git.iko.soy/laop
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.github/workflows/ci.ymldeleted−43
since splitsnapshots
unifiedsplit
@@ -1,43 +0,0 @@
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+2−54
@@ -1,55 +1,3 @@
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
25.stack-work/
1.stack-work/
26−stack.yaml.lock
2+laop.cabal
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]
34−*~
3+*~
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.ymldeleted−47
@@ -1,47 +0,0 @@
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.mddeleted−17
@@ -1,17 +0,0 @@
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.mdnew file+3
@@ -0,0 +1,3 @@
1+# Changelog for laop
2+
3+## Unreleased changes
LICENSE+26−17
@@ -1,21 +1,30 @@
1−MIT License
1+Copyright Armando Santos (c) 2019
2
2
3−Copyright (c) 2020 Armando Santos
3+All rights reserved.
4
4
5−Permission is hereby granted, free of charge, to any person obtaining a copy
5+Redistribution and use in source and binary forms, with or without
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
10−furnished to do so, subject to the following conditions:
6+modification, are permitted provided that the following conditions are met:
11
7
12−The above copyright notice and this permission notice shall be included in all
8+ * Redistributions of source code must retain the above copyright
13−copies or substantial portions of the Software.
9+ notice, this list of conditions and the following disclaimer.
14
10
15−THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
11+ * Redistributions in binary form must reproduce the above
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
16−IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
22+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
17−FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
23+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
18−AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
24+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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
19−LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
28+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
20−OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
29+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
21−SOFTWARE.
30+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
README.md+24−107
@@ -1,11 +1,5 @@
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−
9The LAoP discipline generalises relations and functions treating them as
3The LAoP discipline generalises relations and functions treating them as
10Boolean matrices and in turn consider these as arrows.
4Boolean matrices and in turn consider these as arrows.
11
5
@@ -16,38 +10,33 @@ motivation behind the library, the underlying theory, and implementation details
16This module offers many of the combinators mentioned in the work of
10This module offers many of the combinators mentioned in the work of
17[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).
11[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).
18
12
19−See the package in hackage [here](https://hackage.haskell.org/package/laop-0.1.1.0)
13+See the candidate package in hackage [here](https://hackage.haskell.org/package/laop-0.1.0.0/candidate)
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.
22
14
23## Features
15## Features
24
16
25−This library offers 3 main matrix programming modules:
17+This library offers:
26−
18+
27−- One in which matrices are typed with type level natural numbers;
19+- An internal representation that uses normal datatypes (`Void`, `()`, `Either`) for matrix dimensions and two
28−- One in which matrices are typed with arbitrary generic data types;
20+ type families `FromNat` and `Count` to make it easier to work with these type of matrices.
29−- One in which matrices are regarded as relations, i.e. boolean matrices;
21+- A module that uses generalised types for the dimensions and is just a newtype wrapper
30−
22+ around the internal matrix data type.
31−There's also an experimental module that uses matrices to represent probability
23+- Other that uses type level naturals for the dimensions and is just a newtype wrapper
32−distributions.
24+ around the internal matrix data type.
33−
25+- A very simple Probabilistic Programming module and several functions and data types to
34−The most interesting feature is that matrices are represented as an inductive data type.
26+ make it easier to deal with sample space.
35−Given this formulation, matrix algorithms can be expressed in a much more elegant and
27+
36−calculational style than the traditional vector of vectors representation. This data type
28+Given this, this new matrix formulation compared to other libraries one has much more advantages, such as:
37−guarantees that a matrix will always have valid dimensions. It can also express block matrix
29+
38−computations naturally, which leads to total, efficient and statically typed manipulation
30+- Has an inductive definition enabling writing matrix manipulation functions in a much more elegant and calculational way;
39−and transformation functions.
31+- Statically typed dimensions;
40−
32+- Polymorphic data type dimensions;
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.
33+- Polymorphic matrix content;
42−
34+- Awesome dimension type inference;
43−Given this, this new matrix formulation compared to other libraries:
35+- Fast type natural conversion via `FromNat` type family;
44−
36+- Matrix `Junc` and `Split`-ing in O(1);
45−- Is more compositional and polymorphic and does not have partial matrix manipulation functions (hence less chances for usage errors);
37+- Matrix composition takes advantage of divide-and-conquer and fusion laws.
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.
38+
47−
48−## Known issues
49−
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).
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.
51
40
52## Notes
41## Notes
53
42
@@ -129,73 +118,6 @@ grass_wet = row [0,1] . kp1
129rainning :: Matrix Double (Bool, (Bool, Bool)) One
118rainning :: Matrix Double (Bool, (Bool, Bool)) One
130rainning = row [0,1] . kp2 . kp2
119rainning = row [0,1] . kp2 . kp2
131
120
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−
199main :: IO ()
121main :: IO ()
200main = do
122main = do
201 putStrLn "Monty Hall Problem solution:"
123 putStrLn "Monty Hall Problem solution:"
@@ -210,8 +132,6 @@ main = do
210 prettyPrint (grass_wet . state)
132 prettyPrint (grass_wet . state)
211 putStrLn "\n Probability of rain:"
133 putStrLn "\n Probability of rain:"
212 prettyPrint (rainning . state)
134 prettyPrint (rainning . state)
213− putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
214− print (inv bankStateR)
215```
135```
216
136
217```Shell
137```Shell
@@ -270,7 +190,4 @@ Monty Hall Problem solution:
270┌ ┐
190┌ ┐
271│ 0.2 │
191│ 0.2 │
272└ ┘
192└ ┘
273−
274−Is the arbitrary state a valid state? (Alcuin Puzzle)
275−False
276```
193```
Setup.hsnew file+2
@@ -0,0 +1,2 @@
1+import Distribution.Simple
2+main = defaultMain
app/Main.hsnew file+87
@@ -0,0 +1,87 @@
1+{-# LANGUAGE DeriveGeneric #-}
2+{-# LANGUAGE FlexibleContexts #-}
3+{-# LANGUAGE TypeOperators #-}
4+{-# LANGUAGE TypeApplications #-}
5+{-# LANGUAGE DataKinds #-}
6+
7+module Main where
8+
9+import LAoP.Matrix.Type
10+import LAoP.Utils
11+import LAoP.Dist
12+import GHC.TypeLits
13+import Data.Coerce
14+import GHC.Generics
15+import Control.Category hiding (id)
16+import Prelude hiding ((.))
17+
18+-- Monty Hall Problem
19+data Outcome = Win | Lose
20+ deriving (Bounded, Enum, Eq, Show, Generic)
21+
22+switch :: Outcome -> Outcome
23+switch Win = Lose
24+switch Lose = Win
25+
26+firstChoice :: Matrix Double () Outcome
27+firstChoice = col [1/3, 2/3]
28+
29+secondChoice :: Matrix Double Outcome Outcome
30+secondChoice = fromF' switch
31+
32+-- Dice sum
33+
34+type SS = Natural 1 6 -- Sample Space
35+
36+sumSS :: SS -> SS -> Natural 2 12
37+sumSS = coerceNat (+)
38+
39+sumSSM = fromF' (uncurry sumSS)
40+
41+condition :: (Int, Int) -> Int -> Int
42+condition (fst, snd) thrd = if fst == snd
43+ then fst * 3
44+ else fst + snd + thrd
45+
46+conditionSS :: (SS, SS) -> SS -> Natural 3 18
47+conditionSS = coerceNat2 condition
48+
49+conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die
50+
51+die :: Matrix Double () SS
52+die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]
53+
54+-- Sprinkler
55+
56+rain :: Matrix Double () Bool
57+rain = col [0.8, 0.2]
58+
59+sprinkler :: Matrix Double Bool Bool
60+sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]
61+
62+grass :: Matrix Double (Bool, Bool) Bool
63+grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]
64+
65+state :: Matrix Double () (Bool, (Bool, Bool))
66+state = khatri grass identity . khatri sprinkler identity . rain
67+
68+grass_wet :: Matrix Double (Bool, (Bool, Bool)) One
69+grass_wet = row [0,1] . kp1
70+
71+rainning :: Matrix Double (Bool, (Bool, Bool)) One
72+rainning = row [0,1] . kp2 . kp2
73+
74+main :: IO ()
75+main = do
76+ putStrLn "Monty Hall Problem solution:"
77+ prettyPrint (secondChoice . firstChoice)
78+ putStrLn "\n Sum of dices probability:"
79+ prettyPrint (sumSSM `comp` khatri die die)
80+ putStrLn "\n Conditional dice throw:"
81+ prettyPrint conditionalThrows
82+ putStrLn "\n Checking that the last result is indeed a distribution: "
83+ prettyPrint (bang . sumSSM . khatri die die)
84+ putStrLn "\n Probability of grass being wet:"
85+ prettyPrint (grass_wet . state)
86+ putStrLn "\n Probability of rain:"
87+ prettyPrint (rainning . state)
appveyor.ymldeleted−34
@@ -1,34 +0,0 @@
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.hsdeleted−175
@@ -1,175 +0,0 @@
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.cabaldeleted−89
@@ -1,89 +0,0 @@
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.yamlnew file+48
@@ -0,0 +1,48 @@
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
@@ -36,14 +36,6 @@ 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−
47 -- * Functor instance equivalent functions
39 -- * Functor instance equivalent functions
48 fmapD,
40 fmapD,
49
41
@@ -53,8 +45,6 @@ module LAoP.Dist
53
45
54 -- * Selective equivalent functions
46 -- * Selective equivalent functions
55 selectD,
47 selectD,
56− branchD,
57− ifD,
58
48
59 -- * Monad equivalent functions
49 -- * Monad equivalent functions
60 returnD,
50 returnD,
src/LAoP/Dist/Internal.hs+36−87
@@ -1,5 +1,3 @@
1−{-# LANGUAGE DerivingVia #-}
2−{-# LANGUAGE ConstraintKinds #-}
3{-# LANGUAGE GADTs #-}
1{-# LANGUAGE GADTs #-}
4{-# LANGUAGE AllowAmbiguousTypes #-}
2{-# LANGUAGE AllowAmbiguousTypes #-}
5{-# LANGUAGE TypeApplications #-}
3{-# LANGUAGE TypeApplications #-}
@@ -10,27 +8,18 @@
10{-# LANGUAGE RankNTypes #-}
8{-# LANGUAGE RankNTypes #-}
11
9
12module LAoP.Dist.Internal
10module LAoP.Dist.Internal
13− (
11+ (
14 Dist(..),
12 Dist(..),
15 Prob,
13 Prob,
16
14
17− Countable,
18− CountableN,
19− CountableDimsN,
20− FLN,
21− Liftable,
22− TrivialP,
23−
24 fmapD,
15 fmapD,
25 unitD,
16 unitD,
26 multD,
17 multD,
27 selectD,
18 selectD,
28− branchD,
29− ifD,
30 returnD,
19 returnD,
31 bindD,
20 bindD,
32 (??),
21 (??),
33−
22+
34 choose,
23 choose,
35 shape,
24 shape,
36 linear,
25 linear,
@@ -43,15 +32,11 @@ module LAoP.Dist.Internal
43 )
32 )
44 where
33 where
45
34
46−import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDims, CountableN, CountableDimsN, Liftable, FLN)
35+import LAoP.Matrix.Type
47−import Prelude hiding (id, (.))
48−import qualified LAoP.Matrix.Internal as I
49import LAoP.Utils
36import LAoP.Utils
50import GHC.TypeLits
37import GHC.TypeLits
51import Data.Proxy
38import Data.Proxy
52import Data.List (sortBy)
39import Data.List (sortBy)
53−import Control.DeepSeq
54−import Data.Bool
55
40
56-- | Type synonym for probability value
41-- | Type synonym for probability value
57type Prob = Double
42type Prob = Double
@@ -59,21 +44,17 @@ type Prob = Double
59-- | Type synonym for column vector matrices. This represents a probability
44-- | Type synonym for column vector matrices. This represents a probability
60-- distribution.
45-- distribution.
61newtype Dist a = D (Matrix Prob () a)
46newtype 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)
71
47
72-- | Functor instance
48-- | Functor instance
73fmapD ::
49fmapD ::
74− ( Liftable a b,
50+ ( Bounded a,
75− CountableDimsN a b,
51+ Bounded b,
76− FLN b a
52+ Enum a,
53+ Enum b,
54+ Eq b,
55+ KnownNat (Count (Normalize a)),
56+ KnownNat (Count (Normalize b)),
57+ FromLists Prob (Normalize b) (Normalize a)
77 )
58 )
78 =>
59 =>
79 (a -> b) -> Dist a -> Dist b
60 (a -> b) -> Dist a -> Dist b
@@ -85,58 +66,26 @@ unitD = D (one 1)
85
66
86-- | Applicative/Monoidal instance 'mult' function
67-- | Applicative/Monoidal instance 'mult' function
87multD ::
68multD ::
88− ( CountableDimsN a b,
69+ ( KnownNat (Count (Normalize a)),
89− CountableN (a, b),
70+ KnownNat (Count (Normalize b)),
90− FLN (a, b) a,
71+ KnownNat (Count (Normalize (a, b))),
91− FLN (a, b) b,
72+ FromLists Prob (Normalize (a, b)) (Normalize a),
92− TrivialP a b
73+ FromLists Prob (Normalize (a, b)) (Normalize b),
74+ Normalize (a, b) ~ Normalize (Normalize a, Normalize b)
93 ) => Dist a -> Dist b -> Dist (a, b)
75 ) => Dist a -> Dist b -> Dist (a, b)
94−multD (D a) (D b) = D (kr a b)
76+multD (D a) (D b) = D (khatri a b)
95
77
96-- | Selective instance function
78-- | Selective instance function
97selectD ::
79selectD ::
98− ( FLN b b,
80+ ( Normalize (Either a b) ~ Either (Normalize a) (Normalize b),
99− CountableN b
81+ FromLists Prob (Normalize b) (Normalize b),
82+ KnownNat (Count (Normalize b))
100 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
83 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
101−selectD (D d) m = D (selectM d m)
84+selectD (D d) m = D (junc m identity `comp` d)
102−
85+
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
137
86
138-- | Monad instance 'return' function
87-- | Monad instance 'return' function
139−returnD :: forall a . (Enum a, FLN () a, Countable a) => a -> Dist a
88+returnD :: forall a . (Enum a, FromLists Prob () (Normalize a), KnownNat (Count a)) => a -> Dist a
140returnD a = D (col l)
89returnD a = D (col l)
141 where
90 where
142 i = fromInteger $ natVal (Proxy :: Proxy (Count a))
91 i = fromInteger $ natVal (Proxy :: Proxy (Count a))
@@ -150,8 +99,8 @@ bindD (D d) m = D (m `comp` d)
150-- | Extract probabilities given an Event.
99-- | Extract probabilities given an Event.
151(??) ::
100(??) ::
152 ( Enum a,
101 ( Enum a,
153− Countable a,
102+ KnownNat (Count a),
154− FLN () a
103+ FromLists Prob () (Normalize a)
155 ) => (a -> Bool) -> Dist a -> Prob
104 ) => (a -> Bool) -> Dist a -> Prob
156(??) p d =
105(??) p d =
157 let l = toValues d
106 let l = toValues d
@@ -161,11 +110,11 @@ bindD (D d) m = D (m `comp` d)
161-- Distribution Construction
110-- Distribution Construction
162
111
163-- | Constructs a Bernoulli distribution
112-- | Constructs a Bernoulli distribution
164−choose :: (FLN () a) => Prob -> Dist a
113+choose :: (FromLists Prob () (Normalize a)) => Prob -> Dist a
165choose prob = D (col [prob, 1 - prob])
114choose prob = D (col [prob, 1 - prob])
166
115
167-- | Creates a distribution given a shape function
116-- | Creates a distribution given a shape function
168−shape :: (FLN () a) => (Prob -> Prob) -> [a] -> Dist a
117+shape :: (FromLists Prob () (Normalize a)) => (Prob -> Prob) -> [a] -> Dist a
169shape _ [] = error "Probability.shape: empty list"
118shape _ [] = error "Probability.shape: empty list"
170shape f xs =
119shape f xs =
171 let incr = 1 / fromIntegral (length xs - 1)
120 let incr = 1 / fromIntegral (length xs - 1)
@@ -173,23 +122,23 @@ shape f xs =
173 in fromFreqs (zip xs ps)
122 in fromFreqs (zip xs ps)
174
123
175-- | Constructs a Linear distribution
124-- | Constructs a Linear distribution
176−linear :: (FLN () a) => [a] -> Dist a
125+linear :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
177linear = shape id
126linear = shape id
178
127
179-- | Constructs an Uniform distribution
128-- | Constructs an Uniform distribution
180−uniform :: (FLN () a) => [a] -> Dist a
129+uniform :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
181uniform = shape (const 1)
130uniform = shape (const 1)
182
131
183-- | Constructs an Negative Exponential distribution
132-- | Constructs an Negative Exponential distribution
184−negExp :: (FLN () a) => [a] -> Dist a
133+negExp :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
185negExp = shape (\x -> exp (-x))
134negExp = shape (\x -> exp (-x))
186
135
187-- | Constructs an Normal distribution
136-- | Constructs an Normal distribution
188−normal :: (FLN () a) => [a] -> Dist a
137+normal :: (FromLists Prob () (Normalize a)) => [a] -> Dist a
189normal = shape (normalCurve 0.5 0.5)
138normal = shape (normalCurve 0.5 0.5)
190
139
191-- | Transforms a 'Dist' into a list of pairs.
140-- | Transforms a 'Dist' into a list of pairs.
192−toValues :: forall a . (Enum a, Countable a, FLN () a) => Dist a -> [(a, Prob)]
141+toValues :: forall a . (Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> [(a, Prob)]
193toValues (D d) =
142toValues (D d) =
194 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))
143 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))
195 probs = toList d
144 probs = toList d
@@ -197,7 +146,7 @@ toValues (D d) =
197 in res
146 in res
198
147
199-- | Pretty a distribution
148-- | Pretty a distribution
200−prettyDist :: forall a. (Show a, Enum a, Countable a, FLN () a) => Dist a -> String
149+prettyDist :: forall a. (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> String
201prettyDist d =
150prettyDist d =
202 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)
151 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)
203 w = maximum (map (length . show . fst) values)
152 w = maximum (map (length . show . fst) values)
@@ -209,12 +158,12 @@ prettyDist d =
209 showR n x = show x ++ " "
158 showR n x = show x ++ " "
210
159
211-- | Pretty Print a distribution
160-- | Pretty Print a distribution
212−prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FLN () a) => Dist a -> IO ()
161+prettyPrintDist :: forall a . (Show a, Enum a, KnownNat (Count a), FromLists Prob () (Normalize a)) => Dist a -> IO ()
213prettyPrintDist = putStrLn . prettyDist @a
162prettyPrintDist = putStrLn . prettyDist @a
214
163
215-- Auxiliary functions
164-- Auxiliary functions
216
165
217−fromFreqs :: (FLN () a) => [(a,Prob)] -> Dist a
166+fromFreqs :: (FromLists Prob () (Normalize a)) => [(a,Prob)] -> Dist a
218fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))
167fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))
219 where q = sum $ map snd xs
168 where q = sum $ map snd xs
220
169
src/LAoP/Matrix/Internal.hs+435−477
@@ -1,6 +1,3 @@
1−{-# LANGUAGE StrictData #-}
2−{-# LANGUAGE TypeFamilyDependencies #-}
3−{-# LANGUAGE ConstraintKinds #-}
4{-# LANGUAGE AllowAmbiguousTypes #-}
1{-# LANGUAGE AllowAmbiguousTypes #-}
5{-# LANGUAGE DataKinds #-}
2{-# LANGUAGE DataKinds #-}
6{-# LANGUAGE FlexibleContexts #-}
3{-# LANGUAGE FlexibleContexts #-}
@@ -12,6 +9,7 @@
12{-# LANGUAGE ScopedTypeVariables #-}
9{-# LANGUAGE ScopedTypeVariables #-}
13{-# LANGUAGE StandaloneDeriving #-}
10{-# LANGUAGE StandaloneDeriving #-}
14{-# LANGUAGE TypeApplications #-}
11{-# LANGUAGE TypeApplications #-}
12+{-# LANGUAGE TypeFamilies #-}
15{-# LANGUAGE TypeOperators #-}
13{-# LANGUAGE TypeOperators #-}
16{-# LANGUAGE UndecidableInstances #-}
14{-# LANGUAGE UndecidableInstances #-}
17
15
@@ -30,7 +28,7 @@
30-- motivation behind the library, the underlying theory, and implementation details.
28-- motivation behind the library, the underlying theory, and implementation details.
31--
29--
32-- This module offers many of the combinators mentioned in the work of
30-- This module offers many of the combinators mentioned in the work of
33−-- Macedo (2012) and Oliveira (2012).
31+-- Macedo (2012) and Oliveira (2012).
34--
32--
35-- This is an Internal module and it is no supposed to be imported.
33-- This is an Internal module and it is no supposed to be imported.
