- | Build a matrix out of a list of list of elements. Throws a runtime
238-- error if the dimensions do not match.
266-- error if the dimensions do not match.
239−fromLists :: (I.FromLists Boolean (I.Normalize a) (I.Normalize b)) => [[Boolean]] -> Relation a b
267+fromLists :: (FLN a b) => [[Boolean]] -> Relation a b
240fromLists = R . I.fromLists
268fromLists = R . I.fromLists
241
269
270+-- | Relation builder function. Constructs a relation provided with
271+-- a construction function that operates with indices.
272+relationBuilder' ::
273+ (FLN a b, CountableDimsN a b) =>
274+ ((Int, Int) -> Boolean) -> Relation a b
275+relationBuilder' = R . I.matrixBuilder'
276+
242−-- | Matrix builder function. Constructs a matrix provided with
277+-- | Relation builder function. Constructs a relation provided with
243−-- a construction function.
278+-- a construction function that operates with arbitrary types.
244relationBuilder ::
279relationBuilder ::
245− (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>
280+ ( FLN a b,
246− ((Int, Int) -> Boolean) ->
281+ Enum a,
282+ Enum b,
283+ Bounded a,
284+ Bounded b,
285+ Eq a,
286+ CountableDims a b
247− Relation a b
287+ ) => ((a, b) -> Boolean) -> Relation a b
248relationBuilder = R . I.matrixBuilder
288relationBuilder = R . I.matrixBuilder
249
289
250-- | Lifts functions to matrices with arbitrary dimensions.
290-- | Lifts functions to matrices with arbitrary dimensions.
251--
291--
252-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
292-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
253-- types @a@ or @b@ allows.
293-- types @a@ or @b@ allows.
254−fromF ::
294+fromF' ::
255− ( Bounded a,
295+ ( Liftable a b,
256− Bounded b,
296+ CountableDimsN c d,
257− Enum a,
297+ FLN d c
258− Enum b,
259− Eq b,
260− KnownNat (I.Count (I.Normalize c)),
261− KnownNat (I.Count (I.Normalize d)),
262− I.FromLists Boolean (I.Normalize d) (I.Normalize c)
263 )
298 )
264 => (a -> b) -> Relation c d
299 => (a -> b) -> Relation c d
265−fromF f = R (I.fromFRel f)
300+fromF' f = R (I.fromFRel' f)
266
301
267-- | Lifts functions to matrices with dimensions matching @a@ and @b@
302-- | Lifts functions to matrices with dimensions matching @a@ and @b@
268-- cardinality's.
303-- cardinality's.
269−fromF' ::
304+fromF ::
270− ( Bounded b,
305+ ( Liftable a b,
271− Bounded a,
306+ CountableDimsN a b,
272− Enum a,
307+ FLN b a
273− Enum b,
274− Eq b,
275− KnownNat (I.Count (I.Normalize a)),
276− KnownNat (I.Count (I.Normalize b)),
277− I.FromLists Boolean (I.Normalize b) (I.Normalize a)
278 )
308 )
279 => (a -> b) -> Relation a b
309 => (a -> b) -> Relation a b
280−fromF' f = R (I.fromFRel' f)
310+fromF f = R (I.fromFRel f)
281
311
282-- | Lifts relation functions to 'Relation'
312-- | Lifts relation functions to 'Relation'
283toRel ::
313toRel ::
284− ( Bounded b,
314+ ( Liftable a b,
285− Bounded a,
315+ CountableDims a b,
286− Enum a,
316+ FLN b a
287− Enum b,
288− Eq b,
289− KnownNat (I.Count (I.Normalize a)),
290− KnownNat (I.Count (I.Normalize b)),
291− I.FromLists Boolean (I.Normalize b) (I.Normalize a)
292 )
317 )
293− => (a -> b -> Bool) -> Relation a b
318+ => (a -> b -> Bool) -> Relation a b
294toRel = R . I.toRel
319toRel = R . I.toRel
295
320
321+-- | Lowers a 'Relation' to a function
322+fromRel ::
323+ ( Liftable a b,
324+ Eq a,
325+ CountableDimsN a b,
326+ FLN a One,
327+ FLN b One
328+ )
329+ => Relation a b -> (a -> b -> Bool)
330+fromRel r a b = pointApBool a b r
331+
296-- Conversion
332-- Conversion
297
333
298-- | Converts a matrix to a list of lists of elements.
334-- | Converts a matrix to a list of lists of elements.
@@ -303,106 +339,188 @@ toLists (R m) = I.toLists m
303toList :: Relation a b -> [Boolean]
339toList :: Relation a b -> [Boolean]
304toList (R m) = I.toList m
340toList (R m) = I.toList m
305
341
306−-- | Converts a well typed @'Relation'@ to @'Bool'@.
342+-- | Converts a well typed 'Relation' to 'Bool'.
307toBool :: Relation One One -> Bool
343toBool :: Relation One One -> Bool
308toBool r = case toList r of
344toBool r = case toList r of
309 [Nat 0] -> False
345 [Nat 0] -> False
310 _ -> True
346 _ -> True
311
347
312−-- | Power transpose
348+-- | Power transpose.
349+--
350+-- Maps a relation to a set valued function.
313−pt ::
351+pt ::
314− ( Bounded a,
352+ ( Liftable a b,
315− Bounded b,
316− Enum a,
317− Enum b,
318 Eq a,
353 Eq a,
319− Eq b,
354+ CountableDimsN a b,
320− KnownNat (I.Count (I.Normalize a)),
355+ FLN a One,
321− KnownNat (I.Count (I.Normalize b)),
356+ FLN b One
322− I.FromLists Boolean (I.Normalize a) One,
323− I.FromLists Boolean (I.Normalize b) One
324 )
357 )
325− => Relation a b -> (a -> [b])
358+ => Relation a b -> (a -> List b)
326pt r a =
359pt r a =
327− let lb = [minBound .. maxBound]
360+ let (L lb) = maxBound
328− in [ b | b <- lb, toBool (pointAp a b r) ]
361+ in L [ b | b <- lb, toBool (pointAp a b r) ]
362+
363+-- | Belongs relation
364+belongs ::
365+ ( Bounded a,
366+ Enum a,
367+ Eq a,
368+ CountableDims (List a) a,
369+ FLN a (List a)
370+ )
371+ => Relation (List a) a
372+belongs = toRel elemR
373+ where
374+ elemR (L l) x = x `elem` l
329
375
330-- Zeros Matrix
376-- Zeros Matrix
331
377
332−-- | The zero matrix. A matrix wholly filled with zeros.
378+-- | The zero relation. A relation where no element of type @a@ relates
379+-- with elements of type @b@.
380+--
381+-- Also known as ⊥ (Bottom) Relation.
382+--
383+-- @
384+-- r `.` ⊥ == ⊥ `.` r == ⊥
385+-- ⊥ ``sse`` R && R ``sse`` T == True
386+-- @
333zeros ::
387zeros ::
334− (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>
388+ (FLN a b, CountableDimsN a b) =>
335 Relation a b
389 Relation a b
336−zeros = relationBuilder (const (nat 0))
390+zeros = relationBuilder' (const (nat 0))
337
391
338-- Ones Matrix
392-- Ones Matrix
339
393
340−-- | The ones matrix. A matrix wholly filled with ones.
394+-- | The ones relation. A relation where every element of type @a@ relates
395+-- with every element of type @b@.
341--
396--
342−-- Also known as T (Top) matrix.
397+-- Also known as T (Top) Relation or universal Relation.
398+--
399+-- @
400+-- ⊥ ``sse`` R && R ``sse`` T == True
401+-- @
343ones ::
402ones ::
344− (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>
403+ (FLN a b, CountableDimsN a b) =>
345 Relation a b
404 Relation a b
346−ones = relationBuilder (const (nat 1))
405+ones = relationBuilder' (const (nat 1))
347
406
348-- Bang Matrix
407-- Bang Matrix
349
408
350−-- | The T (Top) row vector matrix.
409+-- | The T (Top) row vector relation.