36--
34--
@@ -43,26 +41,17 @@ module LAoP.Matrix.Internal
43 --
41 --
44 -- There exists two type families that make it easier to write
42 -- There exists two type families that make it easier to write
45 -- matrix dimensions: 'FromNat' and 'Count'. This approach
43 -- matrix dimensions: 'FromNat' and 'Count'. This approach
46− -- leads to a very straightforward implementation
44+ -- leads to a very straightforward implementation
47− -- of LAoP combinators.
45+ -- of LAoP combinators.
48
46
49 -- * Type safe matrix representation
47 -- * Type safe matrix representation
50 Matrix (..),
48 Matrix (..),
51
49
52− -- * Constraint type aliases
53− Countable,
54− CountableDims,
55− CountableN,
56− CountableDimsN,
57− FL,
58− FLN,
59− Liftable,
60− Trivial,
61−
62 -- * Primitives
50 -- * Primitives
51+ empty,
63 one,
52 one,
64− join,
53+ junc,
65− fork,
54+ split,
66
55
67 -- * Auxiliary type families
56 -- * Auxiliary type families
68 FromNat,
57 FromNat,
@@ -75,7 +64,6 @@ module LAoP.Matrix.Internal
75 toLists,
64 toLists,
76 toList,
65 toList,
77 matrixBuilder,
66 matrixBuilder,
78− matrixBuilder',
79 row,
67 row,
80 col,
68 col,
81 zeros,
69 zeros,
@@ -86,38 +74,28 @@ module LAoP.Matrix.Internal
86 -- * Misc
74 -- * Misc
87 -- ** Get dimensions
75 -- ** Get dimensions
88 columns,
76 columns,
89− columns',
90 rows,
77 rows,
91− rows',
92
78
93 -- ** Matrix Transposition
79 -- ** Matrix Transposition
94 tr,
80 tr,
95
81
96− -- ** Scalar multiplication/division of matrices
97− (.|),
98− (./),
99−
100 -- ** Selective operator
82 -- ** Selective operator
101− select,
83+ select,
102− branch,
103
84
104 -- ** McCarthy's Conditional
85 -- ** McCarthy's Conditional
105 cond,
86 cond,
106
87
107 -- ** Matrix "abiding"
88 -- ** Matrix "abiding"
108− abideJF,
89+ abideJS,
109− abideFJ,
90+ abideSJ,
110−
111− -- ** Zip matrices
112− zipWithM,
113
91
114 -- * Biproduct approach
92 -- * Biproduct approach
115− -- ** Fork
93+ -- ** Split
116 (===),
94 (===),
117 -- *** Projections
95 -- *** Projections
118 p1,
96 p1,
119 p2,
97 p2,
120− -- ** Join
98+ -- ** Junc
121 (|||),
99 (|||),
122 -- *** Injections
100 -- *** Injections
123 i1,
101 i1,
@@ -130,13 +108,12 @@ module LAoP.Matrix.Internal
130
108
131 -- | Note that given the restrictions imposed it is not possible to
109 -- | Note that given the restrictions imposed it is not possible to
132 -- implement the standard type classes present in standard Haskell.
110 -- implement the standard type classes present in standard Haskell.
133−
134 -- *** Matrix pairing projections
111 -- *** Matrix pairing projections
135− fstM,
112+ kp1,
136− sndM,
113+ kp2,
137
114
138 -- *** Matrix pairing
115 -- *** Matrix pairing
139− kr,
116+ khatri,
140
117
141 -- * Matrix composition and lifting
118 -- * Matrix composition and lifting
142
119
@@ -144,10 +121,10 @@ module LAoP.Matrix.Internal
144
121
145 -- | Note that given the restrictions imposed it is not possible to
122 -- | Note that given the restrictions imposed it is not possible to
146 -- implement the standard type classes present in standard Haskell.
123 -- implement the standard type classes present in standard Haskell.
147− iden,
124+ identity,
148 comp,
125 comp,
149− fromF',
150 fromF,
126 fromF,
127+ fromF',
151
128
152 -- * Matrix printing
129 -- * Matrix printing
153 pretty,
130 pretty,
@@ -157,39 +134,34 @@ module LAoP.Matrix.Internal
157 toBool,
134 toBool,
158 fromBool,
135 fromBool,
159 compRel,
136 compRel,
160− divR,
161− divL,
162− divS,
163 fromFRel,
137 fromFRel,
164 fromFRel',
138 fromFRel',
165 toRel,
139 toRel,
166− negateM,
167 orM,
140 orM,
168 andM,
141 andM,
169 subM
142 subM
170 )
143 )
171 where
144 where
172
145
173−import LAoP.Utils.Internal
146+import LAoP.Utils
174import Data.Bool
147import Data.Bool
175import Data.Kind
148import Data.Kind
176import Data.List
149import Data.List
177−import Data.Maybe
178import Data.Proxy
150import Data.Proxy
179−import Data.Typeable
180import Data.Void
151import Data.Void
181import GHC.TypeLits
152import GHC.TypeLits
182import Data.Type.Equality
153import Data.Type.Equality
183import GHC.Generics
154import GHC.Generics
184import Control.DeepSeq
155import Control.DeepSeq
156+import Control.Category
185−import Prelude hiding (id, (.))
157+import Prelude hiding ((.))
186−import qualified Prelude (id, (.))
187
158
188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
159-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
189data Matrix e cols rows where
160data Matrix e cols rows where
161+ Empty :: Matrix e Void Void
190 One :: e -> Matrix e () ()
162 One :: e -> Matrix e () ()
191− Join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
163+ Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
192− Fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
164+ Split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
193
165
194deriving instance (Show e) => Show (Matrix e cols rows)
166deriving instance (Show e) => Show (Matrix e cols rows)
195
167
@@ -199,214 +171,221 @@ deriving instance (Show e) => Show (Matrix e cols rows)
199-- 'Generic' instance.
171-- 'Generic' instance.
200type family Count (d :: Type) :: Nat where
172type family Count (d :: Type) :: Nat where
201 Count (Natural n m) = (m - n) + 1
173 Count (Natural n m) = (m - n) + 1
202− Count (List a) = (^) 2 (Count a)
203− Count (Either a b) = (+) (Count a) (Count b)
174+ Count (Either a b) = (+) (Count a) (Count b)
204− Count (a, b) = (*) (Count a) (Count b)
175+ Count (a, b) = (*) (Count a) (Count b)
205− Count (a -> b) = (^) (Count b) (Count a)
176+ Count (a -> b) = (^) (Count b) (Count a)
206 -- Generics
177 -- Generics
207− Count (M1 _ _ f p) = Count (f p)
178+ Count (M1 _ _ f p) = Count (f p)
208− Count (K1 _ _ _) = 1
179+ Count (K1 _ _ _) = 1
209− Count (V1 _) = 0
180+ Count (V1 _) = 0
210− Count (U1 _) = 1
181+ Count (U1 _) = 1
211 Count ((:*:) a b p) = Count (a p) * Count (b p)
182 Count ((:*:) a b p) = Count (a p) * Count (b p)
212 Count ((:+:) a b p) = Count (a p) + Count (b p)
183 Count ((:+:) a b p) = Count (a p) + Count (b p)
213− Count d = Count (Rep d R)
184+ Count d = Count (Rep d R)
214
185
215-- | Type family that computes of a given type dimension from a given natural
186-- | Type family that computes of a given type dimension from a given natural
216--
187--
217-- Thanks to Li-Yao Xia this type family is super fast.
188-- Thanks to Li-Yao Xia this type family is super fast.
218type family FromNat (n :: Nat) :: Type where
189type family FromNat (n :: Nat) :: Type where
190+ FromNat 0 = Void
219 FromNat 1 = ()
191 FromNat 1 = ()
220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
192 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
221
193
222type family FromNat' (b :: Bool) (m :: Type) :: Type where
194type family FromNat' (b :: Bool) (m :: Type) :: Type where
223− FromNat' 'True m = Either m m
195+ FromNat' 'True m = Either m m
224 FromNat' 'False m = Either () (Either m m)
196 FromNat' 'False m = Either () (Either m m)
225
197
226-- | Type family that normalizes the representation of a given data
198-- | Type family that normalizes the representation of a given data
227-- structure
199-- structure
228type family Normalize (d :: Type) :: Type where
200type family Normalize (d :: Type) :: Type where
229− Normalize (Either a b) = Either (Normalize a) (Normalize b)
230− Normalize d = FromNat (Count d)
201+ Normalize d = FromNat (Count d)
231−
202+
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)
240−type Trivial a = FromNat (Count a) ~ a
241−
242−-- | It is possible to implement a constrained version of the category type
203+-- | It isn't possible to implement the 'id' function so it's
243−-- class.
204+-- implementation is 'undefined'. However 'comp' can be and this partial
205+-- class implementation exists just to make the code more readable.
206+--
207+-- Please use 'identity' instead.
244instance (Num e) => Category (Matrix e) where
208instance (Num e) => Category (Matrix e) where
245− type Object (Matrix e) a = (FL a a, Countable a)
209+ id = undefined
246− id = iden
247− (.) = comp
210+ (.) = comp
248
211
249instance NFData e => NFData (Matrix e cols rows) where
212instance NFData e => NFData (Matrix e cols rows) where
213+ rnf Empty = ()
250− rnf (One e) = rnf e
214+ rnf (One e) = rnf e
251− rnf (Join a b) = rnf a `seq` rnf b
215+ rnf (Junc a b) = rnf a `seq` rnf b
252− rnf (Fork a b) = rnf a `seq` rnf b
216+ rnf (Split a b) = rnf a `seq` rnf b
253
217
254instance Eq e => Eq (Matrix e cols rows) where
218instance Eq e => Eq (Matrix e cols rows) where
219+ Empty == Empty = True
255− (One a) == (One b) = a == b
220+ (One a) == (One b) = a == b
256− (Join a b) == (Join c d) = a == c && b == d
221+ (Junc a b) == (Junc c d) = a == c && b == d
257− (Fork a b) == (Fork c d) = a == c && b == d
222+ (Split a b) == (Split c d) = a == c && b == d
258− x@(Fork _ _) == y@(Join _ _) = x == abideJF y
223+ x@(Split _ _) == y@(Junc _ _) = x == abideJS y
259− x@(Join _ _) == y@(Fork _ _) = abideJF x == y
224+ x@(Junc _ _) == y@(Split _ _) = abideJS x == y
260
225
261instance Num e => Num (Matrix e cols rows) where
226instance Num e => Num (Matrix e cols rows) where
262− a + b = zipWithM (+) a b
263
227
264− a - b = zipWithM (-) a b
228+ Empty + Empty = Empty
265−
229+ (One a) + (One b) = One (a + b)
266− a * b = zipWithM (*) a b
230+ (Junc a b) + (Junc c d) = Junc (a + c) (b + d)
267−
231+ (Split a b) + (Split c d) = Split (a + c) (b + d)
232+ x@(Split _ _) + y@(Junc _ _) = x + abideJS y
233+ x@(Junc _ _) + y@(Split _ _) = abideJS x + y
234+
235+ Empty - Empty = Empty
236+ (One a) - (One b) = One (a - b)
237+ (Junc a b) - (Junc c d) = Junc (a - c) (b - d)
238+ (Split a b) - (Split c d) = Split (a - c) (b - d)
239+ x@(Split _ _) - y@(Junc _ _) = x - abideJS y
240+ x@(Junc _ _) - y@(Split _ _) = abideJS x - y
241+
242+ Empty * Empty = Empty
243+ (One a) * (One b) = One (a * b)
244+ (Junc a b) * (Junc c d) = Junc (a * c) (b * d)
245+ (Split a b) * (Split c d) = Split (a * c) (b * d)
246+ x@(Split _ _) * y@(Junc _ _) = x * abideJS y
247+ x@(Junc _ _) * y@(Split _ _) = abideJS x * y
248+
249+ abs Empty = Empty
268− abs (One a) = One (abs a)
250+ abs (One a) = One (abs a)
269− abs (Join a b) = Join (abs a) (abs b)
251+ abs (Junc a b) = Junc (abs a) (abs b)
270− abs (Fork a b) = Fork (abs a) (abs b)
252+ abs (Split a b) = Split (abs a) (abs b)
271−
253+
254+ signum Empty = Empty
272− signum (One a) = One (signum a)
255+ signum (One a) = One (signum a)
273− signum (Join a b) = Join (signum a) (signum b)
256+ signum (Junc a b) = Junc (signum a) (signum b)
274− signum (Fork a b) = Fork (signum a) (signum b)
257+ signum (Split a b) = Split (signum a) (signum b)
275
258
276instance Ord e => Ord (Matrix e cols rows) where
259instance Ord e => Ord (Matrix e cols rows) where
260+ Empty <= Empty = True
277− (One a) <= (One b) = a <= b
261+ (One a) <= (One b) = a <= b
278− (Join a b) <= (Join c d) = (a <= c) && (b <= d)
262+ (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)
279− (Fork a b) <= (Fork c d) = (a <= c) && (b <= d)
263+ (Split a b) <= (Split c d) = (a <= c) && (b <= d)
280− x@(Fork _ _) <= y@(Join _ _) = x <= abideJF y
264+ x@(Split _ _) <= y@(Junc _ _) = x <= abideJS y
281− x@(Join _ _) <= y@(Fork _ _) = abideJF x <= y
265+ x@(Junc _ _) <= y@(Split _ _) = abideJS x <= y
282
266
283-- Primitives
267-- Primitives
284
268
269+-- | Empty matrix constructor
270+empty :: Matrix e Void Void
271+empty = Empty
272+
285-- | Unit matrix constructor
273-- | Unit matrix constructor
286one :: e -> Matrix e () ()
274one :: e -> Matrix e () ()
287one = One
275one = One
288
276
289−-- | Matrix 'Join' constructor
277+-- | Matrix 'Junc' constructor
290−join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
278+junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
291−join = Join
279+junc = Junc
292
280
293infixl 3 |||
281infixl 3 |||
294
282
295−-- | Matrix 'Join' constructor
283+-- | Matrix 'Junc' constructor
296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
284(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
297−(|||) = Join
285+(|||) = Junc
298
286
299−-- | Matrix 'Fork' constructor
287+-- | Matrix 'Split' constructor
300−fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
288+split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
301−fork = Fork
289+split = Split
302
290
303infixl 2 ===
291infixl 2 ===
304
292
305−-- | Matrix 'Fork' constructor
293+-- | Matrix 'Split' constructor
306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
294(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
307−(===) = Fork
295+(===) = Split
308
296
309-- Construction
297-- Construction
310
298
311-- | Type class for defining the 'fromList' conversion function.
299-- | Type class for defining the 'fromList' conversion function.
312--
300--
313-- Given that it is not possible to branch on types at the term level type
301-- Given that it is not possible to branch on types at the term level type
314−-- classes are needed very much like an inductive definition but on types.
302+-- classes are needed bery much like an inductive definition but on types.
315−class FromLists cols rows where
303+class FromLists e cols rows where
316 -- | Build a matrix out of a list of list of elements. Throws a runtime
304 -- | Build a matrix out of a list of list of elements. Throws a runtime
317 -- error if the dimensions do not match.
305 -- error if the dimensions do not match.
318 fromLists :: [[e]] -> Matrix e cols rows
306 fromLists :: [[e]] -> Matrix e cols rows
319
307
320−instance FromLists () () where
308+instance FromLists e Void Void where
309+ fromLists [] = Empty
310+ fromLists _ = error "Wrong dimensions"
311+
312+instance {-# OVERLAPPING #-} FromLists e () () where
321 fromLists [[e]] = One e
313 fromLists [[e]] = One e
322 fromLists _ = error "Wrong dimensions"
314 fromLists _ = error "Wrong dimensions"
323
315
324−instance (FromLists cols ()) => FromLists (Either () cols) () where
316+instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where
325− fromLists [h : t] = Join (One h) (fromLists [t])
317+ fromLists [h : t] = Junc (One h) (fromLists [t])
326 fromLists _ = error "Wrong dimensions"
318 fromLists _ = error "Wrong dimensions"
327
319
328−instance {-# OVERLAPPABLE #-}
329− (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where
320+instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), KnownNat (Count a)) => FromLists e (Either a b) () where
330− fromLists [l] =
321+ fromLists [l] =
331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
322 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
332− in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])
323+ in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])
333 fromLists _ = error "Wrong dimensions"
324 fromLists _ = error "Wrong dimensions"
334
325
335−instance (FromLists () rows) => FromLists () (Either () rows) where
326+instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where
336− fromLists ([h] : t) = Fork (One h) (fromLists t)
327+ fromLists ([h] : t) = Split (One h) (fromLists t)
337 fromLists _ = error "Wrong dimensions"
328 fromLists _ = error "Wrong dimensions"
338
329
339−instance {-# OVERLAPPABLE #-}
340− (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where
330+instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, KnownNat (Count a)) => FromLists e () (Either a b) where
341− fromLists l@([_] : _) =
331+ fromLists l@([_] : _) =
342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
332 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
343− in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))
333+ in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))
344 fromLists _ = error "Wrong dimensions"
334 fromLists _ = error "Wrong dimensions"
345
335
346−instance (FromLists (Either a b) c, FromLists (Either a b) d, Countable c)
336+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
347− => FromLists (Either a b) (Either c d) where
348 fromLists l@(h : t) =
337 fromLists l@(h : t) =
349 let lh = length h
338 let lh = length h
350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
339 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
351− condition = all ((== lh) . length) t
340+ condition = all (== lh) (map length t)
352 in if lh > 0 && condition
341 in if lh > 0 && condition
353− then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))
342+ then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))
354 else error "Not all rows have the same length"
343 else error "Not all rows have the same length"
355
344
356-- | Matrix builder function. Constructs a matrix provided with
345-- | Matrix builder function. Constructs a matrix provided with
357−-- a construction function that operates with indices.
346+-- a construction function.
358−matrixBuilder' ::
347+matrixBuilder ::
359 forall e cols rows.
348 forall e cols rows.
360− ( FL cols rows,
349+ ( FromLists e cols rows,
361− CountableDims cols rows
350+ KnownNat (Count cols),
351+ KnownNat (Count rows)
352+ ) =>
353+ ((Int, Int) -> e) ->
362− ) => ((Int, Int) -> e) -> Matrix e cols rows
354+ Matrix e cols rows
363−matrixBuilder' f =
355+matrixBuilder f =
364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
356 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
357 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
358 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
359 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
368
360
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−
389-- | Constructs a column vector matrix
361-- | Constructs a column vector matrix
390−col :: (FL () rows) => [e] -> Matrix e () rows
362+col :: (FromLists e () rows) => [e] -> Matrix e () rows
391col = fromLists . map (: [])
363col = fromLists . map (: [])
392
364
393-- | Constructs a row vector matrix
365-- | Constructs a row vector matrix
394−row :: (FL cols ()) => [e] -> Matrix e cols ()
366+row :: (FromLists e cols ()) => [e] -> Matrix e cols ()
395row = fromLists . (: [])
367row = fromLists . (: [])
396
368
397-- | Lifts functions to matrices with arbitrary dimensions.
369-- | Lifts functions to matrices with arbitrary dimensions.
398--
370--
399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
371-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
400-- types @a@ or @b@ allows.
372-- types @a@ or @b@ allows.
401−fromF' ::
373+fromF ::
402 forall a b cols rows e.
374 forall a b cols rows e.
403− ( Liftable e a b,
375+ ( Bounded a,
404− CountableDims cols rows,
376+ Bounded b,
377+ Enum a,
378+ Enum b,
379+ Eq b,
380+ Num e,
381+ Ord e,
382+ KnownNat (Count cols),
383+ KnownNat (Count rows),
405− FL rows cols
384+ FromLists e rows cols
406 ) =>
385 ) =>
407 (a -> b) ->
386 (a -> b) ->
408 Matrix e cols rows
387 Matrix e cols rows
409−fromF' f =
388+fromF f =
410 let minA = minBound @a
389 let minA = minBound @a
411 maxA = maxBound @a
390 maxA = maxBound @a
412 minB = minBound @b
391 minB = minBound @b
@@ -416,8 +395,8 @@ fromF' f =
416 elementsA = take ccols [minA .. maxA]
395 elementsA = take ccols [minA .. maxA]
417 elementsB = take rrows [minB .. maxB]
396 elementsB = take rrows [minB .. maxB]
418 combinations = (,) <$> elementsA <*> elementsB
397 combinations = (,) <$> elementsA <*> elementsB
419− combAp = map snd . sort . map (\(a, b) -> if f a == b
398+ combAp = map snd . sort . map (\(a, b) -> if f a == b
420− then ((fromEnum a, fromEnum b), 1)
399+ then ((fromEnum a, fromEnum b), 1)
421 else ((fromEnum a, fromEnum b), 0)) $ combinations
400 else ((fromEnum a, fromEnum b), 0)) $ combinations
422 mList = buildList combAp rrows
401 mList = buildList combAp rrows
423 in tr $ fromLists mList
402 in tr $ fromLists mList
@@ -427,23 +406,48 @@ fromF' f =
427
406
428-- | Lifts functions to matrices with dimensions matching @a@ and @b@
407-- | Lifts functions to matrices with dimensions matching @a@ and @b@
429-- cardinality's.