351bang ::
410bang ::
352− (I.FromLists Boolean (I.Normalize a) One, KnownNat (I.Count (I.Normalize a))) =>
411+ (FLN a One, CountableN a) =>
353 Relation a One
412 Relation a One
354bang = ones
413bang = ones
355
414
356-- | Point constant relation
415-- | Point constant relation
357−point ::
416+point ::
358 ( Bounded a,
417 ( Bounded a,
359 Enum a,
418 Enum a,
360 Eq a,
419 Eq a,
361− KnownNat (I.Count (I.Normalize a)),
420+ CountableN a,
362− I.FromLists Boolean (I.Normalize a) One
421+ FLN a One
363 ) => a -> Relation One a
422 ) => a -> Relation One a
364−point = fromF' . const
423+point = fromF . const
365
424
366−-- Identity Matrix
425+-- iden Matrix
367
426
368−-- | Identity matrix
427+-- | iden matrix
369−identity ::
428+--
370− (I.FromLists Boolean (I.Normalize a) (I.Normalize a), KnownNat (I.Count (I.Normalize a))) =>
429+-- @
430+-- 'iden' `.` r == r == r `.` 'iden'
431+-- @
432+iden ::
371− Relation a a
433+ (FLN a a, CountableN a) => Relation a a
372−identity = relationBuilder (bool (nat 0) (nat 1) . uncurry (==))
434+iden = relationBuilder' (bool (nat 0) (nat 1) . uncurry (==))
373
435
374-- | Relational composition
436-- | Relational composition
437+--
438+-- @
439+-- r `.` (s `.` p) = (r `.` s) `.` p
440+-- @
375comp :: Relation b c -> Relation a b -> Relation a c
441comp :: Relation b c -> Relation a b -> Relation a c
376comp (R a) (R b) = R (I.compRel a b)
442comp (R a) (R b) = R (I.compRel a b)
377
443
378-- | Relational right division
444-- | Relational right division
445+--
446+-- @'divR' x y@ is the largest relation @z@ which,
447+-- pre-composed with @y@, approximates @x@.
379divR :: Relation b c -> Relation b a -> Relation a c
448divR :: Relation b c -> Relation b a -> Relation a c
380divR (R x) (R y) = R (I.divR x y)
449divR (R x) (R y) = R (I.divR x y)
381
450
382-- | Relational left division
451-- | Relational left division
452+--
453+-- The dual division operator:
454+--
455+-- @
456+-- 'divL' y x == 'conv' ('divR' ('conv' x) ('conv' y)
457+-- @
383divL :: Relation c b -> Relation a b -> Relation a c
458divL :: Relation c b -> Relation a b -> Relation a c
384divL (R x) (R y) = R (I.divL x y)
459divL (R x) (R y) = R (I.divL x y)
385
460
386-- | Relational symmetric division
461-- | Relational symmetric division
462+--
463+-- @'pointAp' c b ('divS' s r)@ means that @b@ and @c@
464+-- are related to exactly the same outputs by @r@ and by @s@.
387divS :: Relation c a -> Relation b a -> Relation c b
465divS :: Relation c a -> Relation b a -> Relation c b
388divS (R x) (R y) = R (I.divS x y)
466divS (R x) (R y) = R (I.divS x y)
389
467
468+-- | Relational shrinking.
469+--
470+-- @r ``shrunkBy`` s@ is the largest part of @r@ such that,
471+-- if it yields an output for an input @x@, it must be a maximum,
472+-- with respect to @s@, among all possible outputs of @x@ by @r@.
473+shrunkBy :: Relation b a -> Relation a a -> Relation b a
474+shrunkBy r s = r `intersection` divR s (conv r)
475+
476+-- | Relational overriding.
477+--
478+-- @r ``overriddenBy`` s@ yields the relation which contains the
479+-- whole of @s@ and that part of @r@ where @s@ is undefined.
480+--
481+-- @
482+-- 'zeros' ``overriddenBy`` s == s
483+-- r ``overriddenBy`` 'zeros' == r
484+-- r ``overriddenBy`` r == r
485+-- @
486+overriddenBy ::
487+ ( FLN b b,
488+ CountableN b
489+ ) => Relation a b -> Relation a b -> Relation a b
490+overriddenBy r s = s `union` r `intersection` divR zeros (conv s)
491+
390−-- | Relational application
492+-- | Relational application.
493+--
494+-- If @a@ and @b@ are related by 'Relation' @r@
495+-- then @'pointAp' a b r == 'one' ('nat' 1)@
391pointAp ::
496pointAp ::
392− ( Bounded a,
497+ ( Liftable a b,
393− Bounded b,
394− Enum a,
395− Enum b,
396 Eq a,
498 Eq a,
397− Eq b,
499+ CountableDimsN a b,
398− KnownNat (I.Count (I.Normalize a)),
500+ FLN a One,
399− KnownNat (I.Count (I.Normalize b)),
501+ FLN b One
400− I.FromLists Boolean (I.Normalize a) One,
401− I.FromLists Boolean (I.Normalize b) One
402 ) => a -> b -> Relation a b -> Relation One One
502 ) => a -> b -> Relation a b -> Relation One One
403−pointAp a b r = conv (point b) `comp` r `comp` point a
503+pointAp a b r = conv (point b) . r . point a
504+
505+-- | Relational application
506+--
507+-- The same as 'pointAp' but converts 'Boolean' to 'Bool'
508+pointApBool ::
509+ ( Liftable a b,
510+ Eq a,
511+ CountableDimsN a b,
512+ FLN a One,
513+ FLN b One
514+ ) => a -> b -> Relation a b -> Bool
515+pointApBool a b r = toBool $ conv (point b) . r . point a
404
516
405-- | Relational converse
517-- | Relational converse
518+--
519+-- Given binary 'Relation' r, writing @'pointAp' a b r@
520+-- (read: “@b@ is related to @a@ by @r@”) means the same as
521+-- @'pointAp' b a ('conv' r)@, where @'conv' r@ is said to be
522+-- the converse of @r@.
523+-- In terms of grammar, @'conv' r@ corresponds to the passive voice
406conv :: Relation a b -> Relation b a
524conv :: Relation a b -> Relation b a
407conv (R a) = R (I.tr a)
525conv (R a) = R (I.tr a)
408
526
@@ -420,165 +538,187 @@ iff r s = r == s
420
538
421-- | Relational intersection
539-- | Relational intersection
422--
540--
423−-- Lifts pointwise conjunction.
541+-- Lifts pointwise conjointion.
542+--
543+-- @
544+-- (r ``intersection`` s) ``intersection`` t == r ``intersection`` (s ``intersection`` t)
545+-- x ``sse`` r ``intersection`` s == x ``intersection`` r && x ``intersection`` s
546+-- @
424intersection :: Relation a b -> Relation a b -> Relation a b
547intersection :: Relation a b -> Relation a b -> Relation a b
425intersection a b = a * b
548intersection a b = a * b
426
549
427-- | Relational union
550-- | Relational union
428--
551--
429−-- Lifts pointwise disjunction.
552+-- Lifts pointwise disjointion.
553+--
554+-- @
555+-- (r ``union`` s) ``union`` t == r `'union' (s ``union`` t)
556+-- r ``union`` s ``sse`` x == r ``sse`` x && s ``sse`` x
557+-- r `.` (s ``union`` t) == (r `.` s) ``union`` (r `.` t)
558+-- (s ``union`` t) `.` r == (s `.` r) ``union`` (t `.` r)
559+-- @
430union :: Relation a b -> Relation a b -> Relation a b
560union :: Relation a b -> Relation a b -> Relation a b
431union a b = a + b
561union a b = a + b
432
562
433-- | Relation Kernel
563-- | Relation Kernel
564+--
565+-- @
566+-- 'ker' r == 'conv' r `.` r
567+-- 'ker' r == 'img' ('conv' r)
568+-- @
434ker :: Relation a b -> Relation a a
569ker :: Relation a b -> Relation a a
435−ker r = conv r `comp` r
570+ker r = conv r . r
436
571
437-- | Relation Image
572-- | Relation Image
573+--
574+-- @
575+-- 'img' r == r `.` conv r
576+-- 'img' r == 'ker' ('conv' r)
577+-- @
438img :: Relation a b -> Relation b b
578img :: Relation a b -> Relation b b
439−img r = r `comp` conv r
579+img r = r . conv r
440
580
441−-- | Function division.
581+-- | Function division. Special case of 'divS'.
442--
582--
443-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and
583-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and
444-- 'entire'.
584-- 'entire'.