408-- cardinality's.
430−fromF ::
409+fromF' ::
431 forall a b e.
410 forall a b e.
432− ( Liftable e a b,
411+ ( Bounded a,
433− CountableDimsN a b,
412+ Bounded b,
434− FLN b a
413+ Enum a,
414+ Enum b,
415+ Eq b,
416+ Num e,
417+ Ord e,
418+ KnownNat (Count (Normalize a)),
419+ KnownNat (Count (Normalize b)),
420+ FromLists e (Normalize b) (Normalize a)
435 ) =>
421 ) =>
436 (a -> b) ->
422 (a -> b) ->
437 Matrix e (Normalize a) (Normalize b)
423 Matrix e (Normalize a) (Normalize b)
438−fromF = fromF'
424+fromF' f =
425+ let minA = minBound @a
426+ maxA = maxBound @a
427+ minB = minBound @b
428+ maxB = maxBound @b
429+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
430+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
431+ elementsA = take ccols [minA .. maxA]
432+ elementsB = take rrows [minB .. maxB]
433+ combinations = (,) <$> elementsA <*> elementsB
434+ combAp = map snd . sort . map (\(a, b) -> if f a == b
435+ then ((fromEnum a, fromEnum b), 1)
436+ else ((fromEnum a, fromEnum b), 0)) $ combinations
437+ mList = buildList combAp rrows
438+ in tr $ fromLists mList
439+ where
440+ buildList [] _ = []
441+ buildList l r = take r l : buildList (drop r l) r
439
442
440-- Conversion
443-- Conversion
441
444
442-- | Converts a matrix to a list of lists of elements.
445-- | Converts a matrix to a list of lists of elements.
443toLists :: Matrix e cols rows -> [[e]]
446toLists :: Matrix e cols rows -> [[e]]
447+toLists Empty = []
444toLists (One e) = [[e]]
448toLists (One e) = [[e]]
445−toLists (Fork l r) = toLists l ++ toLists r
449+toLists (Split l r) = toLists l ++ toLists r
446−toLists (Join l r) = zipWith (++) (toLists l) (toLists r)
450+toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)
447
451
448-- | Converts a matrix to a list of elements.
452-- | Converts a matrix to a list of elements.
449toList :: Matrix e cols rows -> [e]
453toList :: Matrix e cols rows -> [e]
@@ -452,38 +456,37 @@ toList = concat . toLists
452-- Zeros Matrix
456-- Zeros Matrix
453
457
454-- | The zero matrix. A matrix wholly filled with zeros.
458-- | The zero matrix. A matrix wholly filled with zeros.
455−zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
459+zeros :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows
456−zeros = matrixBuilder' (const 0)
460+zeros = matrixBuilder (const 0)
457
461
458-- Ones Matrix
462-- Ones Matrix
459
463
460-- | The ones matrix. A matrix wholly filled with ones.
464-- | The ones matrix. A matrix wholly filled with ones.
461--
465--
462-- Also known as T (Top) matrix.
466-- Also known as T (Top) matrix.
463−ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
467+ones :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => Matrix e cols rows
464−ones = matrixBuilder' (const 1)
468+ones = matrixBuilder (const 1)
465
469
466-- Const Matrix
470-- Const Matrix
467
471
468-- | The constant matrix constructor. A matrix wholly filled with a given
472-- | The constant matrix constructor. A matrix wholly filled with a given
469-- value.
473-- value.
470−constant :: (Num e, FL cols rows, CountableDims cols rows) => e -> Matrix e cols rows
474+constant :: (Num e, FromLists e cols rows, KnownNat (Count cols), KnownNat (Count rows)) => e -> Matrix e cols rows
471−constant e = matrixBuilder' (const e)
475+constant e = matrixBuilder (const e)
472
476
473-- Bang Matrix
477-- Bang Matrix
474
478
475-- | The T (Top) row vector matrix.
479-- | The T (Top) row vector matrix.
476−bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()
480+bang :: forall e cols. (Num e, Enum e, FromLists e cols (), KnownNat (Count cols)) => Matrix e cols ()
477bang =
481bang =
478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
482 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
479 in fromLists [take c [1, 1 ..]]
483 in fromLists [take c [1, 1 ..]]
480
484
481−-- iden Matrix
485+-- Identity Matrix
482
486
483−-- | iden matrix.
487+-- | Identity matrix.
484−iden :: forall e cols . (Num e, FL cols cols, Countable cols) => Matrix e cols cols
488+identity :: (Num e, FromLists e cols cols, KnownNat (Count cols)) => Matrix e cols cols
485−iden = matrixBuilder' (bool 0 1 . uncurry (==))
489+identity = matrixBuilder (bool 0 1 . uncurry (==))
486−{-# NOINLINE iden #-}
487
490
488-- Matrix composition (MMM)
491-- Matrix composition (MMM)
489
492
@@ -492,90 +495,60 @@ iden = matrixBuilder' (bool 0 1 . uncurry (==))
492-- This definition takes advantage of divide-and-conquer and fusion laws
495-- This definition takes advantage of divide-and-conquer and fusion laws
493-- from LAoP.
496-- from LAoP.
494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
497comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
498+comp Empty Empty = Empty
495−comp (One a) (One b) = One (a * b)
499+comp (One a) (One b) = One (a * b)
496−comp (Join a b) (Fork c d) = comp a c + comp b d -- Divide-and-conquer law
500+comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law
497−comp (Fork a b) c = Fork (comp a c) (comp b c) -- Fork fusion law
501+comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law
498−comp c (Join a b) = Join (comp c a) (comp c b) -- Join fusion law
502+comp c (Junc a b) = Junc (comp c a) (comp c b) -- Junc 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)
522
503
523-- Projections
504-- Projections
524
505
525-- | Biproduct first component projection
506-- | Biproduct first component projection
526−p1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e (Either m n) m
507+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
527−p1 = join id zeros
508+p1 =
509+ let iden = identity :: Matrix e m m
510+ zero = zeros :: Matrix e n m
511+ in junc iden zero
528
512
529-- | Biproduct second component projection
513-- | Biproduct second component projection
530−p2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e (Either m n) n
514+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
531−p2 = join zeros id
515+p2 =
516+ let iden = identity :: Matrix e n n
517+ zero = zeros :: Matrix e m n
518+ in junc zero iden
532
519
533-- Injections
520-- Injections
534
521
535-- | Biproduct first component injection
522-- | Biproduct first component injection
536−i1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e m (Either m n)
523+i1 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e n m, FromLists e m m) => Matrix e m (Either m n)
537i1 = tr p1
524i1 = tr p1
538
525
539-- | Biproduct second component injection
526-- | Biproduct second component injection
540−i2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e n (Either m n)
527+i2 :: (Num e, KnownNat (Count n), KnownNat (Count m), FromLists e m n, FromLists e n n) => Matrix e n (Either m n)
541i2 = tr p2
528i2 = tr p2
542
529
543-- Dimensions
530-- Dimensions
544
531
545-- | Obtain the number of rows.
532-- | Obtain the number of rows.
546--
533--
547−-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
534+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
548−-- dimensions in constant time. For a version that doesn't require the
535+-- dimensions in constant time.
549−-- constraint see 'rows''.
550−rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int
551−rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
552−
553−-- | Obtain the number of rows in an inefficient manner, but without any
554−-- constraints.
555--
536--
556−-- For a more efficient version see 'rows'.
537+-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
538+-- exchange for performance.
557−rows' :: Matrix e cols rows -> Int
539+rows :: forall e cols rows. (KnownNat (Count rows)) => Matrix e cols rows -> Int
558−rows' (One _) = 1
540+rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
559−rows' (Join lhs _) = rows' lhs
560−rows' (Fork top bottom) = rows' top + rows' bottom
561
541
562-- | Obtain the number of columns.
542-- | Obtain the number of columns.
543+--
544+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
545+-- dimensions in constant time.
563--
546--
564−-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
547+-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
565−-- dimensions in constant time. For a version that doesn't require the
548+-- exchange for performance.
566−-- constraint see 'columns''.
567−columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int
549+columns :: forall e cols rows. (KnownNat (Count cols)) => Matrix e cols rows -> Int
568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
550columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
569
551
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−
579-- Coproduct Bifunctor
552-- Coproduct Bifunctor
580
553
581infixl 5 -|-
554infixl 5 -|-
@@ -584,172 +557,157 @@ infixl 5 -|-
584(-|-) ::
557(-|-) ::
585 forall e n k m j.
558 forall e n k m j.
586 ( Num e,
559 ( Num e,
587− CountableDims j k,
560+ KnownNat (Count j),
588− FL k k,
561+ KnownNat (Count k),
589− FL j k,
562+ FromLists e k k,
590− FL k j,
563+ FromLists e j k,
591− FL j j
564+ FromLists e k j,
565+ FromLists e j j
592 ) =>
566 ) =>
593 Matrix e n k ->
567 Matrix e n k ->
594 Matrix e m j ->
568 Matrix e m j ->
595 Matrix e (Either n m) (Either k j)
569 Matrix e (Either n m) (Either k j)
596−(-|-) a b = Join (i1 . a) (i2 . b)
570+(-|-) a b = Junc (i1 . a) (i2 . b)
597
571
598-- Khatri Rao Product and projections
572-- Khatri Rao Product and projections
599
573
600-- | Khatri Rao product first component projection matrix.
574-- | Khatri Rao product first component projection matrix.
601−fstM ::
575+kp1 ::
602 forall e m k .
576 forall e m k .
603 ( Num e,
577 ( Num e,
604− CountableDims k m,
578+ KnownNat (Count k),
579+ FromLists e (FromNat (Count m * Count k)) m,
580+ KnownNat (Count m),
605− FL (Normalize (m, k)) m,
581+ KnownNat (Count (Normalize (m, k)))
606− CountableN (m, k)
607 ) => Matrix e (Normalize (m, k)) m
582 ) => Matrix e (Normalize (m, k)) m
608−fstM = matrixBuilder' f
583+kp1 = matrixBuilder f
609 where
584 where
610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
585 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
611 f (x, y)
586 f (x, y)
612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
587 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
613− | otherwise = 0
588+ | otherwise = 0
614
589
615-- | Khatri Rao product second component projection matrix.
590-- | Khatri Rao product second component projection matrix.
616−sndM ::
591+kp2 ::
617 forall e m k .
592 forall e m k .
618 ( Num e,
593 ( Num e,
619− CountableDims k m,
594+ KnownNat (Count k),
595+ FromLists e (FromNat (Count m * Count k)) k,
596+ KnownNat (Count m),
620− FL (Normalize (m, k)) k,
597+ KnownNat (Count (Normalize (m, k)))
621− CountableN (m, k)
622 ) => Matrix e (Normalize (m, k)) k
598 ) => Matrix e (Normalize (m, k)) k
623−sndM = matrixBuilder' f
599+kp2 = matrixBuilder f
624 where
600 where
625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
601 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
626 f (x, y)
602 f (x, y)
627 | x == y || mod (y - x) offset == 0 = 1
603 | x == y || mod (y - x) offset == 0 = 1
628− | otherwise = 0
604+ | otherwise = 0
629
605
630-- | Khatri Rao Matrix product also known as matrix pairing.
606-- | Khatri Rao Matrix product also known as matrix pairing.
631--
607--
632-- NOTE: That this is not a true categorical product, see for instance:
608-- NOTE: That this is not a true categorical product, see for instance:
633−--
609+--
634-- @
610-- @
635−-- | fstM . kr a b == a
611+-- | kp1 . khatri a b == a
636−-- kr a b ==> |
612+-- khatri a b ==> |
637−-- | sndM . kr a b == b
613+-- | kp2 . khatri a b == b
638-- @
614-- @
639--
615--
640-- __Emphasis__ on the implication symbol.
616-- __Emphasis__ on the implication symbol.
641−kr ::
617+khatri ::
642− forall e cols a b.
618+ forall e cols a b.
643 ( Num e,
619 ( Num e,
644− CountableDims a b,
620+ KnownNat (Count a),
645− CountableN (a, b),
621+ KnownNat (Count b),
622+ KnownNat (Count (Normalize (a, b))),
646− FL (Normalize (a, b)) a,
623+ FromLists e (Normalize (a, b)) a,
647− FL (Normalize (a, b)) b
624+ FromLists e (Normalize (a, b)) b
648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
625 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
649−kr a b =
626+khatri a b =
650− let fstM' = fstM @e @a @b
627+ let kp1' = kp1 @e @a @b
651− sndM' = sndM @e @a @b
628+ kp2' = kp2 @e @a @b
652− in (tr fstM' . a) * (tr sndM' . b)
629+ in (tr kp1') . a * (tr kp2') . b
653
630
654-- Product Bifunctor (Kronecker)
631-- Product Bifunctor (Kronecker)
655
632
656infixl 4 ><
633infixl 4 ><
657
634
658-- | Matrix product functor also known as kronecker product
635-- | Matrix product functor also known as kronecker product
659−(><) ::
636+(><) ::
660− forall e m p n q.
637+ forall e m p n q.
661 ( Num e,
638 ( Num e,
662− CountableDims m n,
639+ KnownNat (Count m),
663− CountableDims p q,
640+ KnownNat (Count n),
664− CountableDimsN (m, n) (p, q),
641+ KnownNat (Count p),
642+ KnownNat (Count q),
643+ KnownNat (Count (Normalize (m, n))),
665− FL (Normalize (m, n)) m,
644+ FromLists e (Normalize (m, n)) m,
666− FL (Normalize (m, n)) n,
645+ FromLists e (Normalize (m, n)) n,
667− FL (Normalize (p, q)) p,
646+ KnownNat (Count (Normalize (p, q))),
647+ FromLists e (Normalize (p, q)) p,
668− FL (Normalize (p, q)) q
648+ FromLists e (Normalize (p, q)) q
669− )
649+ )
670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
650 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
671(><) a b =
651(><) a b =
672− let fstM' = fstM @e @m @n
652+ let kp1' = kp1 @e @m @n
673− sndM' = sndM @e @m @n
653+ kp2' = kp2 @e @m @n
674− in kr (a . fstM') (b . sndM')
654+ in khatri (a . kp1') (b . kp2')
675
655
676−-- Matrix abide Join Fork
656+-- Matrix abide Junc Split
677
657
678−-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.
658+-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
679−--
659+--
680-- Law:
660-- Law:
681--
661--
682-- @
662-- @
683−-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)
663+-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)
684-- @
664-- @
685−abideJF :: Matrix e cols rows -> Matrix e cols rows
665+abideJS :: Matrix e cols rows -> Matrix e cols rows
686−abideJF (Join (Fork a c) (Fork b d)) = Fork (Join (abideJF a) (abideJF b)) (Join (abideJF c) (abideJF d)) -- Join-Fork abide law
666+abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law
667+abideJS Empty = Empty
687−abideJF (One e) = One e
668+abideJS (One e) = One e
688−abideJF (Join a b) = Join (abideJF a) (abideJF b)
669+abideJS (Junc a b) = Junc (abideJS a) (abideJS b)
689−abideJF (Fork a b) = Fork (abideJF a) (abideJF b)
670+abideJS (Split a b) = Split (abideJS a) (abideJS b)
690
671
691−-- Matrix abide Fork Join
672+-- Matrix abide Split Junc
692
673
693−-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
674+-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
694−--
675+--
695-- @
676-- @
696−-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)
677+-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)
697-- @
678-- @
698−abideFJ :: Matrix e cols rows -> Matrix e cols rows
679+abideSJ :: Matrix e cols rows -> Matrix e cols rows
699−abideFJ (Fork (Join a b) (Join c d)) = Join (Fork (abideFJ a) (abideFJ c)) (Fork (abideFJ b) (abideFJ d)) -- Fork-Join abide law
680+abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law
681+abideSJ Empty = Empty
700−abideFJ (One e) = One e
682+abideSJ (One e) = One e
701−abideFJ (Join a b) = Join (abideFJ a) (abideFJ b)
683+abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)
702−abideFJ (Fork a b) = Fork (abideFJ a) (abideFJ b)
684+abideSJ (Split a b) = Split (abideSJ a) (abideSJ b)
703
685
704-- Matrix transposition
686-- Matrix transposition
705
687
706-- | Matrix transposition.
688-- | Matrix transposition.
707tr :: Matrix e cols rows -> Matrix e rows cols
689tr :: Matrix e cols rows -> Matrix e rows cols
690+tr Empty = Empty
708−tr (One e) = One e
691+tr (One e) = One e
709−tr (Join a b) = Fork (tr a) (tr b)
692+tr (Junc a b) = Split (tr a) (tr b)
710−tr (Fork a b) = Join (tr a) (tr b)
693+tr (Split a b) = Junc (tr a) (tr b)
711
694
712-- Selective 'select' operator
695-- Selective 'select' operator
713
696
714-- | Selective functors 'select' operator equivalent inspired by the
697-- | Selective functors 'select' operator equivalent inspired by the
715-- ArrowMonad solution presented in the paper.
698-- 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− )
735− => Matrix e cols (Either a b) -> Matrix e a c -> Matrix e b c -> Matrix e cols c
699+select :: (Num e, FromLists e b b, KnownNat (Count b)) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
736−branch x l r = f x `select` g l `select` r
700+select m y = junc y identity . m
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
743
701
744-- McCarthy's Conditional
702-- McCarthy's Conditional
745
703
746-- | McCarthy's Conditional expresses probabilistic choice.
704-- | McCarthy's Conditional expresses probabilistic choice.
747cond ::
705cond ::
748− ( Trivial cols,
706+ ( cols ~ FromNat (Count cols),
749− Countable cols,
707+ KnownNat (Count cols),
750− FL () cols,
708+ FromLists e () cols,
751− FL cols (),
709+ FromLists e cols (),
752− FL cols cols,
710+ FromLists e cols cols,
753 Bounded a,
711 Bounded a,
754 Enum a,
712 Enum a,
755 Num e,
713 Num e,
@@ -757,14 +715,14 @@ cond ::
757 )
715 )
758 =>
716 =>
759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
717 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
760−cond p f g = join f g . grd p
718+cond p f g = junc f g . grd p
761−
719+
762−grd ::
720+grd ::
763− ( Trivial q,
721+ ( q ~ FromNat (Count q),
764− Countable q,
722+ KnownNat (Count q),
765− FL () q,
723+ FromLists e () q,
766− FL q (),
724+ FromLists e q (),
767− FL q q,
725+ FromLists e q q,
768 Bounded a,
726 Bounded a,
769 Enum a,
727 Enum a,
770 Num e,
728 Num e,
@@ -772,88 +730,62 @@ grd ::
772 )
730 )
773 =>
731 =>
774 (a -> Bool) -> Matrix e q (Either q q)
732 (a -> Bool) -> Matrix e q (Either q q)
775−grd f = fork (corr f) (corr (not . f))
733+grd f = split (corr f) (corr (not . f))
776−
734+
777−corr ::
735+corr ::
778− forall e a q .
736+ forall e a q .