445--
585--
446−-- @division f g == conv g `'comp'` f@
586+-- @'divisionF' f g == 'conv' g `.` f@
447divisionF :: Relation a c -> Relation b c -> Relation a b
587divisionF :: Relation a c -> Relation b c -> Relation a b
448−divisionF f g = conv g `comp` f
588+divisionF f g = conv g . f
449
589
450-- Taxonomy of binary relations
590-- Taxonomy of binary relations
451
591
452−-- | A 'Relation' @r@ is simple iff @'coreflexive' ('img' r)@
592+-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@
453−simple :: (KnownNat (I.Count (I.Normalize b)), I.FromLists Boolean (I.Normalize b) (I.Normalize b)) => Relation a b -> Bool
593+simple :: (CountableN b, FLN b b) => Relation a b -> Bool
454simple = coreflexive . img
594simple = coreflexive . img
455
595
456−-- | A 'Relation' @r@ is injective iff @'coreflexive' ('ker' r)@
596+-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@
457−injective :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a b -> Bool
597+injective :: (CountableN a, FLN a a) => Relation a b -> Bool
458injective = coreflexive . ker
598injective = coreflexive . ker
459
599
460−-- | A 'Relation' @r@ is entire iff @'reflexive' ('ker' r)@
600+-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@
461−entire :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a b -> Bool
601+entire :: (CountableN a, FLN a a) => Relation a b -> Bool
462entire = reflexive . ker
602entire = reflexive . ker
463
603
464−-- | A 'Relation' @r@ is surjective iff @'reflexive' ('img' r)@
604+-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@
465−surjective :: (KnownNat (I.Count (I.Normalize b)), I.FromLists Boolean (I.Normalize b) (I.Normalize b)) => Relation a b -> Bool
605+surjective :: (CountableN b, FLN b b) => Relation a b -> Bool
466surjective = reflexive . img
606surjective = reflexive . img
467
607
468−-- | A 'Relation' @r@ is a function iff @'simple' r && 'entire' r@
608+-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@
469--
609--
470-- A 'function' @f@ enjoys the following properties, where @r@ and @s@ are binary
610-- A 'function' @f@ enjoys the following properties, where @r@ and @s@ are binary
471-- relations:
611-- relations:
472--
612--
473-- @
613-- @
474−-- f `'comp'` r `'sse'` s == r `'sse'` f `'comp'` s
614+-- f `.` r ``sse`` s == r ``sse`` f `.` s
475−-- r `'comp'` f `'sse'` s == r `'sse'` s `'comp'` f
615+-- r `.` f ``sse`` s == r ``sse`` s `.` f
476-- @
616-- @
477function ::
617function ::
478− ( KnownNat (I.Count (I.Normalize b)),
618+ ( CountableDimsN a b,
479− KnownNat (I.Count (I.Normalize a)),
619+ FLN a a,
480− I.FromLists Boolean (I.Normalize a) (I.Normalize a),
620+ FLN b b
481− I.FromLists Boolean (I.Normalize b) (I.Normalize b)
482 )
621 )
483 => Relation a b -> Bool
622 => Relation a b -> Bool
484function r = simple r && entire r
623function r = simple r && entire r
485
624
486−-- | A 'Relation' @r@ is a representation iff @'injective' r && 'entire' r@
625+-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@
487representation ::
626representation ::
488− ( KnownNat (I.Count (I.Normalize a)),
627+ ( CountableN a,
489− I.FromLists Boolean (I.Normalize a) (I.Normalize a)
628+ FLN a a
490 )
629 )
491 => Relation a b -> Bool
630 => Relation a b -> Bool
492representation r = injective r && entire r
631representation r = injective r && entire r
493
632
494−-- | A 'Relation' @r@ is an abstraction iff @'surjective' r && 'simple' r@
633+-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@
495abstraction ::
634abstraction ::
496− ( KnownNat (I.Count (I.Normalize b)),
635+ ( CountableN b,
497− I.FromLists Boolean (I.Normalize b) (I.Normalize b)
636+ FLN b b
498 )
637 )
499 => Relation a b -> Bool
638 => Relation a b -> Bool
500abstraction r = surjective r && simple r
639abstraction r = surjective r && simple r
501
640
502−-- | A 'Relation' @r@ is a surjection iff @'function' r && 'abstraction' r@
641+-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@
503surjection ::
642surjection ::
504− ( KnownNat (I.Count (I.Normalize b)),
643+ ( CountableDimsN a b,
505− KnownNat (I.Count (I.Normalize a)),
644+ FLN a a,
506− I.FromLists Boolean (I.Normalize a) (I.Normalize a),
645+ FLN b b
507− I.FromLists Boolean (I.Normalize b) (I.Normalize b)
508 )
646 )
509 => Relation a b -> Bool
647 => Relation a b -> Bool
510surjection r = function r && abstraction r
648surjection r = function r && abstraction r
511
649
512−-- | A 'Relation' @r@ is a injection iff @'function' r && 'representation' r@
650+-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@
513injection ::
651injection ::
514− ( KnownNat (I.Count (I.Normalize b)),
652+ ( CountableDimsN a b,
515− KnownNat (I.Count (I.Normalize a)),
653+ FLN a a,
516− I.FromLists Boolean (I.Normalize a) (I.Normalize a),
654+ FLN b b
517− I.FromLists Boolean (I.Normalize b) (I.Normalize b)
518 )
655 )
519 => Relation a b -> Bool
656 => Relation a b -> Bool
520injection r = function r && representation r
657injection r = function r && representation r
521
658
522−-- | A 'Relation' @r@ is an bijection iff @'injection' r && 'surjection' r@
659+-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@
523bijection ::
660bijection ::
524− ( KnownNat (I.Count (I.Normalize b)),
661+ ( CountableDimsN a b,
525− KnownNat (I.Count (I.Normalize a)),
662+ FLN b b,
526− I.FromLists Boolean (I.Normalize b) (I.Normalize b),
663+ FLN a a
527− I.FromLists Boolean (I.Normalize a) (I.Normalize a)
528 ) => Relation a b -> Bool
664 ) => Relation a b -> Bool
529bijection r = injection r && surjection r
665bijection r = injection r && surjection r
530
666
531−
532-- Properties of relations
667-- Properties of relations
533
668
534−-- | A 'Relation' @r@ is reflexive iff @'identity' `'sse'` r@
669+-- | A 'Relation' @r@ is 'reflexive' 'iff' @'id' ``sse`` r@
535−reflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
670+reflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
536−reflexive r = identity <= r
671+reflexive r = id <= r
537
672
538−-- | A 'Relation' @r@ is coreflexive iff @r `'sse'` 'identity'@
673+-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'id'@
539−coreflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
674+coreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
540−coreflexive r = r <= identity
675+coreflexive r = r <= id
541
676
542−-- | A 'Relation' @r@ is transitive iff @(r `'comp'` r) `'sse'` r@
677+-- | A 'Relation' @r@ is 'transitive' 'iff' @(r `.` r) ``sse`` r@
543transitive :: Relation a a -> Bool
678transitive :: Relation a a -> Bool
544−transitive r = (r `comp` r) `sse` r
679+transitive r = (r . r) `sse` r
545
680
546−-- | A 'Relation' @r@ is symmetric iff @r == conv r@
681+-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@
547symmetric :: Relation a a -> Bool
682symmetric :: Relation a a -> Bool
548symmetric r = r == conv r
683symmetric r = r == conv r
549
684
550−-- | A 'Relation' @r@ is anti-symmetric iff @(r `'intersection'` 'conv' r) `'sse'` 'identity'@
685+-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'id'@
551−antiSymmetric :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
686+antiSymmetric :: (CountableN a, FLN a a) => Relation a a -> Bool
552−antiSymmetric r = (r `intersection` conv r) `sse` identity
687+antiSymmetric r = (r `intersection` conv r) `sse` id
553
688
554−-- | A 'Relation' @r@ is irreflexive iff @(r `'intersection'` identity) == 'zeros'@
689+-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'id') == 'zeros'@
555−irreflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
690+irreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
556−irreflexive r = (r `intersection` identity) == zeros
691+irreflexive r = (r `intersection` id) == zeros
557
692
558−-- | A 'Relation' @r@ is connected iff @(r `'union'` 'conv' r) == 'ones'@
693+-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@
559−connected :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
694+connected :: (CountableN a, FLN a a) => Relation a a -> Bool
560connected r = (r `union` conv r) == ones
695connected r = (r `union` conv r) == ones
561
696
562−-- | A 'Relation @r@ is a preorder iff @'reflexive' r && 'transitive' r@
697+-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@
563−preorder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
698+preorder :: (CountableN a, FLN a a) => Relation a a -> Bool
564preorder r = reflexive r && transitive r
699preorder r = reflexive r && transitive r
565
700
566−-- | A 'Relation @r@ is a partial order iff @'antiSymmetric' r && 'preorder' r@
701+-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@
567−partialOrder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
702+partialOrder :: (CountableN a, FLN a a) => Relation a a -> Bool
568partialOrder r = antiSymmetric r && preorder r
703partialOrder r = antiSymmetric r && preorder r
569
704
570−-- | A 'Relation @r@ is a linear order iff @'connected' r && 'partialOrder' r@
705+-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@
571−linearOrder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
706+linearOrder :: (CountableN a, FLN a a) => Relation a a -> Bool
572linearOrder r = connected r && partialOrder r
707linearOrder r = connected r && partialOrder r
573
708
574−-- | A 'Relation @r@ is an equivalence iff @'symmetric' r && 'preorder' r@
709+-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@
575−equivalence :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
710+equivalence :: (CountableN a, FLN a a) => Relation a a -> Bool
576equivalence r = symmetric r && preorder r
711equivalence r = symmetric r && preorder r
577
712
578−-- | A 'Relation @r@ is a partial equivalence iff @'partialOrder' r && 'equivalence' r@
713+-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@
579−partialEquivalence :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
714+partialEquivalence :: (CountableN a, FLN a a) => Relation a a -> Bool
580partialEquivalence r = partialOrder r && equivalence r
715partialEquivalence r = partialOrder r && equivalence r
581
716
717+-- | A 'Relation' @r@ is 'difunctional' or regular wherever
718+-- @r `.` 'conv' r `.` r == r@
719+difunctional :: Relation a b -> Bool
720+difunctional r = r . conv r . r == r
721+
582-- Relational pairing
722-- Relational pairing
583
723
584-- | Relational pairing.