779− ( Trivial q,
737+ ( q ~ FromNat (Count q),
780− Countable q,
738+ KnownNat (Count q),
781− FL () q,
739+ FromLists e () q,
782− FL q (),
740+ FromLists e q (),
783− FL q q,
741+ FromLists e q q,
784− Liftable e a Bool
742+ Bounded a,
743+ Enum a,
744+ Num e,
745+ Ord e
785− )
746+ )
786 => (a -> Bool) -> Matrix e q q
747 => (a -> Bool) -> Matrix e q q
787−corr p = let f = fromF' p :: Matrix e q ()
748+corr p = let f = fromF p :: Matrix e q ()
788− in kr f (id :: Matrix e q q)
749+ in khatri f (identity :: Matrix e q q)
789
750
790-- Pretty print
751-- Pretty print
791
752
792prettyAux :: Show e => [[e]] -> [[e]] -> String
753prettyAux :: Show e => [[e]] -> [[e]] -> String
793prettyAux [] _ = ""
754prettyAux [] _ = ""
794−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
755+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
795 where
756 where
796 v = fmap show m
757 v = fmap show m
797 widest = maximum $ fmap length v
758 widest = maximum $ fmap length v
798 fill str = replicate (widest - length str - 2) ' ' ++ str
759 fill str = replicate (widest - length str - 2) ' ' ++ str
799−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
760+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
800 where
761 where
801− v = fmap show m
762+ v = fmap show m
802− widest = maximum $ fmap length v
763+ widest = maximum $ fmap length v
803 fill str = replicate (widest - length str - 2) ' ' ++ str
764 fill str = replicate (widest - length str - 2) ' ' ++ str
804−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
765+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
805 prettyAux t l
766 prettyAux t l
806 where
767 where
807− v = fmap show l
768+ v = fmap show l
808− widest = maximum $ fmap length v
769+ widest = maximum $ fmap length v
809 fill str = replicate (widest - length str - 2) ' ' ++ str
770 fill str = replicate (widest - length str - 2) ' ' ++ str
810
771
811-- | Matrix pretty printer
772-- | Matrix pretty printer
812−pretty :: (CountableDims cols rows, Show e) => Matrix e cols rows -> String
773+pretty :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> String
813−pretty m = concat
814− [ "┌ ", unwords (replicate (columns m) blank), " ┐\n"
774+pretty m = "┌ " ++ unwords (replicate (columns m) blank) ++ " ┐\n" ++
815− , unlines
775+ prettyAux (toLists m) (toLists m) ++
816− [ "│ " ++ unwords (fmap (\j -> fill $ show $ getElem i j m) [1..columns m]) ++ " │" | i <- [1..rows m] ]
817− , "└ ", unwords (replicate (columns m) blank), " ┘"
776+ "└ " ++ unwords (replicate (columns m) blank) ++ " ┘"
818− ]
819− where
777+ where
820− strings = map show (toList m)
778+ v = fmap show (toList m)
821− widest = maximum $ map length strings
779+ widest = maximum $ fmap length v
822 fill str = replicate (widest - length str) ' ' ++ str
780 fill str = replicate (widest - length str) ' ' ++ str
823− blank = fill ""
781+ 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)
839
782
840-- | Matrix pretty printer
783-- | Matrix pretty printer
841−prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()
784+prettyPrint :: (KnownNat (Count cols), Show e) => Matrix e cols rows -> IO ()
842prettyPrint = putStrLn . pretty
785prettyPrint = putStrLn . pretty
843
786
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−
852-- Relational operators functions
787-- Relational operators functions
853
788
854−type Boolean = Natural 0 1
855−type Relation a b = Matrix Boolean a b
856−
857-- | Helper conversion function
789-- | Helper conversion function
858toBool :: (Num e, Eq e) => e -> Bool
790toBool :: (Num e, Eq e) => e -> Bool
859toBool n
791toBool n
@@ -865,110 +797,136 @@ fromBool :: Bool -> Natural 0 1
865fromBool True = nat 1
797fromBool True = nat 1
866fromBool False = nat 0
798fromBool False = nat 0
867
799
868−-- | Relational negation
869−negateM :: Relation cols rows -> Relation cols rows
870−negateM (One (Nat p)) = One (Nat (negate p))
871−negateM (Join a b) = Join (negateM a) (negateM b)
872−negateM (Fork a b) = Fork (negateM a) (negateM b)
873−
874-- | Relational addition
800-- | Relational addition
875−orM :: Relation cols rows -> Relation cols rows -> Relation cols rows
801+orM :: Matrix (Natural 0 1) cols rows -> Matrix (Natural 0 1) cols rows -> Matrix (Natural 0 1) cols rows
802+orM Empty Empty = Empty
876−orM (One a) (One b) = One (fromBool (toBool a || toBool b))
803+orM (One a) (One b) = One (fromBool (toBool a || toBool b))
877−orM (Join a b) (Join c d) = Join (orM a c) (orM b d)
804+orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)
878−orM (Fork a b) (Fork c d) = Fork (orM a c) (orM b d)
805+orM (Split a b) (Split c d) = Split (orM a c) (orM b d)
879−orM x@(Fork _ _) y@(Join _ _) = orM x (abideJF y)
806+orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)
880−orM x@(Join _ _) y@(Fork _ _) = orM (abideJF x) y
807+orM x@(Junc _ _) y@(Split _ _) = orM (abideJS x) y
881
808
882-- | Relational multiplication
809-- | Relational multiplication
883−andM :: Relation cols rows -> Relation cols rows -> Relation cols rows
810+andM :: Matrix (Natural 0 1) cols rows -> Matrix (Natural 0 1) cols rows -> Matrix (Natural 0 1) cols rows
811+andM Empty Empty = Empty
884−andM (One a) (One b) = One (fromBool (toBool a && toBool b))
812+andM (One a) (One b) = One (fromBool (toBool a && toBool b))
885−andM (Join a b) (Join c d) = Join (andM a c) (andM b d)
813+andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)
886−andM (Fork a b) (Fork c d) = Fork (andM a c) (andM b d)
814+andM (Split a b) (Split c d) = Split (andM a c) (andM b d)
887−andM x@(Fork _ _) y@(Join _ _) = andM x (abideJF y)
815+andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)
888−andM x@(Join _ _) y@(Fork _ _) = andM (abideJF x) y
816+andM x@(Junc _ _) y@(Split _ _) = andM (abideJS x) y
889
817
890-- | Relational subtraction
818-- | Relational subtraction
891−subM :: Relation cols rows -> Relation cols rows -> Relation cols rows
819+subM :: Matrix (Natural 0 1) cols rows -> Matrix (Natural 0 1) cols rows -> Matrix (Natural 0 1) cols rows
820+subM Empty Empty = Empty
892−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
821+subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
893−subM (Join a b) (Join c d) = Join (subM a c) (subM b d)
822+subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)
894−subM (Fork a b) (Fork c d) = Fork (subM a c) (subM b d)
823+subM (Split a b) (Split c d) = Split (subM a c) (subM b d)
895−subM x@(Fork _ _) y@(Join _ _) = subM x (abideJF y)
824+subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)
896−subM x@(Join _ _) y@(Fork _ _) = subM (abideJF x) y
825+subM x@(Junc _ _) y@(Split _ _) = subM (abideJS x) y
897
826
898-- | Matrix relational composition.
827-- | Matrix relational composition.
899−compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows
828+compRel :: Matrix (Natural 0 1) cr rows -> Matrix (Natural 0 1) cols cr -> Matrix (Natural 0 1) cols rows
829+compRel Empty Empty = Empty
900−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
830+compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
901−compRel (Join a b) (Fork c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
831+compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
902−compRel (Fork a b) c = Fork (compRel a c) (compRel b c) -- Fork fusion law
832+compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law
903−compRel c (Join a b) = Join (compRel c a) (compRel c b) -- Join fusion law
833+compRel c (Junc a b) = Junc (compRel c a) (compRel c b) -- Junc fusion law
904−
905−-- | Matrix relational right division
906−divR :: Relation b c -> Relation b a -> Relation a c
907−divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a
908−divR (Join a b) (Join c d) = andM (divR a c) (divR b d)
909−divR (Fork a b) c = Fork (divR a c) (divR b c)
910−divR c (Fork a b) = Join (divR c a) (divR c b)
911−
912−-- | Matrix relational left division
913−divL :: Relation c b -> Relation a b -> Relation a c
914−divL x y = tr (divR (tr y) (tr x))
915−
916−-- | Matrix relational symmetric division
917−divS :: Relation c a -> Relation b a -> Relation c b
918−divS s r = divL r s `intersection` divR (tr r) (tr s)
919− where
920− intersection = andM
921
834
922-- | Lifts functions to relations with arbitrary dimensions.
835-- | Lifts functions to relations with arbitrary dimensions.
923--
836--
924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
837-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
925-- types @a@ or @b@ allows.
838-- types @a@ or @b@ allows.
926−fromFRel' ::
839+fromFRel ::
927 forall a b cols rows.
840 forall a b cols rows.
928− ( Liftable Boolean a b,
841+ ( Bounded a,
929− CountableDims cols rows,
842+ Bounded b,
843+ Enum a,
844+ Enum b,
845+ Eq b,
846+ KnownNat (Count cols),
847+ KnownNat (Count rows),
930− FL rows cols
848+ FromLists (Natural 0 1) rows cols
931 ) =>
849 ) =>
932 (a -> b) ->
850 (a -> b) ->
933− Relation cols rows
851+ Matrix (Natural 0 1) cols rows
934−fromFRel' = fromF'
852+fromFRel f =
853+ let minA = minBound @a
854+ maxA = maxBound @a
855+ minB = minBound @b
856+ maxB = maxBound @b
857+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))
858+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))
859+ elementsA = take ccols [minA .. maxA]
860+ elementsB = take rrows [minB .. maxB]
861+ combinations = (,) <$> elementsA <*> elementsB
862+ combAp = map snd . sort . map (\(a, b) -> if f a == b
863+ then ((fromEnum a, fromEnum b), nat 1)
864+ else ((fromEnum a, fromEnum b), nat 0)) $ combinations
865+ mList = buildList combAp rrows
866+ in tr $ fromLists mList
867+ where
868+ buildList [] _ = []
869+ buildList l r = take r l : buildList (drop r l) r
935
870
936-- | Lifts functions to relations with dimensions matching @a@ and @b@
871-- | Lifts functions to relations with dimensions matching @a@ and @b@
937-- cardinality's.
872-- cardinality's.
938−fromFRel ::
873+fromFRel' ::
939 forall a b.
874 forall a b.
940− ( Liftable Boolean a b,
875+ ( Bounded a,
941− CountableDimsN a b,
876+ Bounded b,
942− FLN b a
877+ Enum a,
878+ Enum b,
879+ Eq b,
880+ KnownNat (Count (Normalize a)),
881+ KnownNat (Count (Normalize b)),
882+ FromLists (Natural 0 1) (Normalize b) (Normalize a)
943 ) =>
883 ) =>
944 (a -> b) ->
884 (a -> b) ->
945− Relation (Normalize a) (Normalize b)
885+ Matrix (Natural 0 1) (Normalize a) (Normalize b)
946−fromFRel = fromFRel'
886+fromFRel' f =
887+ let minA = minBound @a
888+ maxA = maxBound @a
889+ minB = minBound @b
890+ maxB = maxBound @b
891+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
892+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
893+ elementsA = take ccols [minA .. maxA]
894+ elementsB = take rrows [minB .. maxB]
895+ combinations = (,) <$> elementsA <*> elementsB
896+ combAp = map snd . sort . map (\(a, b) -> if f a == b
897+ then ((fromEnum a, fromEnum b), nat 1)
898+ else ((fromEnum a, fromEnum b), nat 0)) $ combinations
899+ mList = buildList combAp rrows
900+ in tr $ fromLists mList
901+ where
902+ buildList [] _ = []
903+ buildList l r = take r l : buildList (drop r l) r
947
904
948-- | Lifts a relation function to a Boolean Matrix
905-- | Lifts a relation function to a Boolean Matrix
949−toRel ::
906+toRel ::
950 forall a b.
907 forall a b.
951 ( Bounded a,
908 ( Bounded a,
952 Bounded b,
909 Bounded b,
953 Enum a,
910 Enum a,
954 Enum b,
911 Enum b,
955 Eq b,
912 Eq b,
956− CountableDims a b,
913+ KnownNat (Count (Normalize a)),
957− FLN b a
914+ KnownNat (Count (Normalize b)),
915+ FromLists (Natural 0 1) (Normalize b) (Normalize a)
958− )
916+ )
959− => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)
917+ => (a -> b -> Bool) -> Matrix (Natural 0 1) (Normalize a) (Normalize b)
960toRel f =
918toRel f =
961 let minA = minBound @a
919 let minA = minBound @a
962 maxA = maxBound @a
920 maxA = maxBound @a
963 minB = minBound @b
921 minB = minBound @b
964 maxB = maxBound @b
922 maxB = maxBound @b
965− ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))
923+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
966− rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))
924+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
967 elementsA = take ccols [minA .. maxA]
925 elementsA = take ccols [minA .. maxA]
968 elementsB = take rrows [minB .. maxB]
926 elementsB = take rrows [minB .. maxB]
969 combinations = (,) <$> elementsA <*> elementsB
927 combinations = (,) <$> elementsA <*> elementsB
970− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
928+ combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
971− then ((fromEnum a, fromEnum b), nat 1)
929+ then ((fromEnum a, fromEnum b), nat 1)
972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
930 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
973 mList = buildList combAp rrows
931 mList = buildList combAp rrows
974 in tr $ fromLists mList
932 in tr $ fromLists mList
src/LAoP/Matrix/Nat.hs+187−194
@@ -1,4 +1,3 @@
1−{-# LANGUAGE ConstraintKinds #-}
2{-# LANGUAGE AllowAmbiguousTypes #-}
1{-# LANGUAGE AllowAmbiguousTypes #-}
3{-# LANGUAGE DataKinds #-}
2{-# LANGUAGE DataKinds #-}
4{-# LANGUAGE DerivingVia #-}
3{-# LANGUAGE DerivingVia #-}
@@ -54,22 +53,11 @@ module LAoP.Matrix.Nat
54 -- * Type safe matrix representation
53 -- * Type safe matrix representation
55 Matrix (..),
54 Matrix (..),
56
55
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−
69 -- * Primitives
56 -- * Primitives
57+ empty,
70 one,
58 one,
71− join,
59+ junc,
72− fork,
60+ split,
73
61
74 -- * Auxiliary type families
62 -- * Auxiliary type families
75 I.FromNat,
63 I.FromNat,
@@ -77,11 +65,11 @@ module LAoP.Matrix.Nat
77 I.Normalize,
65 I.Normalize,
78
66
79 -- * Matrix construction and conversion
67 -- * Matrix construction and conversion
80− I.FL,
68+ I.FromLists,
81 fromLists,
69 fromLists,
82 toLists,
70 toLists,
83 toList,
71 toList,
84− matrixBuilder',
72+ matrixBuilder,
85 row,
73 row,
86 col,
74 col,
87 zeros,
75 zeros,
@@ -97,10 +85,6 @@ module LAoP.Matrix.Nat
97 -- ** Matrix Transposition
85 -- ** Matrix Transposition
98 tr,
86 tr,
99
87
100− -- ** Scalar multiplication/division of matrices
101− (.|),
102− (./),
103−
104 -- ** Selective operator
88 -- ** Selective operator
105 select,
89 select,
106
90
@@ -108,19 +92,16 @@ module LAoP.Matrix.Nat
108 cond,
92 cond,
109
93
110 -- ** Matrix "abiding"
94 -- ** Matrix "abiding"
111− abideJF,
95+ abideJS,
112− abideFJ,
96+ abideSJ,
113−
114− -- ** Zip Matrices
115− zipWithM,
116
97
117 -- * Biproduct approach
98 -- * Biproduct approach
118− -- ** Fork
99+ -- ** Split
119 (===),
100 (===),
120 -- *** Projections
101 -- *** Projections
121 p1,
102 p1,
122 p2,
103 p2,
123− -- ** Join
104+ -- ** Junc
124 (|||),
105 (|||),
125 -- *** Injections
106 -- *** Injections
126 i1,
107 i1,
@@ -134,11 +115,11 @@ module LAoP.Matrix.Nat
134 -- | Note that given the restrictions imposed it is not possible to
115 -- | Note that given the restrictions imposed it is not possible to
135 -- implement the standard type classes present in standard Haskell.
116 -- implement the standard type classes present in standard Haskell.
136 -- *** Matrix pairing projections
117 -- *** Matrix pairing projections
137− fstM,
118+ kp1,
138− sndM,
119+ kp2,
139
120
140 -- *** Matrix pairing
121 -- *** Matrix pairing
141− kr,
122+ khatri,
142
123
143 -- * Matrix composition and lifting
124 -- * Matrix composition and lifting
144
125
@@ -146,10 +127,10 @@ module LAoP.Matrix.Nat
146
127
147 -- | Note that given the restrictions imposed it is not possible to
128 -- | Note that given the restrictions imposed it is not possible to
148 -- implement the standard type classes present in standard Haskell.
129 -- implement the standard type classes present in standard Haskell.
149− iden,
130+ identity,
150 comp,
131 comp,
151− fromF',
152 fromF,
132 fromF,
133+ fromF',
153
134
154 -- * Matrix printing
135 -- * Matrix printing
155 pretty,
136 pretty,
@@ -160,95 +141,107 @@ where
160import Data.Proxy
141import Data.Proxy
161import GHC.TypeLits
142import GHC.TypeLits
162import Control.DeepSeq
143import Control.DeepSeq
144+import qualified Control.Category as C
163import qualified LAoP.Matrix.Internal as I
145import qualified LAoP.Matrix.Internal as I
164
146
165newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))
147newtype Matrix e (cols :: Nat) (rows :: Nat) = M (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))
148 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))
167
149
168−-- | Constraint type synonyms to keep the type signatures less convoluted
150+-- | It isn't possible to implement the 'id' function so it's
169−type Countable a = KnownNat (I.Count a)
151+-- implementation is 'undefined'. However 'comp' can be and this partial
170−type CountableDims a b = (Countable a, Countable b)
152+-- class implementation exists just to make the code more readable.
171−type CountableN a = KnownNat (I.Count (I.FromNat a))
153+--
172−type CountableNz a = KnownNat (I.Count (I.Normalize a))
154+-- Please use 'identity' instead.
173−type CountableDimsN a b = (CountableN a, CountableN b)
155+instance (Num e) => C.Category (Matrix e) where
174−type FLN e a b = I.FL (I.FromNat a) (I.FromNat b)
156+ id = undefined
175−type FLNz e a b = I.FL (I.Normalize a) (I.Normalize b)
157+ (.) = comp
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))
179
158
180-- Primitives
159-- Primitives
181
160
161+empty :: Matrix e 0 0
162+empty = M I.Empty
163+
182one :: e -> Matrix e 1 1
164one :: e -> Matrix e 1 1
183one = M . I.One
165one = M . I.One
184
166
185−join ::
167+junc ::
186− (TrivialE a b) =>
168+ (I.FromNat cols3 ~ Either (I.FromNat cols1) (I.FromNat cols2)) =>
187− Matrix e a rows ->
169+ Matrix e cols1 rows ->
188− Matrix e b rows ->
170+ Matrix e cols2 rows ->
189− Matrix e (a + b) rows
171+ Matrix e cols3 rows
190−join (M a) (M b) = M (I.Join a b)
172+junc (M a) (M b) = M (I.Junc a b)
191
173
192infixl 3 |||
174infixl 3 |||
193(|||) ::
175(|||) ::
194− (TrivialE a b) =>
176+ (I.FromNat cols3 ~ Either (I.FromNat cols1) (I.FromNat cols2)) =>
195− Matrix e a rows ->
177+ Matrix e cols1 rows ->
196− Matrix e b rows ->
178+ Matrix e cols2 rows ->
197− Matrix e (a + b) rows
179+ Matrix e cols3 rows
198−(|||) = join
180+(|||) = junc
199−
181+
200−fork ::
182+split ::
201− (TrivialE a b) =>
183+ (I.FromNat rows3 ~ Either (I.FromNat rows1) (I.FromNat rows2)) =>
202− Matrix e cols a ->
184+ Matrix e cols rows1 ->
203− Matrix e cols b ->
185+ Matrix e cols rows2 ->
204− Matrix e cols (a + b)
186+ Matrix e cols rows3
205−fork (M a) (M b) = M (I.Fork a b)
187+split (M a) (M b) = M (I.Split a b)
206
188
207infixl 2 ===
189infixl 2 ===
208(===) ::
190(===) ::
209− (TrivialE a b) =>
191+ (I.FromNat rows3 ~ Either (I.FromNat rows1) (I.FromNat rows2)) =>
210− Matrix e cols a ->
192+ Matrix e cols rows1 ->
211− Matrix e cols b ->
193+ Matrix e cols rows2 ->
212− Matrix e cols (a + b)
194+ Matrix e cols rows3
213−(===) = fork
195+(===) = split
214
196
215-- Construction
197-- Construction
216
198
217−fromLists :: (FLN e cols rows) => [[e]] -> Matrix e cols rows
199+fromLists :: (I.FromLists e (I.FromNat cols) (I.FromNat rows)) => [[e]] -> Matrix e cols rows
218fromLists = M . I.fromLists
200fromLists = M . I.fromLists
219
201
220−-- | Matrix builder function. Constructs a matrix provided with
221−-- a construction function that operates with indices.