724-- | Relational pairing.
@@ -586,64 +726,81 @@ partialEquivalence r = partialOrder r && equivalence r
586-- NOTE: That this is not a true categorical product, see for instance:
726-- NOTE: That this is not a true categorical product, see for instance:
587--
727--
588-- @
728-- @
589−-- | 'p1' `'comp'` 'splitR' a b `'sse'` a
729+-- | 'fstR' `.` 'splitR' a b ``sse`` a
590−-- 'splitR' a b <=> |
730+-- 'splitR' a b <=> |
591−-- | 'p2' `'comp'` 'splitR' a b `'sse'` b
731+-- | 'sndR' `.` 'splitR' a b ``sse`` b
592-- @
732-- @
593--
733--
594-- __Emphasis__ on the 'sse'.
734-- __Emphasis__ on the 'sse'.
735+--
736+-- @
737+-- 'splitR' r s `.` f == 'splitR' (r `.` f) (s `.` f)
738+-- (R '><' S) `.` 'splitR' p q == 'splitR' (r `.` p) (s `.` q)
739+-- 'conv' ('splitR' r s) `.` 'splitR' x y == ('conv' r `.` x) ``intersection`` ('conv' s `.` y)
740+-- @
741+--
742+-- @
743+-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)
744+-- @
595splitR ::
745splitR ::
596− ( KnownNat (I.Count (I.Normalize a)),
746+ ( CountableDimsN a b,
597− KnownNat (I.Count (I.Normalize b)),
747+ CountableN (a, b),
598− KnownNat (I.Count (I.Normalize (a, b))),
748+ FLN (a, b) a,
599− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize a),
749+ FLN (a, b) b,
600− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize b),
750+ TrivialP a b
601− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
602 )
751 )
603 => Relation c a -> Relation c b -> Relation c (a, b)
752 => Relation c a -> Relation c b -> Relation c (a, b)
604−splitR (R f) (R g) = R (I.khatri f g)
753+splitR (R f) (R g) = R (I.kr f g)
605
754
606-- | Relational pairing first component projection
755-- | Relational pairing first component projection
607−p1 ::
756+--
757+-- @
758+-- 'fstR' `.` 'splitR' r s ``sse`` r
759+-- @
760+fstR ::
608 forall a b .
761 forall a b .
609− ( KnownNat (I.Count (I.Normalize a)),
762+ ( CountableDimsN a b,
610− KnownNat (I.Count (I.Normalize b)),
763+ CountableN (a, b),
611− KnownNat (I.Count (I.Normalize (a, b))),
764+ FLN (a, b) a,
612− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize a),
765+ TrivialP a b
613− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
614 )
766 )
615 => Relation (a, b) a
767 => Relation (a, b) a
616−p1 = R (I.kp1 @Boolean @(I.Normalize a) @(I.Normalize b))
768+fstR = R (I.fstM @Boolean @(I.Normalize a) @(I.Normalize b))
617
769
618-- | Relational pairing second component projection
770-- | Relational pairing second component projection
619−p2 ::
771+--
772+-- @
773+-- 'sndR' `.` 'splitR' r s ``sse`` s
774+-- @
775+sndR ::
620 forall a b .
776 forall a b .
621− ( KnownNat (I.Count (I.Normalize b)),
777+ ( CountableDimsN a b,
622− KnownNat (I.Count (I.Normalize a)),
778+ CountableN (a, b),
623− KnownNat (I.Count (I.Normalize (a, b))),
779+ FLN (a, b) b,
624− I.FromLists Boolean (I.Normalize (a, b)) (I.Normalize b),
780+ TrivialP a b
625− I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
626 )
781 )
627 => Relation (a, b) b
782 => Relation (a, b) b
628−p2 = R (I.kp2 @Boolean @(I.Normalize a) @(I.Normalize b))
783+sndR = R (I.sndM @Boolean @(I.Normalize a) @(I.Normalize b))
629
784
630-- Relational pairing functor
785-- Relational pairing functor
631
786
632infixl 4 ><
787infixl 4 ><
633-- | Relational pairing functor
788-- | Relational pairing functor
789+--
790+-- @
791+-- r '><' s == 'splitR' (r `.` fstR) (s `.` sndR)
792+-- (r '><' s) `.` (p '><' q) == (r `.` p) '><' (s `.` q)
793+-- @
634(><) ::
794(><) ::
635− ( KnownNat (I.Count (I.Normalize a)),
795+ ( CountableDimsN a b,
636− KnownNat (I.Count (I.Normalize b)),
796+ CountableDimsN c d,
637− KnownNat (I.Count (I.Normalize c)),
797+ CountableDimsN (a, c) (b, d),
638− KnownNat (I.Count (I.Normalize d)),
798+ FLN (a, c) a,
639− KnownNat (I.Count (I.Normalize (a, c))),
799+ FLN (a, c) c,
640− KnownNat (I.Count (I.Normalize (b, d))),
800+ FLN (b, d) b,
641− I.FromLists Boolean (I.Normalize (a, c)) (I.Normalize a),
801+ FLN (b, d) d,
642− I.FromLists Boolean (I.Normalize (a, c)) (I.Normalize c),
802+ TrivialP a c,
643− I.FromLists Boolean (I.Normalize (b, d)) (I.Normalize b),
803+ TrivialP b d
644− I.FromLists Boolean (I.Normalize (b, d)) (I.Normalize d),
645− I.Normalize (a, c) ~ I.Normalize (I.Normalize a, I.Normalize c),
646− I.Normalize (b, d) ~ I.Normalize (I.Normalize b, I.Normalize d)
647 )
804 )
648 => Relation a b -> Relation c d -> Relation (a, c) (b, d)
805 => Relation a b -> Relation c d -> Relation (a, c) (b, d)
649(><) (R a) (R b) = R ((I.><) a b)
806(><) (R a) (R b) = R ((I.><) a b)
@@ -651,28 +808,47 @@ infixl 4 ><
651-- Relational co-products
808-- Relational co-products
652
809
653-- | Relational coproduct.
810-- | Relational coproduct.