222−matrixBuilder' ::
202+matrixBuilder ::
223− (FLN e cols rows, CountableN cols, CountableN rows)
203+ (I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
204+ ((Int, Int) -> e) ->
224− => ((Int, Int) -> e) -> Matrix e cols rows
205+ Matrix e cols rows
225−matrixBuilder' = M . I.matrixBuilder'
206+matrixBuilder = M . I.matrixBuilder
226
207
227−col :: (I.FL () (I.FromNat rows)) => [e] -> Matrix e 1 rows
208+col :: (I.FromLists e () (I.FromNat rows)) => [e] -> Matrix e 1 rows
228col = M . I.col
209col = M . I.col
229
210
230−row :: (I.FL (I.FromNat cols) ()) => [e] -> Matrix e cols 1
211+row :: (I.FromLists e (I.FromNat cols) ()) => [e] -> Matrix e cols 1
231row = M . I.row
212row = M . I.row
232
213
233−fromF' ::
214+fromF ::
234− ( Liftable e a b,
215+ ( Bounded a,
235− CountableN cols,
216+ Bounded b,
236− CountableN rows,
217+ Enum a,
237− FLN e rows cols
218+ Enum b,
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)
238 ) =>
225 ) =>
239 (a -> b) ->
226 (a -> b) ->
240 Matrix e cols rows
227 Matrix e cols rows
241−fromF' = M . I.fromF'
228+fromF = M . I.fromF
242
229
243−fromF ::
230+fromF' ::
244− ( Liftable e a b,
231+ ( Bounded a,
245− CountableNz a,
232+ Bounded b,
246− CountableNz b,
233+ Enum a,
247− FLNz e b a
234+ Enum b,
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)
248 ) =>
241 ) =>
249 (a -> b) ->
242 (a -> b) ->
250 Matrix e (I.Count a) (I.Count b)
243 Matrix e (I.Count a) (I.Count b)
251−fromF = undefined -- M . I.fromF
244+fromF' = M . I.fromF'
252
245
253-- Conversion
246-- Conversion
254
247
@@ -261,21 +254,21 @@ toList (M m) = I.toList m
261-- Zeros Matrix
254-- Zeros Matrix
262
255
263zeros ::
256zeros ::
264− (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
257+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
265 Matrix e cols rows
258 Matrix e cols rows
266zeros = M I.zeros
259zeros = M I.zeros
267
260
268-- Ones Matrix
261-- Ones Matrix
269
262
270ones ::
263ones ::
271− (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
264+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
272 Matrix e cols rows
265 Matrix e cols rows
273ones = M I.ones
266ones = M I.ones
274
267
275-- Const Matrix
268-- Const Matrix
276
269
277constant ::
270constant ::
278− (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
271+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat rows), KnownNat (I.Count (I.FromNat cols)), KnownNat (I.Count (I.FromNat rows))) =>
279 e ->
272 e ->
280 Matrix e cols rows
273 Matrix e cols rows
281constant = M . I.constant
274constant = M . I.constant
@@ -284,52 +277,40 @@ constant = M . I.constant
284
277
285bang ::
278bang ::
286 forall e cols.
279 forall e cols.
287− (Num e, Enum e, I.FL (I.FromNat cols) (), CountableN cols) =>
280+ (Num e, Enum e, I.FromLists e (I.FromNat cols) (), KnownNat (I.Count (I.FromNat cols))) =>
288 Matrix e cols 1
281 Matrix e cols 1
289bang = M I.bang
282bang = M I.bang
290
283
291−-- iden Matrix
284+-- Identity Matrix
292
285
293−iden ::
286+identity ::
294− (Num e, FLN e cols cols, CountableN cols) =>
287+ (Num e, I.FromLists e (I.FromNat cols) (I.FromNat cols), KnownNat (I.Count (I.FromNat cols))) =>
295 Matrix e cols cols
288 Matrix e cols cols
296−iden = M I.iden
289+identity = M I.identity
297
290
298-- Matrix composition (MMM)
291-- Matrix composition (MMM)
299
292
300comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
293comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
301comp (M a) (M b) = M (I.comp a b)
294comp (M a) (M b) = M (I.comp a b)
302
295
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−
317p1 ::
296p1 ::
318 ( Num e,
297 ( Num e,
319− CountableDimsN n m,
298+ KnownNat (I.Count (I.FromNat n)),
320− FLN e n m,
299+ KnownNat (I.Count (I.FromNat m)),
321− FLN e m m,
300+ I.FromLists e (I.FromNat n) (I.FromNat m),
322− TrivialE m n
301+ I.FromLists e (I.FromNat m) (I.FromNat m),
302+ I.FromNat (m + n) ~ Either (I.FromNat m) (I.FromNat n)
323 ) =>
303 ) =>
324 Matrix e (m + n) m
304 Matrix e (m + n) m
325p1 = M I.p1
305p1 = M I.p1
326
306
327p2 ::
307p2 ::
328 ( Num e,
308 ( Num e,
329− CountableDimsN n m,
309+ KnownNat (I.Count (I.FromNat n)),
330− FLN e m n,
310+ KnownNat (I.Count (I.FromNat m)),
331− FLN e n n,
311+ I.FromLists e (I.FromNat m) (I.FromNat n),
332− TrivialE m n
312+ I.FromLists e (I.FromNat n) (I.FromNat n),
313+ I.FromNat (m + n) ~ Either (I.FromNat m) (I.FromNat n)
333 ) =>
314 ) =>
334 Matrix e (m + n) n
315 Matrix e (m + n) n
335p2 = M I.p2
316p2 = M I.p2
@@ -338,125 +319,138 @@ p2 = M I.p2
338
319
339i1 ::
320i1 ::
340 ( Num e,
321 ( Num e,
341− CountableDimsN n rows,
322+ KnownNat (I.Count (I.FromNat n)),
342− FLN e n rows,
323+ KnownNat (I.Count (I.FromNat rows)),
343− FLN e rows rows,
324+ I.FromLists e (I.FromNat n) (I.FromNat rows),
344− TrivialE rows n
325+ I.FromLists e (I.FromNat rows) (I.FromNat rows),
326+ I.FromNat (rows + n) ~ Either (I.FromNat rows) (I.FromNat n)
345 ) =>
327 ) =>
346 Matrix e rows (rows + n)
328 Matrix e rows (rows + n)
347i1 = tr p1
329i1 = tr p1
348
330
349i2 ::
331i2 ::
350 ( Num e,
332 ( Num e,
351− CountableDimsN rows m,
333+ KnownNat (I.Count (I.FromNat rows)),
352− FLN e m rows,
334+ KnownNat (I.Count (I.FromNat m)),
353− FLN e rows rows,
335+ I.FromLists e (I.FromNat m) (I.FromNat rows),
354− TrivialE m rows
336+ I.FromLists e (I.FromNat rows) (I.FromNat rows),
337+ I.FromNat (m + rows) ~ Either (I.FromNat m) (I.FromNat rows)
355 ) =>
338 ) =>
356 Matrix e rows (m + rows)
339 Matrix e rows (m + rows)
357i2 = tr p2
340i2 = tr p2
358
341
359-- Dimensions
342-- Dimensions
360
343
361−rows :: (CountableN rows) => Matrix e cols rows -> Int
344+rows :: (KnownNat (I.Count (I.FromNat rows))) => Matrix e cols rows -> Int
362rows (M m) = I.rows m
345rows (M m) = I.rows m
363
346
364−columns :: (CountableN cols) => Matrix e cols rows -> Int
347+columns :: (KnownNat (I.Count (I.FromNat cols))) => Matrix e cols rows -> Int
365columns (M m) = I.columns m
348columns (M m) = I.columns m
366
349
350+-- Coproduct Bifunctor
351+
367infixl 5 -|-
352infixl 5 -|-
368
353
369−-- | Coproduct Bifunctor (Direct sum)
370(-|-) ::
354(-|-) ::
371 ( Num e,
355 ( Num e,
372− CountableDimsN j k,
356+ KnownNat (I.Count (I.FromNat j)),
373− FLN e k k,
357+ KnownNat (I.Count (I.FromNat k)),
374− FLN e j k,
358+ I.FromLists e (I.FromNat k) (I.FromNat k),
375− FLN e k j,
359+ I.FromLists e (I.FromNat j) (I.FromNat k),
376− FLN e j j,
360+ I.FromLists e (I.FromNat k) (I.FromNat j),
377− TrivialE n m,
361+ I.FromLists e (I.FromNat j) (I.FromNat j),
378− TrivialE k j
362+ I.FromNat (n + m) ~ Either (I.FromNat n) (I.FromNat m),
363+ I.FromNat (k + j) ~ Either (I.FromNat k) (I.FromNat j)
379 ) =>
364 ) =>
380 Matrix e n k ->
365 Matrix e n k ->
381 Matrix e m j ->
366 Matrix e m j ->
382 Matrix e (n + m) (k + j)
367 Matrix e (n + m) (k + j)
383(-|-) (M a) (M b) = M ((I.-|-) a b)
368(-|-) (M a) (M b) = M ((I.-|-) a b)
384
369
385−-- | Khatri Rao Product first projection
370+-- Khatri Rao Product and projections
386−fstM ::
371+
372+kp1 ::
387 forall e m k .
373 forall e m k .
388 ( Num e,
374 ( Num e,
389− CountableDimsN m k,
375+ KnownNat (I.Count (I.FromNat m)),
390− CountableN (m * k),
376+ KnownNat (I.Count (I.FromNat k)),
391− FLN e (m * k) m,
377+ KnownNat (I.Count (I.FromNat (m * k))),
392− TrivialP m k
378+ I.FromLists e (I.FromNat (m * k)) (I.FromNat m),
379+ I.FromNat (m * k) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat k))
393 ) => Matrix e (m * k) m
380 ) => Matrix e (m * k) m
394−fstM = M (I.fstM @e @(I.FromNat m) @(I.FromNat k))
381+kp1 = M (I.kp1 @e @(I.FromNat m) @(I.FromNat k))
395
382
396−-- | Khatri Rao Product second projection
383+kp2 ::
397−sndM ::
398 forall e m k.
384 forall e m k.
399 ( Num e,
385 ( Num e,
400− CountableDimsN k m,
386+ KnownNat (I.Count (I.FromNat k)),
401− FLN e (m * k) k,
387+ I.FromLists e (I.FromNat (m * k)) (I.FromNat k),
402− CountableN (m * k),
388+ KnownNat (I.Count (I.FromNat m)),
403− TrivialP m k
389+ KnownNat (I.Count (I.FromNat (m * k))),
390+ I.FromNat (m * k) ~ I.FromNat (I.Count (I.FromNat m) * I.Count (I.FromNat k))
404 ) => Matrix e (m * k) k
391 ) => Matrix e (m * k) k
405−sndM = M (I.sndM @e @(I.FromNat m) @(I.FromNat k))
392+kp2 = M (I.kp2 @e @(I.FromNat m) @(I.FromNat k))
406
393
407−-- | Khatri Rao Product
394+khatri ::
408−kr ::
409 forall e cols a b.
395 forall e cols a b.
410 ( Num e,
396 ( Num e,
411− CountableDimsN a b,
397+ KnownNat (I.Count (I.FromNat a)),
412− CountableN (a * b),
398+ KnownNat (I.Count (I.FromNat b)),
413− FLN e (a * b) a,
399+ KnownNat (I.Count (I.FromNat (a * b))),
414− FLN e (a * b) b,
400+ I.FromLists e (I.FromNat (a * b)) (I.FromNat a),
415− TrivialP a b
401+ I.FromLists e (I.FromNat (a * b)) (I.FromNat b),
402+ I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))
416 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)
403 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)
417−kr a b =
404+khatri a b =
418− let fstM' = fstM @e @a @b
405+ let kp1' = kp1 @e @a @b
419− sndM' = sndM @e @a @b
406+ kp2' = kp2 @e @a @b
420− in comp (tr fstM') a * comp (tr sndM') b
407+ in comp (tr kp1') a * comp (tr kp2') b
408+
409+-- Product Bifunctor (Kronecker)
421
410
422−-- | Product Bifunctor (Kronecker)
423infixl 4 ><
411infixl 4 ><
424
412
425(><) ::
413(><) ::
426 forall e m p n q.
414 forall e m p n q.
427 ( Num e,
415 ( Num e,
428− CountableDimsN m n,
416+ KnownNat (I.Count (I.FromNat m)),
429− CountableDimsN p q,
417+ KnownNat (I.Count (I.FromNat n)),
430− CountableDimsN (m * n) (p * q),
418+ KnownNat (I.Count (I.FromNat p)),
431− FLN e (m * n) m,
419+ KnownNat (I.Count (I.FromNat q)),
432− FLN e (m * n) n,
420+ KnownNat (I.Count (I.FromNat (m * n))),
433− FLN e (p * q) p,
421+ KnownNat (I.Count (I.FromNat (p * q))),
434− FLN e (p * q) q,
422+ I.FromLists e (I.FromNat (m * n)) (I.FromNat m),
435− TrivialP m n,
423+ I.FromLists e (I.FromNat (m * n)) (I.FromNat n),
436− TrivialP p q
424+ I.FromLists e (I.FromNat (p * q)) (I.FromNat p),
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))
437 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)
428 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)
438(><) a b =
429(><) a b =
439− let fstM' = fstM @e @m @n
430+ let kp1' = kp1 @e @m @n
440− sndM' = sndM @e @m @n
431+ kp2' = kp2 @e @m @n
441− in kr (comp a fstM') (comp b sndM')
432+ in khatri (comp a kp1') (comp b kp2')
433+
434+-- Matrix abide Junc Split
442
435
443−-- | Matrix abide Join Fork
444−abideJF :: Matrix e cols rows -> Matrix e cols rows
436+abideJS :: Matrix e cols rows -> Matrix e cols rows
445−abideJF (M m) = M (I.abideJF m)
437+abideJS (M m) = M (I.abideJS m)
446
438
447−-- | Matrix abide Fork Join
439+-- Matrix abide Split Junc
440+
448−abideFJ :: Matrix e cols rows -> Matrix e cols rows
441+abideSJ :: Matrix e cols rows -> Matrix e cols rows
449−abideFJ (M m) = M (I.abideFJ m)
442+abideSJ (M m) = M (I.abideSJ m)
443+
444+-- Matrix transposition
450
445
451−-- | Matrix transposition
452tr :: Matrix e cols rows -> Matrix e rows cols
446tr :: Matrix e cols rows -> Matrix e rows cols
453tr (M m) = M (I.tr m)
447tr (M m) = M (I.tr m)
454
448
455-- Selective 'select' operator
449-- Selective 'select' operator
456select ::
450select ::
457 ( Num e,
451 ( Num e,
458− FLN e rows1 rows1,
452+ I.FromLists e (I.FromNat rows1) (I.FromNat rows1),
459− CountableN rows1,
453+ KnownNat (I.Count (I.FromNat rows1)),
460 I.FromNat rows2 ~ I.FromNat rows1,
454 I.FromNat rows2 ~ I.FromNat rows1,
461 I.FromNat cols1 ~ I.FromNat cols2,
455 I.FromNat cols1 ~ I.FromNat cols2,
462 I.FromNat rows3 ~ Either (I.FromNat cols3) (I.FromNat rows1)
456 I.FromNat rows3 ~ Either (I.FromNat cols3) (I.FromNat rows1)
@@ -467,11 +461,14 @@ select (M m) (M y) = M (I.select m y)
467
461
468cond ::
462cond ::
469 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,
463 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,
470− CountableN cols,
464+ KnownNat (I.Count (I.FromNat cols)),
471− I.FL () (I.FromNat cols),
465+ I.FromLists e () (I.FromNat cols),
472− I.FL (I.FromNat cols) (),
466+ I.FromLists e (I.FromNat cols) (),
473− FLN e cols cols,
467+ I.FromLists e (I.FromNat cols) (I.FromNat cols),
474− Liftable e a Bool
468+ Bounded a,
469+ Enum a,
470+ Num e,
471+ Ord e
475 )
472 )
476 =>
473 =>
477 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
474 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
@@ -479,12 +476,8 @@ cond p (M a) (M b) = M (I.cond p a b)
479
476
480-- Pretty print
477-- Pretty print
481
478
482−pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String
479+pretty :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> String
483pretty (M m) = I.pretty m
480pretty (M m) = I.pretty m
484
481
485−prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()
482+prettyPrint :: (KnownNat (I.Count (I.FromNat cols)), Show e) => Matrix e cols rows -> IO ()
486prettyPrint (M m) = I.prettyPrint m
483prettyPrint (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+240−283
@@ -1,11 +1,10 @@
1−{-# LANGUAGE ConstraintKinds #-}
2−{-# LANGUAGE TypeFamilies #-}
3{-# LANGUAGE AllowAmbiguousTypes #-}
1{-# LANGUAGE AllowAmbiguousTypes #-}
4{-# LANGUAGE DataKinds #-}
2{-# LANGUAGE DataKinds #-}
5{-# LANGUAGE DerivingVia #-}
3{-# LANGUAGE DerivingVia #-}
6{-# LANGUAGE FlexibleContexts #-}
4{-# LANGUAGE FlexibleContexts #-}
7{-# LANGUAGE FlexibleInstances #-}
5{-# LANGUAGE FlexibleInstances #-}
8{-# LANGUAGE GADTs #-}
6{-# LANGUAGE GADTs #-}
7+{-# LANGUAGE KindSignatures #-}
9{-# LANGUAGE MultiParamTypeClasses #-}
8{-# LANGUAGE MultiParamTypeClasses #-}
10{-# LANGUAGE NoStarIsType #-}
9{-# LANGUAGE NoStarIsType #-}
11{-# LANGUAGE RankNTypes #-}
10{-# LANGUAGE RankNTypes #-}
@@ -56,24 +55,15 @@ module LAoP.Matrix.Type
56 -- * Type safe matrix representation
55 -- * Type safe matrix representation
57 Matrix (..),
56 Matrix (..),
58
57
59− -- * Constraint type aliases
60− Countable,
61− CountableDims,
62− CountableN,
63− CountableDimsN,
64− FLN,
65− Liftable,
66− Trivial,
67− TrivialP,
68−
69 -- * Type aliases
58 -- * Type aliases
70 Zero,
59 Zero,
71 One,
60 One,
72
61
73 -- * Primitives
62 -- * Primitives
63+ empty,
74 one,
64 one,
75− join,
65+ junc,
76− fork,
66+ split,
77
67
78 -- * Auxiliary type families
68 -- * Auxiliary type families
79 I.FromNat,
69 I.FromNat,
@@ -85,27 +75,24 @@ module LAoP.Matrix.Type
85 fromLists,
75 fromLists,
86 toLists,
76 toLists,
87 toList,
77 toList,
88− matrixBuilder',
89 matrixBuilder,
78 matrixBuilder,
90 row,
79 row,
91 col,
80 col,
92 zeros,
81 zeros,
93 ones,
82 ones,
94 bang,
83 bang,
95− point,
96 constant,
84 constant,
97
85
98 -- * Functor instance equivalent function
86 -- * Functor instance equivalent function
99 fmapM,
87 fmapM,
100− bimapM,
101
88
102 -- * Applicative/Monoidal instance equivalent functions
89 -- * Applicative/Monoidal instance equivalent functions
103 unitM,
90 unitM,
104 multM,
91 multM,
105
92
106 -- * Selective equivalent instance function
93 -- * Selective equivalent instance function
107− selectM,
94+ selectM,
108−
95+
109 -- * Monad equivalent instance function
96 -- * Monad equivalent instance function
110 returnM,
97 returnM,
111 bindM,
98 bindM,
@@ -113,34 +100,25 @@ module LAoP.Matrix.Type
113 -- * Misc
100 -- * Misc
114 -- ** Get dimensions
101 -- ** Get dimensions
115 columns,
102 columns,
116− columns',
117 rows,
103 rows,
118− rows',
119
104
120 -- ** Matrix Transposition
105 -- ** Matrix Transposition
121 tr,
106 tr,
122
107
123− -- ** Scalar multiplication/division of matrices
124− (.|),
125− (./),
126−
127 -- ** McCarthy's Conditional
108 -- ** McCarthy's Conditional
128 cond,
109 cond,
129
110
130 -- ** Matrix "abiding"
111 -- ** Matrix "abiding"
131− abideJF,
112+ abideJS,
132− abideFJ,
113+ abideSJ,
133−
134− -- ** Zip Matrices
135− zipWithM,
136
114
137 -- * Biproduct approach
115 -- * Biproduct approach
138− -- ** Fork
116+ -- ** Split
139 (===),
117 (===),
140 -- *** Projections
118 -- *** Projections
141 p1,
119 p1,
142 p2,
120 p2,
143− -- ** Join
121+ -- ** Junc
144 (|||),
122 (|||),
145 -- *** Injections
123 -- *** Injections
146 i1,
124 i1,
@@ -154,11 +132,11 @@ module LAoP.Matrix.Type
154 -- | Note that given the restrictions imposed it is not possible to
132 -- | Note that given the restrictions imposed it is not possible to
155 -- implement the standard type classes present in standard Haskell.
133 -- implement the standard type classes present in standard Haskell.
156 -- *** Matrix pairing projections
134 -- *** Matrix pairing projections
157− fstM,
135+ kp1,
158− sndM,
136+ kp2,
159
137
160 -- *** Matrix pairing
138 -- *** Matrix pairing
161− kr,
139+ khatri,
162
140
163 -- * Matrix composition and lifting
141 -- * Matrix composition and lifting
164
142
@@ -166,10 +144,10 @@ module LAoP.Matrix.Type
166
144
167 -- | Note that given the restrictions imposed it is not possible to
145 -- | Note that given the restrictions imposed it is not possible to
168 -- implement the standard type classes present in standard Haskell.
146 -- implement the standard type classes present in standard Haskell.
169− iden,
147+ identity,
170 comp,
148 comp,
171− fromF',
172 fromF,
149 fromF,
150+ fromF',
173
151
174 -- * Relation
152 -- * Relation
175 toRel,
153 toRel,
@@ -183,44 +161,23 @@ where
183import Data.Void
161import Data.Void
184import Data.Proxy
162import Data.Proxy
185import Data.Kind
163import Data.Kind
186−import GHC.TypeLits
164+import GHC.TypeLits
187import Control.DeepSeq
165import Control.DeepSeq
188import LAoP.Utils
166import LAoP.Utils
167+import qualified Control.Category as C
189import qualified LAoP.Matrix.Internal as I
168import qualified LAoP.Matrix.Internal as I
190−import Prelude hiding (id, (.))