654−eitherR :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))
811+--
812+-- @
813+-- | 'eitherR' a b `.` 'i1' == a
814+-- 'eitherR' a b <=> |
815+-- | 'eitherR' a b `.` 'i2' == b
816+-- @
817+--
818+-- @
819+-- 'eitherR' r s `.` 'conv' ('eitherR' t u) == (r `.` 'conv' t) ``union`` (s `.` 'conv' u)
820+-- @
821+--
822+-- @
823+-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)
824+-- @
655− => Relation a c -> Relation b c -> Relation (Either a b) c
825+eitherR :: Relation a c -> Relation b c -> Relation (Either a b) c
656−eitherR = junc
826+eitherR = join
657
827
658-- | Relational coproduct first component injection
828-- | Relational coproduct first component injection
829+--
830+-- @
831+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'
832+-- 'i1' `.` 'i2' = 'zeros'
833+-- @
659i1 ::
834i1 ::
660− ( KnownNat (I.Count (I.Normalize b)),
835+ ( CountableDimsN a b,
661− KnownNat (I.Count (I.Normalize a)),
836+ FLN b a,
662− I.FromLists Boolean (I.Normalize b) (I.Normalize a),
837+ FLN a a
663− I.FromLists Boolean (I.Normalize a) (I.Normalize a),
664− I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)
665 )
838 )
666 => Relation a (Either a b)
839 => Relation a (Either a b)
667i1 = R I.i1
840i1 = R I.i1
668
841
669-- | Relational coproduct second component injection
842-- | Relational coproduct second component injection
843+--
844+-- @
845+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'
846+-- 'i1' `.` 'i2' = 'zeros'
847+-- @
670i2 ::
848i2 ::
671− ( KnownNat (I.Count (I.Normalize b)),
849+ ( CountableDimsN a b,
672− KnownNat (I.Count (I.Normalize a)),
850+ FLN a b,
673− I.FromLists Boolean (I.Normalize a) (I.Normalize b),
851+ FLN b b
674− I.FromLists Boolean (I.Normalize b) (I.Normalize b),
675− I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)
676 )
852 )
677 => Relation b (Either a b)
853 => Relation b (Either a b)
678i2 = R I.i2
854i2 = R I.i2
@@ -680,15 +856,16 @@ i2 = R I.i2
680infixl 5 -|-
856infixl 5 -|-
681
857
682-- | Relational coproduct functor.
858-- | Relational coproduct functor.
859+--
860+-- @
861+-- r '-|-' s == 'eitherR' ('i1' `.` r) ('i2' `.` s)
862+-- @
683(-|-) ::
863(-|-) ::
684− ( KnownNat (I.Count (I.Normalize d)),
864+ ( CountableDimsN b d,
685− KnownNat (I.Count (I.Normalize b)),
865+ FLN b b,
686− I.FromLists Boolean (I.Normalize b) (I.Normalize b),
866+ FLN d b,
687− I.FromLists Boolean (I.Normalize d) (I.Normalize b),
867+ FLN b d,
688− I.FromLists Boolean (I.Normalize b) (I.Normalize d),
868+ FLN d d
689− I.FromLists Boolean (I.Normalize d) (I.Normalize d),
690− I.Normalize (Either a c) ~ Either (I.Normalize a) (I.Normalize c),
691− I.Normalize (Either b d) ~ Either (I.Normalize b) (I.Normalize d)
692 )
869 )
693 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)
870 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)
694(-|-) (R a) (R b) = R ((I.-|-) a b)
871(-|-) (R a) (R b) = R ((I.-|-) a b)
@@ -696,49 +873,136 @@ infixl 5 -|-
696-- Relational "Currying"
873-- Relational "Currying"
697
874
698-- | Relational 'trans'
875-- | Relational 'trans'
876+--
877+-- Every n-ary relation can be expressed as a binary relation through
878+-- 'trans'/'untrans';
879+-- more-over, where each particular attribute is placed (input/output) is irrelevant.
699trans ::
880trans ::
700− ( KnownNat (I.Count (I.Normalize (a, b))),
881+ ( CountableDimsN a b,
701− KnownNat (I.Count (I.Normalize a)),
882+ CountableN c,
702− KnownNat (I.Count (I.Normalize (c, b))),
883+ CountableDimsN (a, b) (c, b),
703− KnownNat (I.Count (I.Normalize b)),
884+ FLN (c, b) c,
704− KnownNat (I.Count (I.Normalize c)),
885+ FLN (c, b) b,
705− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count c)),
886+ FLN (a, b) a,
706− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count b)),
887+ FLN (a, b) b,
707− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count b)),
888+ Trivial (a, b),
708− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count a)),
889+ Trivial (c, b),
709− I.Normalize (a, b) ~ I.Normalize (I.Normalize (a, b)),
890+ TrivialP a b,
710− I.Normalize (c, b) ~ I.Normalize (I.Normalize (c, b)),
891+ TrivialP c b
711− I.Normalize (I.Normalize a, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (a, b))),
712− I.Normalize (I.Normalize c, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (c, b)))
713 )
892 )
714 => Relation (a, b) c -> Relation a (c, b)
893 => Relation (a, b) c -> Relation a (c, b)
715−trans r = splitR r p2 `comp` conv p1
894+trans r = splitR r sndR . conv fstR
716
895
717-- | Relational 'untrans'
896-- | Relational 'untrans'
897+--
898+-- Every n-ary relation can be expressed as a binary relation through
899+-- 'trans'/'untrans';
900+-- more-over, where each particular attribute is placed (input/output) is irrelevant.
718untrans ::
901untrans ::
719− ( KnownNat (I.Count (I.Normalize (a, b))),
902+ ( CountableDimsN a b,
720− KnownNat (I.Count (I.Normalize a)),
903+ CountableN c,
721− KnownNat (I.Count (I.Normalize (c, b))),
904+ CountableDimsN (a, b) (c, b),
722− KnownNat (I.Count (I.Normalize b)),
905+ FLN (c, b) c,
723− KnownNat (I.Count (I.Normalize c)),
906+ FLN (c, b) b,
724− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count c)),
907+ FLN (a, b) b,
725− I.FromLists Boolean (I.Normalize (c, b)) (I.FromNat (I.Count b)),
908+ FLN (a, b) a,
726− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count b)),
909+ Trivial (a, b),
727− I.FromLists Boolean (I.Normalize (a, b)) (I.FromNat (I.Count a)),
910+ Trivial (c, b),
728− I.Normalize (a, b) ~ I.Normalize (I.Normalize (a, b)),
911+ TrivialP a b,
729− I.Normalize (c, b) ~ I.Normalize (I.Normalize (c, b)),
912+ TrivialP c b
730− I.Normalize (I.Normalize a, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (a, b))),
731− I.Normalize (I.Normalize c, I.Normalize b) ~ I.FromNat (I.Count (I.Normalize (c, b)))
732 )
913 )
733 => Relation a (c, b) -> Relation (a, b) c
914 => Relation a (c, b) -> Relation (a, b) c
734−untrans s = p1 `comp` conv (splitR (conv s) p2)
915+untrans s = fstR . conv (splitR (conv s) sndR)
916+
917+-- | Transforms predicate @p@ into a correflexive relation.
918+--
919+-- @
920+-- 'predR' ('const' True) == 'id'
921+-- 'predR' ('const' False) == 'zeros'
922+-- @
923+--
924+-- @
925+-- 'predR' q `.` 'predR' p == 'predR' q ``union`` 'predR' p
926+-- @
927+predR ::
928+ ( Bounded a,
929+ Enum a,
930+ Countable a,
931+ CountableN a,
932+ FLN a a,
933+ FLN Bool a
934+ )
935+ => Relation a Bool -> Relation a a
936+predR p = id `intersection` divisionF (fromF (const True)) p
937+
938+-- | Equalizes functions @f@ and @g@.
939+-- That is, @'equalizer' f g@ is the largest coreflexive
940+-- that restricts @g@ so that @f@ and @g@ yield the same outputs.
941+--
942+-- @
943+-- 'equalizer' r r == 'id'
944+-- 'equalizer' ('point' True) ('point' False) == 'zeros'
945+-- @
946+equalizer ::
947+ ( CountableN a,
948+ FLN a a
949+ )
950+ => Relation a b -> Relation a b -> Relation a a
951+equalizer f g = id `intersection` divisionF f g
952+
953+-- | Relational conditional guard.
954+--
955+-- @
956+-- 'guard' p = 'i2' ``overriddenBy`` 'i1' `.` 'predR' p
957+-- @
958+guard ::
959+ ( Bounded b,
960+ Enum b,
961+ CountableN b,
962+ Countable b,
963+ FLN b b,
964+ FLN Bool b
965+ ) => Relation b Bool -> Relation b (Either b b)
966+guard p = conv (eitherR (predR p) (predR (negate p)))
967+
968+-- | Relational McCarthy's conditional.
969+cond ::
970+ ( Bounded b,
971+ Enum b,
972+ Countable b,
973+ CountableN b,
974+ FLN b b,
975+ FLN Bool b
976+ )
977+ => Relation b Bool -> Relation b c -> Relation b c -> Relation b c
978+cond p r s = eitherR r s . guard p
979+
980+-- | Relational domain.