191
169
192newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))
170newtype Matrix e (cols :: Type) (rows :: Type) = M (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))
171 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))
194
172
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−
207−-- | It is possible to implement a constrained version of the category type
173+-- | It isn't possible to implement the 'id' function so it's
208−-- class.
174+-- implementation is 'undefined'. However 'comp' can be and this partial
175+-- class implementation exists just to make the code more readable.
176+--
177+-- Please use 'identity' instead.
209−instance (Num e) => Category (Matrix e) where
178+instance (Num e) => C.Category (Matrix e) where
210− type Object (Matrix e) a = (FLN a a, CountableN a)
179+ id = undefined
211− id = iden
212− (.) = comp
180+ (.) = 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)
224
181
225-- | Zero type alias
182-- | Zero type alias
226type Zero = Void
183type Zero = Void
@@ -230,75 +187,92 @@ type One = ()
230
187
231-- Primitives
188-- Primitives
232
189
190+-- | Empty matrix constructor
191+empty :: Matrix e Zero Zero
192+empty = M I.Empty
193+
233-- | Unit matrix constructor
194-- | Unit matrix constructor
234one :: e -> Matrix e One One
195one :: e -> Matrix e One One
235one = M . I.One
196one = M . I.One
236
197
237−-- | Matrix 'Join' constructor
198+-- | Matrix 'Junc' constructor
238−join ::
199+junc ::
200+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
239 Matrix e a rows ->
201 Matrix e a rows ->
240 Matrix e b rows ->
202 Matrix e b rows ->
241 Matrix e (Either a b) rows
203 Matrix e (Either a b) rows
242−join (M a) (M b) = M (I.Join a b)
204+junc (M a) (M b) = M (I.Junc a b)
243
205
244infixl 3 |||
206infixl 3 |||
245−-- | Matrix 'Join' constructor
207+-- | Matrix 'Junc' constructor
246(|||) ::
208(|||) ::
209+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
247 Matrix e a rows ->
210 Matrix e a rows ->
248 Matrix e b rows ->
211 Matrix e b rows ->
249 Matrix e (Either a b) rows
212 Matrix e (Either a b) rows
250−(|||) = join
213+(|||) = junc
251
214
252−-- | Matrix 'Fork' constructor
215+-- | Matrix 'Split' constructor
253−fork ::
216+split ::
217+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
254 Matrix e cols a ->
218 Matrix e cols a ->
255 Matrix e cols b ->
219 Matrix e cols b ->
256 Matrix e cols (Either a b)
220 Matrix e cols (Either a b)
257−fork (M a) (M b) = M (I.Fork a b)
221+split (M a) (M b) = M (I.Split a b)
258
222
259infixl 2 ===
223infixl 2 ===
260−-- | Matrix 'Fork' constructor
224+-- | Matrix 'Split' constructor
261(===) ::
225(===) ::
226+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
262 Matrix e cols a ->
227 Matrix e cols a ->
263 Matrix e cols b ->
228 Matrix e cols b ->
264 Matrix e cols (Either a b)
229 Matrix e cols (Either a b)
265−(===) = fork
230+(===) = split
266
231
267-- Functor hierarchy
232-- Functor hierarchy
268
233
269-- | Functor instance equivalent function
234-- | Functor instance equivalent function
270−fmapM ::
235+fmapM ::
271− ( Liftable e a b,
236+ ( Bounded a,
272− CountableDimsN a b,
237+ Bounded b,
273− FLN b a
238+ Enum 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)
274 )
246 )
275 =>
247 =>
276 (a -> b) -> Matrix e c a -> Matrix e c b
248 (a -> b) -> Matrix e c a -> Matrix e c b
277−fmapM f m = fromF f . m
249+fmapM f m = fromF' f `comp` m
278
250
279-- | Applicative instance equivalent 'unit' function,
251-- | Applicative instance equivalent 'unit' function,
280unitM :: (Num e) => Matrix e () ()
252unitM :: (Num e) => Matrix e () ()
281unitM = one 1
253unitM = one 1
282
254
283-- | Applicative instance equivalent 'unit' function,
255-- | Applicative instance equivalent 'unit' function,
284−multM ::
256+multM ::
285− ( CountableDimsN a b,
257+ ( KnownNat (I.Count (I.Normalize a)),
286− CountableN (a, b),
258+ KnownNat (I.Count (I.Normalize b)),
259+ KnownNat (I.Count (I.Normalize (a, b))),
287 Num e,
260 Num e,
288− FLN (a, b) a,
261+ I.FromLists e (I.Normalize (a, b)) (I.Normalize a),
289− FLN (a, b) b,
262+ I.FromLists e (I.Normalize (a, b)) (I.Normalize b),
290− TrivialP a b
263+ I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
291 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)
264 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)
292−multM = kr
265+multM a b = khatri a b
266+
293
267
294-- | Monad instance equivalent 'return' function,
268-- | Monad instance equivalent 'return' function,
295−returnM ::
269+returnM ::
296− forall e a .
270+ forall e a .
297− ( Num e,
271+ ( Num e,
298− Enum e,
272+ Enum e,
299− Enum a,
273+ Enum a,
300− FLN () a,
274+ I.FromLists e () (I.Normalize a),
301− Countable a
275+ KnownNat (I.Count a)
302 ) => a -> Matrix e One a
276 ) => a -> Matrix e One a
303returnM a = col l
277returnM a = col l
304 where
278 where
@@ -314,64 +288,73 @@ bindM = flip comp
314
288
315-- | Build a matrix out of a list of list of elements. Throws a runtime
289-- | Build a matrix out of a list of list of elements. Throws a runtime
316-- error if the dimensions do not match.
290-- error if the dimensions do not match.
317−fromLists :: (FLN cols rows) => [[e]] -> Matrix e cols rows
291+fromLists :: (I.FromLists e (I.Normalize cols) (I.Normalize rows)) => [[e]] -> Matrix e cols rows
318fromLists = M . I.fromLists
292fromLists = M . I.fromLists
319
293
320-- | Matrix builder function. Constructs a matrix provided with
294-- | Matrix builder function. Constructs a matrix provided with
321−-- a construction function that operates with indices.
295+-- a construction function.
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.
329matrixBuilder ::
296matrixBuilder ::
330− ( FLN a b,
297+ (I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
331− Enum a,
298+ ((Int, Int) -> e) ->
332− Enum b,
333− Bounded a,
334− Bounded b,
335− Eq a,
336− Countable b
337− ) => ((a, b) -> e) -> Matrix e a b
299+ Matrix e cols rows
338−matrixBuilder f = M (I.matrixBuilder f)
300+matrixBuilder = M . I.matrixBuilder
339
301
340-- | Constructs a column vector matrix
302-- | Constructs a column vector matrix
341−col :: (FLN () rows) => [e] -> Matrix e One rows
303+col :: (I.FromLists e () (I.Normalize rows)) => [e] -> Matrix e One rows
342col = M . I.col
304col = M . I.col
343
305
344-- | Constructs a row vector matrix
306-- | Constructs a row vector matrix
345−row :: (FLN cols ()) => [e] -> Matrix e cols One
307+row :: (I.FromLists e (I.Normalize cols) ()) => [e] -> Matrix e cols One
346row = M . I.row
308row = M . I.row
347
309
348-- | Lifts functions to matrices with arbitrary dimensions.
310-- | Lifts functions to matrices with arbitrary dimensions.
349--
311--
350-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
312-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
351-- types @a@ or @b@ allows.
313-- types @a@ or @b@ allows.
352−fromF' ::
314+fromF ::
353− ( Liftable e a b,
315+ ( Bounded a,
354− CountableDimsN cols rows,
316+ Bounded b,
355− FLN rows cols
317+ Enum a,
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)
356 ) =>
325 ) =>
326+ (a -> b) ->
357− (a -> b) -> Matrix e cols rows
327+ Matrix e cols rows
358−fromF' = M . I.fromF'
328+fromF = M . I.fromF
359
329
360-- | Lifts functions to matrices with dimensions matching @a@ and @b@
330-- | Lifts functions to matrices with dimensions matching @a@ and @b@
361-- cardinality's.
331-- cardinality's.
362−fromF ::
332+fromF' ::
363− ( Liftable e a b,
333+ ( Bounded a,
364− CountableDimsN a b,
334+ Bounded b,
365− FLN b a
335+ Enum 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)
366 ) =>
343 ) =>
344+ (a -> b) ->
367− (a -> b) -> Matrix e a b
345+ Matrix e a b
368−fromF = M . I.fromF
346+fromF' = M . I.fromF'
369
347
370-- | Lifts relation functions to Boolean Matrix
348-- | Lifts relation functions to Boolean Matrix
371toRel ::
349toRel ::
372− ( Liftable (Natural 0 1) a b,
350+ ( Bounded a,
373− CountableDims a b,
351+ Bounded b,
374− FLN b a
352+ Enum 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)
375 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b
358 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b
376toRel = M . I.toRel
359toRel = M . I.toRel
377
360
@@ -389,8 +372,8 @@ toList (M m) = I.toList m
389
372
390-- | The zero matrix. A matrix wholly filled with zeros.
373-- | The zero matrix. A matrix wholly filled with zeros.
391zeros ::
374zeros ::
392− (Num e, FLN cols rows, CountableDimsN cols rows)
375+ (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
393− => Matrix e cols rows
376+ Matrix e cols rows
394zeros = M I.zeros
377zeros = M I.zeros
395
378
396-- Ones Matrix
379-- Ones Matrix
@@ -399,8 +382,8 @@ zeros = M I.zeros
399--
382--
400-- Also known as T (Top) matrix.
383-- Also known as T (Top) matrix.
401ones ::
384ones ::
402− (Num e, FLN cols rows, CountableDimsN cols rows)
385+ (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
403− => Matrix e cols rows
386+ Matrix e cols rows
404ones = M I.ones
387ones = M I.ones
405
388
406-- Const Matrix
389-- Const Matrix
@@ -408,8 +391,9 @@ ones = M I.ones
408-- | The constant matrix constructor. A matrix wholly filled with a given
391-- | The constant matrix constructor. A matrix wholly filled with a given
409-- value.
392-- value.
410constant ::
393constant ::
411− (Num e, FLN cols rows, CountableDimsN cols rows)
394+ (Num e, I.FromLists e (I.Normalize cols) (I.Normalize rows), KnownNat (I.Count (I.Normalize cols)), KnownNat (I.Count (I.Normalize rows))) =>
395+ e ->
412− => e -> Matrix e cols rows
396+ Matrix e cols rows
413constant = M . I.constant
397constant = M . I.constant
414
398
415-- Bang Matrix
399-- Bang Matrix
@@ -417,30 +401,17 @@ constant = M . I.constant
417-- | The T (Top) row vector matrix.
401-- | The T (Top) row vector matrix.
418bang ::
402bang ::
419 forall e cols.
403 forall e cols.
420− (Num e, Enum e, FLN cols (), CountableN cols) =>
404+ (Num e, Enum e, I.FromLists e (I.Normalize cols) (), KnownNat (I.Count (I.Normalize cols))) =>
421 Matrix e cols One
405 Matrix e cols One
422bang = M I.bang
406bang = M I.bang
423
407
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−
436−-- iden Matrix
408+-- Identity Matrix
437
409
438−-- | iden matrix
410+-- | Identity matrix
439−iden ::
411+identity ::
440− (Num e, FLN a a, CountableN a) =>
412+ (Num e, I.FromLists e (I.Normalize a) (I.Normalize a), KnownNat (I.Count (I.Normalize a))) =>
441 Matrix e a a
413 Matrix e a a
442−iden = M I.iden
414+identity = M I.identity
443−{-# NOINLINE iden #-}
444
415
445-- Matrix composition (MMM)
416-- Matrix composition (MMM)
446
417
@@ -450,32 +421,15 @@ iden = M I.iden
450-- from LAoP.
421-- from LAoP.
451comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
422comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
452comp (M a) (M b) = M (I.comp a b)
423comp (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)
472
424
473-- | Biproduct first component projection
425-- | Biproduct first component projection
474p1 ::
426p1 ::
475 ( Num e,
427 ( Num e,
476− CountableDimsN n m,
428+ KnownNat (I.Count (I.Normalize n)),
477− FLN n m,
429+ KnownNat (I.Count (I.Normalize m)),
478− FLN m m
430+ I.FromLists e (I.Normalize n) (I.Normalize m),
431+ I.FromLists e (I.Normalize m) (I.Normalize m),
432+ I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
479 ) =>
433 ) =>
480 Matrix e (Either m n) m
434 Matrix e (Either m n) m
481p1 = M I.p1
435p1 = M I.p1
@@ -483,9 +437,11 @@ p1 = M I.p1
483-- | Biproduct second component projection
437-- | Biproduct second component projection
484p2 ::
438p2 ::
485 ( Num e,
439 ( Num e,
486− CountableDimsN n m,
440+ KnownNat (I.Count (I.Normalize n)),
487− FLN m n,
441+ KnownNat (I.Count (I.Normalize m)),
488− FLN n n
442+ I.FromLists e (I.Normalize m) (I.Normalize n),
443+ I.FromLists e (I.Normalize n) (I.Normalize n),
444+ I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
489 ) =>
445 ) =>
490 Matrix e (Either m n) n
446 Matrix e (Either m n) n
491p2 = M I.p2
447p2 = M I.p2
@@ -495,9 +451,11 @@ p2 = M I.p2
495-- | Biproduct first component injection
451-- | Biproduct first component injection
496i1 ::
452i1 ::
497 ( Num e,
453 ( Num e,
498− CountableDimsN n m,
454+ KnownNat (I.Count (I.Normalize n)),
499− FLN n m,
455+ KnownNat (I.Count (I.Normalize m)),
500− FLN m m
456+ I.FromLists e (I.Normalize n) (I.Normalize m),
457+ I.FromLists e (I.Normalize m) (I.Normalize m),
458+ I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
501 ) =>
459 ) =>
502 Matrix e m (Either m n)
460 Matrix e m (Either m n)
503i1 = tr p1
461i1 = tr p1
@@ -505,9 +463,11 @@ i1 = tr p1
505-- | Biproduct second component injection
463-- | Biproduct second component injection
506i2 ::
464i2 ::
507 ( Num e,
465 ( Num e,
508− CountableDimsN n m,
466+ KnownNat (I.Count (I.Normalize n)),
509− FLN m n,
467+ KnownNat (I.Count (I.Normalize m)),
510− FLN n n
468+ I.FromLists e (I.Normalize m) (I.Normalize n),
469+ I.FromLists e (I.Normalize n) (I.Normalize n),
470+ I.Normalize (Either m n) ~ Either (I.Normalize m) (I.Normalize n)
511 ) =>
471 ) =>
512 Matrix e n (Either m n)
472 Matrix e n (Either m n)
513i2 = tr p2
473i2 = tr p2
@@ -516,34 +476,24 @@ i2 = tr p2
516
476
517-- | Obtain the number of rows.
477-- | Obtain the number of rows.
518--
478--
519−-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
479+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
520−-- dimensions in constant time. For a version that doesn't require the
480+-- dimensions in constant time.
521−-- constraint see 'rows''.
522−rows :: (CountableN rows) => Matrix e cols rows -> Int
523−rows (M m) = I.rows m
524−
525−-- | Obtain the number of rows in an inefficient manner, but without any
526−-- constraints.
527--
481--
528−-- For a more efficient version see 'rows'.
482+-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
483+-- exchange for performance.
529−rows' :: Matrix e cols rows -> Int
484+rows :: (KnownNat (I.Count (I.Normalize rows))) => Matrix e cols rows -> Int
530−rows' (M m) = I.rows' m
485+rows (M m) = I.rows m
531
486
532-- | Obtain the number of columns.
487-- | Obtain the number of columns.
488+--
489+-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
490+-- dimensions in constant time.
533--
491--
534−-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
492+-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
535−-- dimensions in constant time. For a version that doesn't require the
493+-- exchange for performance.
536−-- constraint see 'columns''.
537−columns :: (CountableN cols) => Matrix e cols rows -> Int
494+columns :: (KnownNat (I.Count (I.Normalize cols))) => Matrix e cols rows -> Int
538columns (M m) = I.columns m
495columns (M m) = I.columns m
539
496
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−
547-- Coproduct Bifunctor
497-- Coproduct Bifunctor
548
498
549infixl 5 -|-
499infixl 5 -|-
@@ -551,11 +501,14 @@ infixl 5 -|-
551-- | Matrix coproduct functor also known as matrix direct sum.
501-- | Matrix coproduct functor also known as matrix direct sum.
552(-|-) ::
502(-|-) ::
553 ( Num e,
503 ( Num e,
554− CountableDimsN j k,
504+ KnownNat (I.Count (I.Normalize j)),
555− FLN k k,
505+ KnownNat (I.Count (I.Normalize k)),
556− FLN j k,
506+ I.FromLists e (I.Normalize k) (I.Normalize k),
557− FLN k j,
507+ I.FromLists e (I.Normalize j) (I.Normalize k),
558− FLN j j
508+ I.FromLists e (I.Normalize k) (I.Normalize 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)
559 ) =>
512 ) =>
560 Matrix e n k ->
513 Matrix e n k ->
561 Matrix e m j ->
514 Matrix e m j ->
@@ -565,98 +518,104 @@ infixl 5 -|-
565-- Khatri Rao Product and projections
518-- Khatri Rao Product and projections
566
519
567-- | Khatri Rao product first component projection matrix.
520-- | Khatri Rao product first component projection matrix.
568−fstM ::
521+kp1 ::
569 forall e m k .
522 forall e m k .
570 ( Num e,
523 ( Num e,
571− CountableDimsN m k,
524+ KnownNat (I.Count (I.Normalize m)),
572− CountableN (m, k),
525+ KnownNat (I.Count (I.Normalize k)),
573− FLN (m, k) m,
526+ KnownNat (I.Count (I.Normalize (m, k))),
574− TrivialP m k
527+ I.FromLists e (I.Normalize (m, k)) (I.Normalize m),
528+ I.Normalize (m, k) ~ I.Normalize (I.Normalize m, I.Normalize k)
575 ) => Matrix e (m, k) m
529 ) => Matrix e (m, k) m
576−fstM = M (I.fstM @e @(I.Normalize m) @(I.Normalize k))
530+kp1 = M (I.kp1 @e @(I.Normalize m) @(I.Normalize k))
577
531
578-- | Khatri Rao product second component projection matrix.
532-- | Khatri Rao product second component projection matrix.
579−sndM ::
533+kp2 ::
580 forall e m k.
534 forall e m k.
581 ( Num e,
535 ( Num e,
582− CountableDimsN k m,
536+ KnownNat (I.Count (I.Normalize k)),
583− CountableN (m, k),
537+ KnownNat (I.Count (I.Normalize m)),
584− FLN (m, k) k,
538+ KnownNat (I.Count (I.Normalize (m, k))),
585− TrivialP m k
539+ I.FromLists e (I.Normalize (m, k)) (I.Normalize k),
540+ I.Normalize (m, k) ~ I.Normalize (I.Normalize m, I.Normalize k)
586 ) => Matrix e (m, k) k
541 ) => Matrix e (m, k) k
587−sndM = M (I.sndM @e @(I.Normalize m) @(I.Normalize k))
542+kp2 = M (I.kp2 @e @(I.Normalize m) @(I.Normalize k))
588
543
589-- | Khatri Rao Matrix product also known as matrix pairing.
544-- | Khatri Rao Matrix product also known as matrix pairing.
590--
545--
591-- NOTE: That this is not a true categorical product, see for instance:
546-- NOTE: That this is not a true categorical product, see for instance:
592−--
547+--
593-- @
548-- @
594−-- | fstM . kr a b == a
549+-- | kp1 `comp` khatri a b == a
595−-- kr a b ==> |
550+-- khatri a b ==> |
596−-- | sndM . kr a b == b
551+-- | kp2 `comp` khatri a b == b
597-- @
552-- @
598--
553--
599-- __Emphasis__ on the implication symbol.
554-- __Emphasis__ on the implication symbol.
600−kr ::
555+khatri ::
601 forall e cols a b.
556 forall e cols a b.
602 ( Num e,
557 ( Num e,
603− CountableDimsN a b,
558+ KnownNat (I.Count (I.Normalize a)),
604− CountableN (a, b),
559+ KnownNat (I.Count (I.Normalize b)),
605− FLN (a, b) a,
560+ KnownNat (I.Count (I.Normalize (a, b))),
606− FLN (a, b) b,
561+ I.FromLists e (I.Normalize (a, b)) (I.Normalize a),
607− TrivialP a b
562+ I.FromLists e (I.Normalize (a, b)) (I.Normalize b),
563+ I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
608 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)
564 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)
609−kr a b =
565+khatri a b =
610− let fstM' = fstM @e @a @b
566+ let kp1' = kp1 @e @a @b
611− sndM' = sndM @e @a @b
567+ kp2' = kp2 @e @a @b
612− in comp (tr fstM') a * comp (tr sndM') b
568+ in comp (tr kp1') a * comp (tr kp2') b
613
569
614-- Product Bifunctor (Kronecker)
570-- Product Bifunctor (Kronecker)
615
571
616infixl 4 ><
572infixl 4 ><
617
573
618−-- | Matrix product functor also known as Kronecker product
574+-- | Matrix product functor also known as kronecker product
619(><) ::
575(><) ::
620 forall e m p n q.