981+--
982+-- For injective relations, 'domain' and 'ker'nel coincide,
983+-- since @'ker' r ``sse`` 'id'@ in such situations.
984+domain ::
985+ ( CountableN a,
986+ FLN a a
987+ ) => Relation a b -> Relation a a
988+domain r = ker r `intersection` id
989+
990+-- | Relational range.
991+--
992+-- For functions, 'range' and 'img' (image) coincide,
993+-- since @'img' f ``sse`` id@ for any @f@.
994+range ::
995+ ( CountableN b,
996+ FLN b b
997+ ) => Relation a b -> Relation b b
998+range r = img r `intersection` id
735
999
736-- Relation pretty print
1000-- Relation pretty print
737
1001
738-- | Relation pretty printing
1002-- | Relation pretty printing
739−pretty :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> String
1003+pretty :: (CountableDimsN a b) => Relation a b -> String
740pretty (R a) = I.pretty a
1004pretty (R a) = I.pretty a
741
1005
742-- | Relation pretty printing
1006-- | Relation pretty printing
743−prettyPrint :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> IO ()
1007+prettyPrint :: (CountableDimsN a b) => Relation a b -> IO ()
744prettyPrint (R a) = I.prettyPrint a
1008prettyPrint (R a) = I.prettyPrint a
src/LAoP/Utils.hs+6
@@ -94,6 +94,12 @@ module LAoP.Utils
94 coerceNat,
94 coerceNat,
95 coerceNat2,
95 coerceNat2,
96 coerceNat3,
96 coerceNat3,
97+
98+ -- * List data type
99+ List (..),
100+
101+ -- * Category type-class
102+ Category(..)
97 )
103 )
98where
104where
99
105
src/LAoP/Utils/Internal.hs+70−73
@@ -1,82 +1,13 @@
1+{-# LANGUAGE TypeFamilies #-}
2+{-# LANGUAGE DeriveGeneric #-}
1{-# LANGUAGE TypeApplications #-}
3{-# LANGUAGE TypeApplications #-}
2{-# LANGUAGE DataKinds #-}
4{-# LANGUAGE DataKinds #-}
3{-# LANGUAGE FlexibleInstances #-}
5{-# LANGUAGE FlexibleInstances #-}
4{-# LANGUAGE GeneralizedNewtypeDeriving #-}
6{-# LANGUAGE GeneralizedNewtypeDeriving #-}
5−{-# LANGUAGE KindSignatures #-}
6{-# LANGUAGE ScopedTypeVariables #-}
7{-# LANGUAGE ScopedTypeVariables #-}
7
8
8module LAoP.Utils.Internal
9module LAoP.Utils.Internal
9− ( -- | This is an internal module and it is not meant to be imported.
10+ (
10− --
11− -- Utility module that provides the 'Natural' data type.
12− -- The semantic associated with this data type is that
13− -- it's meant to be a restricted 'Int' value. For example
14− -- the type @Natural 6@ can only be instanciated with @Nat n@
15− -- where @0 <= n <= 6@. Why, You might ask, because with normal
16− -- 'Int's it is not possible to have a decent @Enum (Int, Int)@
17− -- instance. See the following probabilistic programming model as and
18− -- example:
19− --
20− -- We want to calculate the probability of the sum of two dice throws.
21− -- To do this we start by defining the sample space:
22− --
23− -- @
24− -- type SampleSpace = Int -- We think 'Int' are enough
25− --
26− -- die :: Dist Int 6
27− -- die = unifrom [1..6]
28− --
29− -- -- Promote 'Int' addition to a matrix
30− -- addM = fromF (uncurry (+)) -- Impossible
31− -- @
32− --
33− -- The last line is impossible because @(Int, Int)@ does not have
34− -- a good 'Enum' instance: @[(0, 1), (0, 2), .. (0, maxBound), (1, 0),
35− -- ..]@. And we'd like the addition matrix to be of 36 columns by 12
36− -- rows but limited to integers up to @6@!
37− --
38− -- One way to solve this issue is by defining and auxilary data type to
39− -- represent the sample space:
40− --
41− -- @
42− -- data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6
43− -- deriving (Show, Eq, Enum, Bounded) -- Enum and Bounded are
44− -- important
45− -- @
46− --
47− -- And write the sample space addition function:
48− --
49− -- @
50− -- ssAdd :: SampleSpace -> SampleSpace -> Int
51− -- ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)
52− -- @
53− --
54− -- And then promote that function to matrix and everything is alright:
55− --
56− -- @
57− -- ssAddM = fromF' (uncurry ssAdd)
58− --
59− -- dieSumProb = ssAddM `comp` (khatri die die)
60− -- @
61− --
62− -- This is a nice solution for small sample spaces. But for larger ones
63− -- it is not feasible to write a data type with hundreds of constructors
64− -- and then write manipulation functions that need to deal with them.
65− -- To mitigate this limitation the 'Natural' type comes a long way and
66− -- allows one to model the sample in an easier way. See for instance:
67− --
68− -- @
69− -- ssAdd :: Natural 6 -> Natural 6 -> Natural 12
70− -- ssAdd = coerceNat (+)
71− --
72− -- ssAddM = fromF' (uncurry sumSS)
73− --
74− -- die :: Dist (Natural 6) 6
75− -- die = uniform [nat @6 1 .. nat 6]
76− --
77− -- dieSumProb = ssAddM `comp` (khatri die die)
78− -- @
79− --
80 -- * 'Natural' data type
11 -- * 'Natural' data type
81 Natural(..),
12 Natural(..),
82 nat,
13 nat,
@@ -86,19 +17,31 @@ module LAoP.Utils.Internal
86 coerceNat,
17 coerceNat,
87 coerceNat2,
18 coerceNat2,
88 coerceNat3,
19 coerceNat3,
20+
21+ -- * 'List' data type
22+ List (..),
23+
24+ -- * Category type class
25+ Category(..)
89 )
26 )
90where
27where
91
28
92import Data.Coerce
29import Data.Coerce
93import Data.Proxy
30import Data.Proxy
31+import Data.List
32+import Data.Maybe
94import GHC.TypeLits
33import GHC.TypeLits
95import Control.DeepSeq
34import Control.DeepSeq
35+import GHC.Generics
36+import Data.Kind
37+import Prelude hiding (id, (.))
38+import qualified Prelude
96
39
97-- | Wrapper around 'Int's that have a restrictive semantic associated.
40-- | Wrapper around 'Int's that have a restrictive semantic associated.
98-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'
41-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'
99-- @i@ that's @n <= i <= m@.
42-- @i@ that's @n <= i <= m@.
100newtype Natural (start :: Nat) (end :: Nat) = Nat Int
43newtype Natural (start :: Nat) (end :: Nat) = Nat Int
101− deriving (Show, Read, Eq, Ord, NFData)
44+ deriving (Show, Read, Eq, Ord, NFData, Generic)
102
45
103-- | Throws a runtime error if any of the operations overflows or
46-- | Throws a runtime error if any of the operations overflows or
104-- underflows.
47-- underflows.
@@ -197,3 +140,57 @@ instance
197
140
198 fromEnum (Left a) = fromEnum a
141 fromEnum (Left a) = fromEnum a
199 fromEnum (Right b) = fromEnum (maxBound :: a) + fromEnum b + 1
142 fromEnum (Right b) = fromEnum (maxBound :: a) + fromEnum b + 1
143+
144+-- | Powerset data type.
145+--
146+-- This data type is a newtype wrapper around '[]'. This exists in order to
147+-- implement an 'Enum' and 'Bounded' instance that cannot be harmful for the outside.
148+newtype List a = L [a]
149+ deriving (Eq, Show, Read)
150+
151+powerset :: [a] -> [[a]]
152+powerset [] = [[]]
153+powerset (x:xs) = powerset xs ++ [x:ps | ps <- powerset xs]
154+
155+instance
156+ ( Enum a,
157+ Bounded a
158+ ) => Bounded (List a)
159+ where
160+ minBound = L []
161+ maxBound = L [minBound .. maxBound]
162+
163+instance
164+ ( Bounded a,
165+ Enum a,
166+ Eq a
167+ ) => Enum (List a)
168+ where
169+ toEnum i =
170+ let as = [minBound .. maxBound]
171+ in L (powerset as !! i)
172+
173+ fromEnum (L []) = 0
174+ fromEnum (L x) =
175+ let as = [minBound .. maxBound]
176+ in fromMaybe (error "Does not exist") $ elemIndex x (powerset as)
177+
178+-- | Constrained category instance
179+class Category k where
180+ type Object k o :: Constraint
181+ type Object k o = ()
182+ id :: Object k a => k a a
183+ (.) :: k b c -> k a b -> k a c
184+
185+{-# RULES
186+"identity/left" forall p .