576 forall e m p n q.
621 ( Num e,
577 ( Num e,
622− CountableDimsN m n,
578+ KnownNat (I.Count (I.Normalize m)),
623− CountableDimsN p q,
579+ KnownNat (I.Count (I.Normalize n)),
624− CountableDimsN (m, n) (p, q),
580+ KnownNat (I.Count (I.Normalize p)),
625− FLN (m, n) m,
581+ KnownNat (I.Count (I.Normalize q)),
626− FLN (m, n) n,
582+ KnownNat (I.Count (I.Normalize (m, n))),
627− FLN (p, q) p,
583+ KnownNat (I.Count (I.Normalize (p, q))),
628− FLN (p, q) q,
584+ I.FromLists e (I.Normalize (m, n)) (I.Normalize m),
629− TrivialP m n,
585+ I.FromLists e (I.Normalize (m, n)) (I.Normalize n),
630− TrivialP p q
586+ I.FromLists e (I.Normalize (p, q)) (I.Normalize p),
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)
631 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)
590 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)
632(><) a b =
591(><) a b =
633− let fstM' = fstM @e @m @n
592+ let kp1' = kp1 @e @m @n
634− sndM' = sndM @e @m @n
593+ kp2' = kp2 @e @m @n
635− in kr (comp a fstM') (comp b sndM')
594+ in khatri (comp a kp1') (comp b kp2')
636
595
637−-- Matrix abide Join Fork
596+-- Matrix abide Junc Split
638
597
639−-- | Matrix "abiding" followin the 'Join'-'Fork' abide law.
598+-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
640−--
599+--
641-- Law:
600-- Law:
642--
601--
643-- @
602-- @
644−-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)
603+-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)
645-- @
604-- @
646−abideJF :: Matrix e cols rows -> Matrix e cols rows
605+abideJS :: Matrix e cols rows -> Matrix e cols rows
647−abideJF (M m) = M (I.abideJF m)
606+abideJS (M m) = M (I.abideJS m)
648
607
649−-- Matrix abide Fork Join
608+-- Matrix abide Split Junc
650
609
651−-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
610+-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
652−--
611+--
653-- Law:
612-- Law:
654--
613--
655-- @
614-- @
656−-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)
615+-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)
657-- @
616-- @
658−abideFJ :: Matrix e cols rows -> Matrix e cols rows
617+abideSJ :: Matrix e cols rows -> Matrix e cols rows
659−abideFJ (M m) = M (I.abideFJ m)
618+abideSJ (M m) = M (I.abideSJ m)
660
619
661-- Matrix transposition
620-- Matrix transposition
662
621
@@ -668,10 +627,11 @@ tr (M m) = M (I.tr m)
668
627
669-- | Selective functors 'select' operator equivalent inspired by the
628-- | Selective functors 'select' operator equivalent inspired by the
670-- ArrowMonad solution presented in the paper.
629-- ArrowMonad solution presented in the paper.
671−selectM ::
630+selectM ::
672 ( Num e,
631 ( Num e,
673− FLN b b,
632+ I.FromLists e (I.Normalize b) (I.Normalize b),
674− CountableN b
633+ KnownNat (I.Count (I.Normalize b)),
634+ I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)
675 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
635 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
676selectM (M m) (M y) = M (I.select m y)
636selectM (M m) (M y) = M (I.select m y)
677
637
@@ -679,14 +639,15 @@ selectM (M m) (M y) = M (I.select m y)
679
639
680-- | McCarthy's Conditional expresses probabilistic choice.
640-- | McCarthy's Conditional expresses probabilistic choice.
681cond ::
641cond ::
682− ( Trivial a,
642+ ( I.Normalize (I.Normalize a) ~ I.Normalize a,
683− Trivial2 a,
643+ KnownNat (I.Count (I.Normalize a)),
684− Trivial3 a,
644+ I.FromLists e () (I.Normalize a),
685− CountableN a,
645+ I.FromLists e (I.Normalize a) (),
686− FLN () a,
646+ I.FromLists e (I.Normalize a) (I.Normalize a),
687− FLN a (),
647+ Bounded a,
688− FLN a a,
648+ Enum a,
689− Liftable e a Bool
649+ Num e,
650+ Ord e
690 )
651 )
691 =>
652 =>
692 (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b
653 (a -> Bool) -> Matrix e a b -> Matrix e a b -> Matrix e a b
@@ -695,13 +656,9 @@ cond p (M a) (M b) = M (I.cond p a b)
695-- Pretty print
656-- Pretty print
696
657
697-- | Matrix pretty printer
658-- | Matrix pretty printer
698−pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String
659+pretty :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> String
699pretty (M m) = I.pretty m
660pretty (M m) = I.pretty m
700
661
701-- | Matrix pretty printer
662-- | Matrix pretty printer
702−prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()
663+prettyPrint :: (KnownNat (I.Count (I.Normalize cols)), Show e) => Matrix e cols rows -> IO ()
703prettyPrint (M m) = I.prettyPrint m
664prettyPrint (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+9−76
@@ -30,22 +30,11 @@ 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−
43 -- * Primitives
33 -- * Primitives
34+ empty,
44 one,
35 one,
45− join,
36+ junc,
46− (|||),
37+ split,
47− fork,
48− (===),
49
38
50 -- * Auxiliary type families
39 -- * Auxiliary type families
51 FromNat,
40 FromNat,
@@ -55,73 +44,27 @@ module LAoP.Relation
55 -- * Matrix construction and conversion
44 -- * Matrix construction and conversion
56 FromLists,
45 FromLists,
57 fromLists,
46 fromLists,
58− fromF,
59− fromF',
60− toRel,
61 toLists,
47 toLists,
62 toList,
48 toList,
63− toBool,
49+ matrixBuilder,
64− pt,
65− belongs,
66− relationBuilder,
67 zeros,
50 zeros,
68 ones,
51 ones,
69 bang,
52 bang,
70− point,
71
53
72 -- * Relational operations
54 -- * Relational operations
73 conv,
55 conv,
74 intersection,
56 intersection,
75 union,
57 union,
76 sse,
58 sse,
77− implies,
78− iff,
79 ker,
59 ker,
80 img,
60 img,
81
61
82 -- * Taxonomy of binary relations
62 -- * Taxonomy of binary relations
83− injective,
84 entire,
63 entire,
85− simple,
64+ injective,
86 surjective,
65 surjective,
87− representation,
66+ simple,
88− function,
67+ bijective,
89− abstraction,
90− injection,
91− surjection,
92− bijection,
93− domain,
94− range,
95−
96− -- * Function division
97− divisionF,
98−
99− -- * Relation division
100− divR,
101− divL,
102− divS,
103− shrunkBy,
104− overriddenBy,
105−
106− -- * Relational pairing
107− splitR,
108−
109− -- ** Projections
110− fstR,
111− sndR,
112− -- ** Bifunctor
113− (><),
114− -- * Relational coproduct
115− eitherR,
116− -- ** Injections
117− i1,
118− i2,
119− -- ** Bifunctor
120− (-|-),
121−
122− -- * Relational "currying"
123− trans,
124− untrans,
125
68
126 -- * (Endo-)Relational properties
69 -- * (Endo-)Relational properties
127 reflexive,
70 reflexive,
@@ -136,26 +79,16 @@ module LAoP.Relation
136 linearOrder,
79 linearOrder,
137 equivalence,
80 equivalence,
138 partialEquivalence,
81 partialEquivalence,
139− difunctional,
140−
141− -- * Conditionals
142− equalizer,
143
82
144 -- ** McCarthy's Conditional
83 -- ** McCarthy's Conditional
145− predR,
146− guard,
147− cond,
84+ -- cond,
148
85
149 -- * Relational composition and lifting
86 -- * Relational composition and lifting
150− iden,
87+ identity,
151 comp,
88 comp,
152 fromF,
89 fromF,
153 fromF',
90 fromF',
154
91
155− -- ** Relational application
156− pointAp,
157− pointApBool,
158−
159 -- * Matrix printing
92 -- * Matrix printing
160 pretty,
93 pretty,
161 prettyPrint
94 prettyPrint
src/LAoP/Relation/Internal.hs+148−747
@@ -1,9 +1,6 @@
1−{-# LANGUAGE TypeFamilies #-}
2−{-# LANGUAGE ConstraintKinds #-}
3{-# LANGUAGE GADTs #-}
1{-# LANGUAGE GADTs #-}
4{-# LANGUAGE UndecidableInstances #-}
2{-# LANGUAGE UndecidableInstances #-}
5{-# LANGUAGE FlexibleContexts #-}
3{-# LANGUAGE FlexibleContexts #-}
6−{-# LANGUAGE TypeApplications #-}
7{-# LANGUAGE ScopedTypeVariables #-}
4{-# LANGUAGE ScopedTypeVariables #-}
8{-# LANGUAGE StandaloneDeriving #-}
5{-# LANGUAGE StandaloneDeriving #-}
9{-# LANGUAGE DerivingVia #-}
6{-# LANGUAGE DerivingVia #-}
@@ -29,7 +26,7 @@
29
26
30module LAoP.Relation.Internal
27module LAoP.Relation.Internal
31 ( -- | This definition makes use of the fact that 'Void' is
28 ( -- | This definition makes use of the fact that 'Void' is
32− -- isomorphic to 0 and 'One' to 1 and captures matrix
29+ -- isomorphic to 0 and '()' to 1 and captures matrix
33 -- dimensions as stacks of 'Either's.
30 -- dimensions as stacks of 'Either's.
34 --
31 --
35 -- There exists two type families that make it easier to write
32 -- There exists two type families that make it easier to write
@@ -41,22 +38,11 @@ module LAoP.Relation.Internal
41 Relation (..),
38 Relation (..),
42 Boolean,
39 Boolean,
43
40
44− -- * Constraint type synonyms
45− Countable,
46− CountableDims,
47− CountableN,
48− CountableDimsN,
49− FLN,
50− Liftable,
51− Trivial,
52− TrivialP,
53−
54 -- * Primitives
41 -- * Primitives
42+ empty,
55 one,
43 one,
56− join,
44+ junc,
57− (|||),
45+ split,
58− fork,
59− (===),
60
46
61 -- * Auxiliary type families
47 -- * Auxiliary type families
62 I.FromNat,
48 I.FromNat,
@@ -66,74 +52,30 @@ module LAoP.Relation.Internal
66 -- * Matrix construction and conversion
52 -- * Matrix construction and conversion
67 I.FromLists,
53 I.FromLists,
68 fromLists,
54 fromLists,
69− fromF',
70 fromF,
55 fromF,
56+ fromF',
71 toRel,
57 toRel,
72 toLists,
58 toLists,
73 toList,
59 toList,
74− toBool,
60+ matrixBuilder,
75− pt,
76− belongs,
77− relationBuilder',
78− relationBuilder,
79 zeros,
61 zeros,
80 ones,
62 ones,
81 bang,
63 bang,
82− point,
83
64
84 -- * Relational operations
65 -- * Relational operations
85 conv,
66 conv,
86 intersection,
67 intersection,
87 union,
68 union,
88 sse,
69 sse,
89− implies,
90− iff,
91 ker,
70 ker,
92 img,
71 img,
93
72
94 -- * Taxonomy of binary relations
73 -- * Taxonomy of binary relations
95− injective,
96 entire,
74 entire,
97− simple,
75+ injective,
98 surjective,
76 surjective,
99− representation,
77+ simple,
100− function,
78+ bijective,
101− abstraction,
102− injection,
103− surjection,
104− bijection,
105− domain,
106− range,
107−
108− -- * Function division
109− divisionF,
110−
111− -- * Relation division
112− divR,
113− divL,
114− divS,
115− shrunkBy,
116− overriddenBy,
117−
118− -- * Relational pairing
119− splitR,
120−
121− -- ** Projections
122− fstR,
123− sndR,
124− -- ** Bifunctor
125− (><),
126− -- * Relational coproduct
127− eitherR,
128− -- ** Injections
129− i1,
130− i2,
131− -- ** Bifunctor
132− (-|-),
133−
134− -- * Relational "currying"
135− trans,
136− untrans,
137
79
138 -- * (Endo-)Relational properties
80 -- * (Endo-)Relational properties
139 reflexive,
81 reflexive,
@@ -148,25 +90,15 @@ module LAoP.Relation.Internal
148 linearOrder,
90 linearOrder,
149 equivalence,
91 equivalence,
150 partialEquivalence,
92 partialEquivalence,
151− difunctional,
152−
153− -- * Conditionals
154− equalizer,
155
93
156 -- ** McCarthy's Conditional
94 -- ** McCarthy's Conditional
157− predR,
158− guard,
159− cond,
95+ -- cond,
160
96
161 -- * Relational composition and lifting
97 -- * Relational composition and lifting
162− iden,
98+ identity,
163 comp,
99 comp,
164− fromF',
165 fromF,
100 fromF,
166−
101+ fromF',
167− -- ** Relational application
168− pointAp,
169− pointApBool,
170
102
171 -- * Matrix printing
103 -- * Matrix printing
172 pretty,
104 pretty,
@@ -176,15 +108,15 @@ module LAoP.Relation.Internal
176
108
177import Data.Void
109import Data.Void
178import qualified LAoP.Matrix.Internal as I
110import qualified LAoP.Matrix.Internal as I
179−import LAoP.Utils.Internal
111+import LAoP.Utils
180import Control.DeepSeq
112import Control.DeepSeq
113+import qualified Control.Category as C
114+import Data.Coerce
181import Data.Bool
115import Data.Bool
182import GHC.TypeLits
116import GHC.TypeLits
183−import Prelude hiding (id, (.))
184
117
185-- | Boolean type synonym for working with boolean matrices
118-- | Boolean type synonym for working with boolean matrices
186type Boolean = Natural 0 1
119type Boolean = Natural 0 1
187−type Powerset a = List (List a)
188
120
189-- | Relation data type.
121-- | Relation data type.
190newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))
122newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))
@@ -192,143 +124,128 @@ newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))
192
124
193deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)
125deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)
194
126
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−
205−-- | It is possible to implement a constrained version of the category type
127+-- | It isn't possible to implement the 'id' function so it's
206−-- class.
128+-- implementation is 'undefined'. However 'comp' can be and this partial
129+-- class implementation exists just to make the code more readable.
130+--
131+-- Please use 'identity' instead.
207−instance Category Relation where
132+instance C.Category Relation where
208− type Object Relation a = (FLN a a, CountableN a)
133+ id = undefined
209− id = iden
210− (.) = comp
134+ (.) = comp
211
135
212instance Num (Relation a b) where
136instance Num (Relation a b) where
213− -- | Matrix addition becomes Boolean matrix disjointion
137+ -- | Matrix addition becomes Boolean matrix disjunction
214 (R a) + (R b) = R (I.orM a b)
138 (R a) + (R b) = R (I.orM a b)
215
139
216 -- | Matrix subtraction becomes Relational subtraction
140 -- | Matrix subtraction becomes Relational subtraction
217 (R a) - (R b) = R (I.subM a b)
141 (R a) - (R b) = R (I.subM a b)
218
142
219− -- | Matrix multiplication becomes Boolean matrix conjointion
143+ -- | Matrix multiplication becomes Boolean matrix conjunction
220 (R a) * (R b) = R (I.andM a b)
144 (R a) * (R b) = R (I.andM a b)
221
145
222− -- | Matrix negation becomes Boolean matrix negation
223− negate (R a) = R (I.negateM a)
224−
225-- Type alias
146-- Type alias
226type Zero = Void
147type Zero = Void
227type One = ()
148type One = ()
228
149
229-- Primitives
150-- Primitives
230
151
152+-- | Empty matrix constructor
153+empty :: Relation Zero Zero
154+empty = R I.Empty
155+
231-- | Unit matrix constructor
156-- | Unit matrix constructor
232one :: Boolean -> Relation One One
157one :: Boolean -> Relation One One
233one = R . I.One
158one = R . I.One
234
159
235−-- | Boolean Matrix 'Join' constructor, also known as relational coproduct.
160+-- | Boolean Matrix 'Junc' constructor
236−--
161+junc :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))
237−-- See 'eitherR'.
238−join :: Relation a c -> Relation b c -> Relation (Either a b) c
162+ => Relation a c -> Relation b c -> Relation (Either a b) c
239−join (R a) (R b) = R (I.Join a b)
163+junc (R a) (R b) = R (I.Junc a b)
240
164
241infixl 3 |||
165infixl 3 |||
242−-- | Boolean Matrix 'Join' constructor
166+-- | Boolean Matrix 'Junc' constructor
243−--
244−-- See 'eitherR'.
245(|||) ::
167(|||) ::
168+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
246 Relation a c ->
169 Relation a c ->
247 Relation b c ->
170 Relation b c ->
248 Relation (Either a b) c
171 Relation (Either a b) c
249−(|||) = join
172+(|||) = junc
250
173
251−-- | Boolean Matrix 'Fork' constructor, also known as relational product.
174+-- | Boolean Matrix 'Split' constructor
175+split :: (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b))
252−fork :: Relation c a -> Relation c b -> Relation c (Either a b)
176+ => Relation c a -> Relation c b -> Relation c (Either a b)
253−fork (R a) (R b) = R (I.Fork a b)
177+split (R a) (R b) = R (I.Split a b)
254
178
255infixl 2 ===
179infixl 2 ===
256−-- | Boolean Matrix 'Fork' constructor
180+-- | Boolean Matrix 'Split' constructor
257(===) ::
181(===) ::
182+ (I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)) =>
258 Relation c a ->
183 Relation c a ->
259 Relation c b ->
184 Relation c b ->
260 Relation c (Either a b)
185 Relation c (Either a b)
261−(===) = fork
186+(===) = split
262
187
263-- Construction
188-- Construction
264
189
265-- | Build a matrix out of a list of list of elements. Throws a runtime
190-- | Build a matrix out of a list of list of elements. Throws a runtime
266-- error if the dimensions do not match.
191-- error if the dimensions do not match.
267−fromLists :: (FLN a b) => [[Boolean]] -> Relation a b
192+fromLists :: (I.FromLists Boolean (I.Normalize a) (I.Normalize b)) => [[Boolean]] -> Relation a b
268fromLists = R . I.fromLists
193fromLists = R . I.fromLists
269
194
270−-- | Relation builder function. Constructs a relation provided with
195+-- | Matrix builder function. Constructs a matrix provided with
271−-- a construction function that operates with indices.
196+-- a construction function.
272−relationBuilder' ::
197+matrixBuilder ::
273− (FLN a b, CountableDimsN a b) =>
198+ (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>
274− ((Int, Int) -> Boolean) -> Relation a b
199+ ((Int, Int) -> Boolean) ->
275−relationBuilder' = R . I.matrixBuilder'
276−
277−-- | Relation builder function. Constructs a relation provided with
278−-- a construction function that operates with arbitrary types.
279−relationBuilder ::
280− ( FLN a b,
281− Enum a,
282− Enum b,
283− Bounded a,
284− Bounded b,
285− Eq a,
286− CountableDims a b
287− ) => ((a, b) -> Boolean) -> Relation a b
200+ Relation a b
288−relationBuilder = R . I.matrixBuilder
201+matrixBuilder = R . I.matrixBuilder
289
202
290-- | Lifts functions to matrices with arbitrary dimensions.
203-- | Lifts functions to matrices with arbitrary dimensions.
291--
204--
292-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
205-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
293-- types @a@ or @b@ allows.
206-- types @a@ or @b@ allows.
294−fromF' ::
207+fromF ::
295− ( Liftable a b,
208+ ( Bounded a,
296− CountableDimsN c d,
209+ Bounded b,
297− FLN d c
210+ Enum a,
211+ Enum b,
212+ Eq b,
213+ KnownNat (I.Count (I.Normalize c)),
214+ KnownNat (I.Count (I.Normalize d)),
215+ I.FromLists Boolean (I.Normalize d) (I.Normalize c)
298 )
216 )
299 => (a -> b) -> Relation c d
217 => (a -> b) -> Relation c d
300−fromF' f = R (I.fromFRel' f)
218+fromF f = R (I.fromFRel f)
301
219
302-- | Lifts functions to matrices with dimensions matching @a@ and @b@
220-- | Lifts functions to matrices with dimensions matching @a@ and @b@
303-- cardinality's.
221-- cardinality's.