187+ id . p = p
188+"identity/right" forall p .
189+ p . id = p
190+"association" forall p q r .
191+ (p . q) . r = p . (q . r)
192+ #-}
193+
194+instance Category (->) where
195+ id = Prelude.id
196+ (.) = (Prelude..)
stack-8.8.3.yamlnew file+1
@@ -0,0 +1 @@
1+resolver: lts-15.5
stack.yaml+1−66
@@ -1,66 +1 @@
1−# This file was automatically generated by 'stack init'
2−#
3−# Some commonly used options have been documented as comments in this file.
4−# For advanced use and comprehensive documentation of the format, please see:
5−# https://docs.haskellstack.org/en/stable/yaml_configuration/
6−
7−# Resolver to choose a 'specific' stackage snapshot or a compiler version.
8−# A snapshot resolver dictates the compiler version and the set of packages
9−# to be used for project dependencies. For example:
10−#
11−# resolver: lts-3.5
12−# resolver: nightly-2015-09-21
13−# resolver: ghc-7.10.2
14−#
15−# The location of a snapshot can be provided as a file or url. Stack assumes
16−# a snapshot provided as a file might change, whereas a url resource does not.
17−#
18−# resolver: ./custom-snapshot.yaml
19−# resolver: https://example.com/snapshots/2018-01-01.yaml
20−resolver: lts-14.18
1+resolver: lts-14.3
21−
22−# User packages to be built.
23−# Various formats can be used as shown in the example below.
24−#
25−# packages:
26−# - some-directory
27−# - https://example.com/foo/bar/baz-0.0.2.tar.gz
28−# subdirs:
29−# - auto-update
30−# - wai
31−packages:
32−- .
33−# Dependency packages to be pulled from upstream that are not in the resolver.
34−# These entries can reference officially published versions as well as
35−# forks / in-progress versions pinned to a git hash. For example:
36−#
37−# extra-deps:
38−# - acme-missiles-0.3
39−# - git: https://github.com/commercialhaskell/stack.git
40−# commit: e7b331f14bcffb8367cd58fbfc8b40ec7642100a
41−#
42−# extra-deps: []
43−
44−# Override default flag values for local packages and extra-deps
45−# flags: {}
46−
47−# Extra package databases containing global packages
48−# extra-package-dbs: []
49−
50−# Control whether we use the GHC we find on the path
51−# system-ghc: true
52−#
53−# Require a specific version of stack, using version ranges
54−# require-stack-version: -any # Default
55−# require-stack-version: ">=2.1"
56−#
57−# Override the architecture used by stack, especially useful on Windows
58−# arch: i386
59−# arch: x86_64
60−#
61−# Extra directories used by stack for building
62−# extra-include-dirs: [/path/to/dir]
63−# extra-lib-dirs: [/path/to/dir]
64−#
65−# Allow a newer minor version of GHC than the snapshot specifies
66−# compiler-check: newer-minor
stack.yaml.lock+4−4
@@ -6,7 +6,7 @@
6packages: []
6packages: []
7snapshots:
7snapshots:
8- completed:
8- completed:
9− size: 524789
9+ size: 523878
10− url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/18.yaml
10+ url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/3.yaml
11− sha256: 646be71223e08234131c6989912e6011e01b9767bc447b6d466a35e14360bdf2
11+ sha256: 470c46c27746a48c7c50f829efc0cf00112787a7804ee4ac7a27754658f6d92c
12− original: lts-14.18
12+ original: lts-14.3
test/Examples/Quantum.hsnew file+31
@@ -0,0 +1,31 @@
1+{-# LANGUAGE StandaloneDeriving #-}
2+module Examples.Quantum where
3+
4+import LAoP.Utils
5+import LAoP.Matrix.Type
6+import Data.Complex
7+import Prelude hiding (id, (.))
8+
9+deriving instance Ord a => Ord (Complex a)
10+
11+xor :: (Bool, Bool) -> Bool
12+xor (False, b) = b
13+xor (True, b) = not b
14+
15+ker :: Num e => Matrix e a b -> Matrix e a a
16+ker m = tr m . m
17+
18+cnot :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)
19+cnot = kr fstM (fromF xor)
20+
21+ccnot :: (Num e, Ord e) => Matrix e ((Bool, Bool), Bool) ((Bool, Bool), Bool)
22+ccnot = kr fstM (fromF f)
23+ where
24+ f = xor . tp (uncurry (&&)) id
25+ tp f g (a,b) = (f a, g b)
26+
27+had :: Matrix (Complex Double) Bool Bool
28+had = (1/sqrt 2) .| fromLists [[1, 1], [1, -1]]
29+
30+bell :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)
31+bell = cnot . (had >< iden)
app/Main.hs → test/Examples/Readme.hsrenamed+61−33
@@ -1,10 +1,13 @@
1+{-# LANGUAGE GADTs #-}
1{-# LANGUAGE DeriveGeneric #-}
2{-# LANGUAGE DeriveGeneric #-}
2{-# LANGUAGE FlexibleContexts #-}
3{-# LANGUAGE FlexibleContexts #-}
3−{-# LANGUAGE TypeOperators #-}
4{-# LANGUAGE TypeApplications #-}
4{-# LANGUAGE TypeApplications #-}
5{-# LANGUAGE DataKinds #-}
5{-# LANGUAGE DataKinds #-}
6
6
7+module Examples.Readme
8+ ( exec
9+ )
7−module Main where
10+ where
8
11
9import LAoP.Matrix.Type
12import LAoP.Matrix.Type
10import qualified LAoP.Relation as R
13import qualified LAoP.Relation as R
@@ -12,13 +15,12 @@ import LAoP.Utils
12import LAoP.Dist
15import LAoP.Dist
13import GHC.TypeLits
16import GHC.TypeLits
14import Data.Coerce
17import Data.Coerce
15−import GHC.Generics
18+import qualified GHC.Generics as G
16−import Control.Category hiding (id)
19+import Prelude hiding (id, (.))
17−import Prelude hiding ((.))
18
20
19-- Monty Hall Problem
21-- Monty Hall Problem
20data Outcome = Win | Lose
22data Outcome = Win | Lose
21− deriving (Bounded, Enum, Eq, Show, Generic)
23+ deriving (Bounded, Enum, Eq, Show, G.Generic)
22
24
23switch :: Outcome -> Outcome
25switch :: Outcome -> Outcome
24switch Win = Lose
26switch Win = Lose
@@ -28,7 +30,7 @@ firstChoice :: Matrix Double () Outcome
28firstChoice = col [1/3, 2/3]
30firstChoice = col [1/3, 2/3]
29
31
30secondChoice :: Matrix Double Outcome Outcome
32secondChoice :: Matrix Double Outcome Outcome
31−secondChoice = fromF' switch
33+secondChoice = fromF switch
32
34
33-- Dice sum
35-- Dice sum
34
36
@@ -37,7 +39,7 @@ type SS = Natural 1 6 -- Sample Space
37sumSS :: SS -> SS -> Natural 2 12
39sumSS :: SS -> SS -> Natural 2 12
38sumSS = coerceNat (+)
40sumSS = coerceNat (+)
39
41
40−sumSSM = fromF' (uncurry sumSS)
42+sumSSM = fromF (uncurry sumSS)
41
43
42condition :: (Int, Int) -> Int -> Int
44condition :: (Int, Int) -> Int -> Int
43condition (fst, snd) thrd = if fst == snd
45condition (fst, snd) thrd = if fst == snd
@@ -47,38 +49,64 @@ condition (fst, snd) thrd = if fst == snd
47conditionSS :: (SS, SS) -> SS -> Natural 3 18
49conditionSS :: (SS, SS) -> SS -> Natural 3 18
48conditionSS = coerceNat2 condition
50conditionSS = coerceNat2 condition
49
51
50−conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die
52+conditionalThrows = fromF (uncurry conditionSS) . kr (kr die die) die
51
53
52die :: Matrix Double () SS
54die :: Matrix Double () SS
53die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]
55die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]
54
56
55-- Sprinkler
57-- Sprinkler
58+data G = Dry | Wet
59+ deriving (Bounded, Enum, Eq, Show, G.Generic)
60+
61+data S = Off | On
62+ deriving (Bounded, Enum, Eq, Show, G.Generic)
56
63
64+data R = No | Yes
65+ deriving (Bounded, Enum, Eq, Show, G.Generic)
66+
57−rain :: Matrix Double () Bool
67+rain :: Matrix Double () R
58−rain = col [0.8, 0.2]
68+rain = matrixBuilder gen
69+ where
70+ gen (_, No) = 0.8
71+ gen (_, Yes) = 0.2
59
72
60−sprinkler :: Matrix Double Bool Bool
73+sprinkler :: Matrix Double R S
61−sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]
74+sprinkler = matrixBuilder gen
75+ where
76+ gen (No, Off) = 0.6
77+ gen (No, On) = 0.4
78+ gen (Yes, Off) = 0.99
79+ gen (Yes, On) = 0.01
80+
81+grass :: Matrix Double (S, R) G
82+grass = matrixBuilder gen
83+ where
84+ gen ((Off, No), Dry) = 1
85+ gen ((Off, Yes), Dry) = 0.2
86+ gen ((On, No), Dry) = 0.1
87+ gen ((On, Yes), Dry) = 0.01
88+ gen ((Off, No), Wet) = 0
89+ gen ((Off, Yes), Wet) = 0.8
90+ gen ((On, No), Wet) = 0.9
91+ gen ((On, Yes), Wet) = 0.99
62
92
63−grass :: Matrix Double (Bool, Bool) Bool
93+tag f = kr f id
64−grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]
65
94
66−state :: Matrix Double () (Bool, (Bool, Bool))
95+state g s r = tag g . tag s . r
67−state = khatri grass identity . khatri sprinkler identity . rain
68
96
69−grass_wet :: Matrix Double (Bool, (Bool, Bool)) One
97+grass_wet :: Matrix Double () (G, (S, R)) -> Matrix Double One One
70−grass_wet = row [0,1] . kp1
98+grass_wet state = row [0,1] . fstM . state
71
99
72−rainning :: Matrix Double (Bool, (Bool, Bool)) One
100+rainning :: Matrix Double (G, (S, R)) One
73−rainning = row [0,1] . kp2 . kp2
101+rainning = row [0,1] . sndM . sndM
74
102
75-- Alcuin Puzzle
103-- Alcuin Puzzle
76
104
77data Being = Farmer | Fox | Goose | Beans
105data Being = Farmer | Fox | Goose | Beans
78− deriving (Bounded, Enum, Eq, Show, Generic)
106+ deriving (Bounded, Enum, Eq, Show, G.Generic)
79
107
80data Bank = LeftB | RightB
108data Bank = LeftB | RightB
81− deriving (Bounded, Enum, Eq, Show, Generic)
109+ deriving (Bounded, Enum, Eq, Show, G.Generic)
82
110
83eats :: Being -> Being -> Bool
111eats :: Being -> Being -> Bool
84eats Fox Goose = True
112eats Fox Goose = True
@@ -93,21 +121,21 @@ cross LeftB = RightB
93cross RightB = LeftB
121cross RightB = LeftB
94
122
95crossR :: R.Relation Bank Bank
123crossR :: R.Relation Bank Bank
96−crossR = R.fromF' cross
124+crossR = R.fromF cross
97
125
98-- | Initial state, everyone in the left bank
126-- | Initial state, everyone in the left bank
99locationLeft :: Being -> Bank
127locationLeft :: Being -> Bank
100locationLeft _ = LeftB
128locationLeft _ = LeftB
101
129
102locationLeftR :: R.Relation Being Bank
130locationLeftR :: R.Relation Being Bank
103−locationLeftR = R.fromF' locationLeft
131+locationLeftR = R.fromF locationLeft
104
132
105-- | Initial state, everyone in the right bank
133-- | Initial state, everyone in the right bank
106locationRight :: Being -> Bank
134locationRight :: Being -> Bank
107locationRight _ = RightB
135locationRight _ = RightB
108
136
109locationRightR :: R.Relation Being Bank
137locationRightR :: R.Relation Being Bank
110−locationRightR = R.fromF' locationRight
138+locationRightR = R.fromF locationRight
111
139
112-- Properties
140-- Properties
113
141
@@ -124,7 +152,7 @@ inv :: R.Relation Being Bank -> Bool
124inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)
152inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)
125 where
153 where
126 farmer :: R.Relation Being Being
154 farmer :: R.Relation Being Being
127− farmer = R.fromF' (const Farmer)
155+ farmer = R.fromF (const Farmer)
128
156
129-- Arbitrary state
157-- Arbitrary state
130bankState :: Being -> Bank -> Bool
158bankState :: Being -> Bank -> Bool
@@ -138,19 +166,19 @@ bankStateR :: R.Relation Being Bank
138bankStateR = R.toRel bankState
166bankStateR = R.toRel bankState
139
167
140-- Main
168-- Main
141−main :: IO ()
169+exec :: IO ()
142−main = do
170+exec = do
143 putStrLn "Monty Hall Problem solution:"
171 putStrLn "Monty Hall Problem solution:"
144 prettyPrint (secondChoice . firstChoice)
172 prettyPrint (secondChoice . firstChoice)
145 putStrLn "\n Sum of dices probability:"
173 putStrLn "\n Sum of dices probability:"
146− prettyPrint (sumSSM `comp` khatri die die)
174+ prettyPrint (sumSSM . kr die die)
147 putStrLn "\n Conditional dice throw:"
175 putStrLn "\n Conditional dice throw:"
148 prettyPrint conditionalThrows
176 prettyPrint conditionalThrows
149 putStrLn "\n Checking that the last result is indeed a distribution: "
177 putStrLn "\n Checking that the last result is indeed a distribution: "
150− prettyPrint (bang . sumSSM . khatri die die)
178+ prettyPrint (bang . sumSSM . kr die die)
151 putStrLn "\n Probability of grass being wet:"
179 putStrLn "\n Probability of grass being wet:"
152− prettyPrint (grass_wet . state)
180+ prettyPrint (grass_wet (state grass sprinkler rain))
153 putStrLn "\n Probability of rain:"
181 putStrLn "\n Probability of rain:"
154− prettyPrint (rainning . state)
182+ prettyPrint (rainning . state grass sprinkler rain)
155 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
183 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
156 print (inv bankStateR)
184 print (inv bankStateR)
test/Examples/Spreadsheets.hsnew file+47
@@ -0,0 +1,47 @@
1+{-# LANGUAGE DeriveGeneric #-}
2+
3+module Examples.Spreadsheets
4+
5+import LAoP.Utils
6+import LAoP.Matrix.Type
7+import GHC.Generics
8+import Prelude hiding (id, (.))
9+
10+data Student = Student1 | Student2 | Student3 | Student4
11+ deriving (Eq, Show, Enum, Bounded, Generic)
12+
13+data Question = Question1 | Question2 | Question3 | Question4
14+ deriving (Eq, Show, Enum, Bounded, Generic)
15+
16+data Results = Exam | Test | Final
17+ deriving (Eq, Show, Enum, Bounded, Generic)
18+
19+test :: Matrix Float One Results
20+test = point Test
21+
22+exam :: Matrix Float One Results
23+exam = point Exam
24+
25+final :: Matrix Float One Results
26+final = point Final
27+
28+m :: Matrix Float Question Student
29+m = fromLists [[95, 90, 100, 40], [20, 90, 90, 0], [30, 20, 95, 0], [50, 80, 100, 30]]
30+
31+w :: Matrix Float Question One
32+w = fromLists [[0.2, 0.3, 0.2, 0.3]]
33+
34+xls :: Matrix Float Question One
35+ -> Matrix Float Question Student
36+ -> Matrix Float Student One
37+ -> Matrix Float (Either Question Results) (Either One Student)
38+xls w m t = join (fork w m) (fork zeros r)
39+ where
40+ rExam = m . tr w
41+ rTest = tr t
42+ rFinal = rTest `maxPW` rExam
43+ r = (rExam . tr exam) + (rTest . tr test) + (rFinal . tr final)
44+
45+-- | Overloaded, point-wise 'max' function
46+maxPW :: Ord e => Matrix e a b -> Matrix e a b -> Matrix e a b
47+maxPW = zipWithM max
test/Spec.hs+7−1
@@ -1,2 +1,8 @@
1+module Main (main) where
2+
3+import qualified Examples.Readme as RD
4+
1main :: IO ()
5main :: IO ()
2−main = putStrLn "Test suite not yet implemented"
6+main = do
7+ putStrLn ("Testing if README works" :: String)
8+ RD.exec