304−fromF ::
222+fromF' ::
305− ( Liftable a b,
223+ ( Bounded b,
306− CountableDimsN a b,
224+ Bounded a,
307− FLN b a
225+ Enum a,
226+ Enum b,
227+ Eq b,
228+ KnownNat (I.Count (I.Normalize a)),
229+ KnownNat (I.Count (I.Normalize b)),
230+ I.FromLists Boolean (I.Normalize b) (I.Normalize a)
308 )
231 )
309 => (a -> b) -> Relation a b
232 => (a -> b) -> Relation a b
310−fromF f = R (I.fromFRel f)
233+fromF' f = R (I.fromFRel' f)
311
234
312-- | Lifts relation functions to 'Relation'
235-- | Lifts relation functions to 'Relation'
313toRel ::
236toRel ::
314− ( Liftable a b,
237+ ( Bounded b,
315− CountableDims a b,
238+ Bounded a,
316− FLN b a
239+ Enum a,
240+ Enum b,
241+ Eq b,
242+ KnownNat (I.Count (I.Normalize a)),
243+ KnownNat (I.Count (I.Normalize b)),
244+ I.FromLists Boolean (I.Normalize b) (I.Normalize a)
317 )
245 )
318− => (a -> b -> Bool) -> Relation a b
246+ => (a -> b -> Bool) -> Relation a b
319toRel = R . I.toRel
247toRel = R . I.toRel
320
248
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−
332-- Conversion
249-- Conversion
333
250
334-- | Converts a matrix to a list of lists of elements.
251-- | Converts a matrix to a list of lists of elements.
@@ -339,188 +256,46 @@ toLists (R m) = I.toLists m
339toList :: Relation a b -> [Boolean]
256toList :: Relation a b -> [Boolean]
340toList (R m) = I.toList m
257toList (R m) = I.toList m
341
258
342−-- | Converts a well typed 'Relation' to 'Bool'.
343−toBool :: Relation One One -> Bool
344−toBool r = case toList r of
345− [Nat 0] -> False
346− _ -> True
347−
348−-- | Power transpose.
349−--
350−-- Maps a relation to a set valued function.
351−pt ::
352− ( Liftable a b,
353− Eq a,
354− CountableDimsN a b,
355− FLN a One,
356− FLN b One
357− )
358− => Relation a b -> (a -> List b)
359−pt r a =
360− let (L lb) = maxBound
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
375−
376-- Zeros Matrix
259-- Zeros Matrix
377
260
378−-- | The zero relation. A relation where no element of type @a@ relates
261+-- | The zero matrix. A matrix wholly filled with zeros.
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−-- @
387zeros ::
262zeros ::
388− (FLN a b, CountableDimsN a b) =>
263+ (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>
389 Relation a b
264 Relation a b
390−zeros = relationBuilder' (const (nat 0))
265+zeros = matrixBuilder (const (nat 0))
391
266
392-- Ones Matrix
267-- Ones Matrix
393
268
394−-- | The ones relation. A relation where every element of type @a@ relates
269+-- | The ones matrix. A matrix wholly filled with ones.
395−-- with every element of type @b@.
396−--
397−-- Also known as T (Top) Relation or universal Relation.
398--
270--
399−-- @
271+-- Also known as T (Top) matrix.
400−-- ⊥ ``sse`` R && R ``sse`` T == True
401−-- @
402ones ::
272ones ::
403− (FLN a b, CountableDimsN a b) =>
273+ (I.FromLists Boolean (I.Normalize a) (I.Normalize b), KnownNat (I.Count (I.Normalize a)), KnownNat (I.Count (I.Normalize b))) =>
404 Relation a b
274 Relation a b
405−ones = relationBuilder' (const (nat 1))
275+ones = matrixBuilder (const (nat 1))
406
276
407-- Bang Matrix
277-- Bang Matrix
408
278
409−-- | The T (Top) row vector relation.
279+-- | The T (Top) row vector matrix.
410bang ::
280bang ::
411− (FLN a One, CountableN a) =>
281+ forall e a.
282+ (I.FromLists Boolean (I.Normalize a) (), KnownNat (I.Count (I.Normalize a))) =>
412 Relation a One
283 Relation a One
413bang = ones
284bang = ones
414
285
415−-- | Point constant relation
416−point ::
417− ( Bounded a,
418− Enum a,
419− Eq a,
420− CountableN a,
421− FLN a One
422− ) => a -> Relation One a
423−point = fromF . const
424−
425−-- iden Matrix
286+-- Identity Matrix
426
287
427−-- | iden matrix
288+-- | Identity matrix
428−--
289+identity ::
429−-- @
290+ (I.FromLists Boolean (I.Normalize a) (I.Normalize a), KnownNat (I.Count (I.Normalize a))) =>
430−-- 'iden' `.` r == r == r `.` 'iden'
431−-- @
432−iden ::
433− (FLN a a, CountableN a) => Relation a a
291+ Relation a a
434−iden = relationBuilder' (bool (nat 0) (nat 1) . uncurry (==))
292+identity = matrixBuilder (bool (nat 0) (nat 1) . uncurry (==))
435
293
436-- | Relational composition
294-- | Relational composition
437−--
438−-- @
439−-- r `.` (s `.` p) = (r `.` s) `.` p
440−-- @
441−comp :: Relation b c -> Relation a b -> Relation a c
295+comp :: forall a b c . Relation b c -> Relation a b -> Relation a c
442comp (R a) (R b) = R (I.compRel a b)
296comp (R a) (R b) = R (I.compRel a b)
443
297
444−-- | Relational right division
445−--
446−-- @'divR' x y@ is the largest relation @z@ which,
447−-- pre-composed with @y@, approximates @x@.
448−divR :: Relation b c -> Relation b a -> Relation a c
449−divR (R x) (R y) = R (I.divR x y)
450−
451−-- | Relational left division
452−--
453−-- The dual division operator:
454−--
455−-- @
456−-- 'divL' y x == 'conv' ('divR' ('conv' x) ('conv' y)
457−-- @
458−divL :: Relation c b -> Relation a b -> Relation a c
459−divL (R x) (R y) = R (I.divL x y)
460−
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@.
465−divS :: Relation c a -> Relation b a -> Relation c b
466−divS (R x) (R y) = R (I.divS x y)
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−
492−-- | Relational application.
493−--
494−-- If @a@ and @b@ are related by 'Relation' @r@
495−-- then @'pointAp' a b r == 'one' ('nat' 1)@
496−pointAp ::
497− ( Liftable a b,
498− Eq a,
499− CountableDimsN a b,
500− FLN a One,
501− FLN b One
502− ) => a -> b -> Relation a b -> Relation One One
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
516−
517-- | Relational converse
298-- | 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
524conv :: Relation a b -> Relation b a
299conv :: Relation a b -> Relation b a
525conv (R a) = R (I.tr a)
300conv (R a) = R (I.tr a)
526
301
@@ -528,481 +303,107 @@ conv (R a) = R (I.tr a)
528sse :: Relation a b -> Relation a b -> Bool
303sse :: Relation a b -> Relation a b -> Bool
529sse a b = a <= b
304sse a b = a <= b
530
305
531−-- | Relational implication (the same as @'sse'@)
532−implies :: Relation a b -> Relation a b -> Relation a b
533−implies r s = negate r `union` s
534−
535−-- | Relational bi-implication
536−iff :: Relation a b -> Relation a b -> Bool
537−iff r s = r == s
538−
539-- | Relational intersection
306-- | Relational intersection
540--
307--
541−-- Lifts pointwise conjointion.
308+-- @M `inter` N = M * N@
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−-- @
547intersection :: Relation a b -> Relation a b -> Relation a b
309intersection :: Relation a b -> Relation a b -> Relation a b
548intersection a b = a * b
310intersection a b = a * b
549
311
550-- | Relational union
312-- | Relational union
551--
313--
552−-- Lifts pointwise disjointion.
314+-- @M `union` N = M + N - M * N
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−-- @
560union :: Relation a b -> Relation a b -> Relation a b
315union :: Relation a b -> Relation a b -> Relation a b
561−union a b = a + b
316+union a b = a + b - a * b
562
317
563-- | Relation Kernel
318-- | Relation Kernel
564−--
565−-- @
566−-- 'ker' r == 'conv' r `.` r
567−-- 'ker' r == 'img' ('conv' r)
568−-- @
569ker :: Relation a b -> Relation a a
319ker :: Relation a b -> Relation a a
570−ker r = conv r . r
320+ker r = conv r `comp` r
571
321
572-- | Relation Image
322-- | Relation Image
573−--
574−-- @
575−-- 'img' r == r `.` conv r
576−-- 'img' r == 'ker' ('conv' r)
577−-- @
578img :: Relation a b -> Relation b b
323img :: Relation a b -> Relation b b
579−img r = r . conv r
324+img r = r `comp` conv r
580
325
581−-- | Function division. Special case of 'divS'.
326+-- | Properties of relations
582−--
327+
583−-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and
328+-- | A 'Relation' @r@ is reflexive iff @'identity' `'sse'` r@
584−-- 'entire'.
329+reflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
585−--
330+reflexive r = identity <= r
586−-- @'divisionF' f g == 'conv' g `.` f@
587−divisionF :: Relation a c -> Relation b c -> Relation a b
588−divisionF f g = conv g . f
589
331
590−-- Taxonomy of binary relations
332+-- | A 'Relation' @r@ is coreflexive iff @r `'sse'` 'identity'@
333+coreflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
334+coreflexive r = r <= identity
591
335
592−-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@
336+-- | A 'Relation' @r@ is simple iff @'coreflexive' ('img' r)@
593−simple :: (CountableN b, FLN b b) => Relation a b -> Bool
337+simple :: (KnownNat (I.Count (I.Normalize b)), I.FromLists Boolean (I.Normalize b) (I.Normalize b)) => Relation a b -> Bool
594simple = coreflexive . img
338simple = coreflexive . img
595
339
596−-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@
340+-- | A 'Relation' @r@ is injective iff @'coreflexive' ('ker' r)@
597−injective :: (CountableN a, FLN a a) => Relation a b -> Bool
341+injective :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a b -> Bool
598injective = coreflexive . ker
342injective = coreflexive . ker
599
343
600−-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@
344+-- | A 'Relation' @r@ is entire iff @'reflexive' ('ker' r)@
601−entire :: (CountableN a, FLN a a) => Relation a b -> Bool
345+entire :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a b -> Bool
602entire = reflexive . ker
346entire = reflexive . ker
603
347
604−-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@
348+-- | A 'Relation' @r@ is surjective iff @'reflexive' ('img' r)@
605−surjective :: (CountableN b, FLN b b) => Relation a b -> Bool
349+surjective :: (KnownNat (I.Count (I.Normalize b)), I.FromLists Boolean (I.Normalize b) (I.Normalize b)) => Relation a b -> Bool
606surjective = reflexive . img
350surjective = reflexive . img
607
351
608−-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@
352+-- | A 'Relation' @r@ is bijective iff @'injective' r && 'surjective' r@
609−--
353+bijective ::
610−-- A 'function' @f@ enjoys the following properties, where @r@ and @s@ are binary
354+ ( KnownNat (I.Count (I.Normalize b)),
611−-- relations:
355+ KnownNat (I.Count (I.Normalize a)),
612−--
356+ I.FromLists Boolean (I.Normalize b) (I.Normalize b),
613−-- @
357+ I.FromLists Boolean (I.Normalize a) (I.Normalize a)
614−-- f `.` r ``sse`` s == r ``sse`` f `.` s
615−-- r `.` f ``sse`` s == r ``sse`` s `.` f
616−-- @
617−function ::
618− ( CountableDimsN a b,
619− FLN a a,
620− FLN b b
621− )
622− => Relation a b -> Bool
623−function r = simple r && entire r
624−
625−-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@
626−representation ::
627− ( CountableN a,
628− FLN a a
629− )
630− => Relation a b -> Bool
631−representation r = injective r && entire r
632−
633−-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@
634−abstraction ::
635− ( CountableN b,
636− FLN b b
637− )
638− => Relation a b -> Bool
639−abstraction r = surjective r && simple r
640−
641−-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@
642−surjection ::
643− ( CountableDimsN a b,
644− FLN a a,
645− FLN b b
646− )
647− => Relation a b -> Bool
648−surjection r = function r && abstraction r
649−
650−-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@
651−injection ::
652− ( CountableDimsN a b,
653− FLN a a,
654− FLN b b
655− )
656− => Relation a b -> Bool
657−injection r = function r && representation r
658−
659−-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@
660−bijection ::
661− ( CountableDimsN a b,
662− FLN b b,
663− FLN a a
664 ) => Relation a b -> Bool
358 ) => Relation a b -> Bool
665−bijection r = injection r && surjection r
359+bijective r = injective r && surjective r
666−
667−-- Properties of relations
668
360
669−-- | A 'Relation' @r@ is 'reflexive' 'iff' @'id' ``sse`` r@
670−reflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
671−reflexive r = id <= r
672−
673−-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'id'@
674−coreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
675−coreflexive r = r <= id
676−
677−-- | A 'Relation' @r@ is 'transitive' 'iff' @(r `.` r) ``sse`` r@
361+-- | A 'Relation' @r@ is transitive iff @(r `'comp'` r) `'sse'` r@
678transitive :: Relation a a -> Bool
362transitive :: Relation a a -> Bool
679−transitive r = (r . r) `sse` r
363+transitive r = (r `comp` r) `sse` r
680
364
681−-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@
365+-- | A 'Relation' @r@ is symmetric iff @r == conv r@
682symmetric :: Relation a a -> Bool
366symmetric :: Relation a a -> Bool
683symmetric r = r == conv r
367symmetric r = r == conv r
684
368
685−-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'id'@
369+-- | A 'Relation' @r@ is anti-symmetric iff @(r `'intersection'` 'conv' r) `'sse'` 'identity'@
686−antiSymmetric :: (CountableN a, FLN a a) => Relation a a -> Bool
370+antiSymmetric :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
687−antiSymmetric r = (r `intersection` conv r) `sse` id
371+antiSymmetric r = (r `intersection` conv r) `sse` identity
688
372
689−-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'id') == 'zeros'@
373+-- | A 'Relation' @r@ is irreflexive iff @(r `'intersection'` identity) == 'zeros'@
690−irreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
374+irreflexive :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
691−irreflexive r = (r `intersection` id) == zeros
375+irreflexive r = (r `intersection` identity) == zeros
692
376
693−-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@
377+-- | A 'Relation' @r@ is connected iff @(r `'union'` 'conv' r) == 'ones'@
694−connected :: (CountableN a, FLN a a) => Relation a a -> Bool
378+connected :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
695connected r = (r `union` conv r) == ones
379connected r = (r `union` conv r) == ones
696
380
697−-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@
381+-- | A 'Relation @r@ is a preorder iff @'reflexive' r && 'transitive' r@
698−preorder :: (CountableN a, FLN a a) => Relation a a -> Bool
382+preorder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
699preorder r = reflexive r && transitive r
383preorder r = reflexive r && transitive r
700
384
701−-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@
385+-- | A 'Relation @r@ is a partial order iff @'antiSymmetric' r && 'preorder' r@
702−partialOrder :: (CountableN a, FLN a a) => Relation a a -> Bool
386+partialOrder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
703partialOrder r = antiSymmetric r && preorder r
387partialOrder r = antiSymmetric r && preorder r
704
388
705−-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@
389+-- | A 'Relation @r@ is a linear order iff @'connected' r && 'partialOrder' r@
706−linearOrder :: (CountableN a, FLN a a) => Relation a a -> Bool
390+linearOrder :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
707linearOrder r = connected r && partialOrder r
391linearOrder r = connected r && partialOrder r
708
392
709−-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@
393+-- | A 'Relation @r@ is an equivalence iff @'symmetric' r && 'preorder' r@
710−equivalence :: (CountableN a, FLN a a) => Relation a a -> Bool
394+equivalence :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
711equivalence r = symmetric r && preorder r
395equivalence r = symmetric r && preorder r
712
396
713−-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@
397+-- | A 'Relation @r@ is a partial equivalence iff @'partialOrder' r && 'equivalence' r@
714−partialEquivalence :: (CountableN a, FLN a a) => Relation a a -> Bool
398+partialEquivalence :: (KnownNat (I.Count (I.Normalize a)), I.FromLists Boolean (I.Normalize a) (I.Normalize a)) => Relation a a -> Bool
715partialEquivalence r = partialOrder r && equivalence r
399partialEquivalence r = partialOrder r && equivalence r
716
400
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−
722−-- Relational pairing
723−
724−-- | Relational pairing.
725−--
726−-- NOTE: That this is not a true categorical product, see for instance:
727−--
728−-- @
729−-- | 'fstR' `.` 'splitR' a b ``sse`` a
730−-- 'splitR' a b <=> |
731−-- | 'sndR' `.` 'splitR' a b ``sse`` b
732−-- @
733−--
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−-- @
745−splitR ::
746− ( CountableDimsN a b,
747− CountableN (a, b),
748− FLN (a, b) a,
749− FLN (a, b) b,
750− TrivialP a b
751− )
752− => Relation c a -> Relation c b -> Relation c (a, b)
753−splitR (R f) (R g) = R (I.kr f g)
754−
755−-- | Relational pairing first component projection
756−--
757−-- @
758−-- 'fstR' `.` 'splitR' r s ``sse`` r
759−-- @
760−fstR ::
761− forall a b .
762− ( CountableDimsN a b,
763− CountableN (a, b),
764− FLN (a, b) a,
765− TrivialP a b
766− )
767− => Relation (a, b) a
768−fstR = R (I.fstM @Boolean @(I.Normalize a) @(I.Normalize b))
769−
770−-- | Relational pairing second component projection
771−--
772−-- @
773−-- 'sndR' `.` 'splitR' r s ``sse`` s
774−-- @
775−sndR ::
776− forall a b .
777− ( CountableDimsN a b,
778− CountableN (a, b),
779− FLN (a, b) b,
780− TrivialP a b
781− )
782− => Relation (a, b) b
783−sndR = R (I.sndM @Boolean @(I.Normalize a) @(I.Normalize b))
784−
785−-- Relational pairing functor
786−
787−infixl 4 ><
788−-- | Relational pairing functor
789−--
790−-- @
791−-- r '><' s == 'splitR' (r `.` fstR) (s `.` sndR)
792−-- (r '><' s) `.` (p '><' q) == (r `.` p) '><' (s `.` q)
793−-- @
794−(><) ::
795− ( CountableDimsN a b,
796− CountableDimsN c d,
797− CountableDimsN (a, c) (b, d),
798− FLN (a, c) a,
799− FLN (a, c) c,
800− FLN (b, d) b,
801− FLN (b, d) d,
802− TrivialP a c,
803− TrivialP b d
804− )
805− => Relation a b -> Relation c d -> Relation (a, c) (b, d)
806−(><) (R a) (R b) = R ((I.><) a b)
807−
808−-- Relational co-products
809−
810−-- | Relational coproduct.
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−-- @
825−eitherR :: Relation a c -> Relation b c -> Relation (Either a b) c
826−eitherR = join
827−
828−-- | Relational coproduct first component injection
829−--
830−-- @
831−-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'
832−-- 'i1' `.` 'i2' = 'zeros'
833−-- @
834−i1 ::
835− ( CountableDimsN a b,
836− FLN b a,
837− FLN a a
838− )
839− => Relation a (Either a b)
840−i1 = R I.i1
841−
842−-- | Relational coproduct second component injection
843−--
844−-- @
845−-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'
846−-- 'i1' `.` 'i2' = 'zeros'
847−-- @
848−i2 ::
849− ( CountableDimsN a b,
850− FLN a b,
851− FLN b b
852− )
853− => Relation b (Either a b)
854−i2 = R I.i2
855−
856−infixl 5 -|-
857−
858−-- | Relational coproduct functor.
859−--
860−-- @
861−-- r '-|-' s == 'eitherR' ('i1' `.` r) ('i2' `.` s)
862−-- @
863−(-|-) ::
864− ( CountableDimsN b d,
865− FLN b b,
866− FLN d b,
867− FLN b d,
868− FLN d d
869− )
870− => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)
871−(-|-) (R a) (R b) = R ((I.-|-) a b)
872−
873−-- Relational "Currying"
874−
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.
880−trans ::
881− ( CountableDimsN a b,
882− CountableN c,
883− CountableDimsN (a, b) (c, b),
884− FLN (c, b) c,
885− FLN (c, b) b,
886− FLN (a, b) a,
887− FLN (a, b) b,
888− Trivial (a, b),
889− Trivial (c, b),
890− TrivialP a b,
891− TrivialP c b
892− )
893− => Relation (a, b) c -> Relation a (c, b)
894−trans r = splitR r sndR . conv fstR
895−
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.
901−untrans ::
902− ( CountableDimsN a b,
903− CountableN c,
904− CountableDimsN (a, b) (c, b),
905− FLN (c, b) c,
906− FLN (c, b) b,
907− FLN (a, b) b,
908− FLN (a, b) a,
909− Trivial (a, b),
910− Trivial (c, b),
911− TrivialP a b,
912− TrivialP c b
913− )
914− => Relation a (c, b) -> Relation (a, b) c
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
999−
1000-- Relation pretty print
401-- Relation pretty print
1001
402
1002-- | Relation pretty printing
403-- | Relation pretty printing
1003−pretty :: (CountableDimsN a b) => Relation a b -> String
404+pretty :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> String
1004pretty (R a) = I.pretty a
405pretty (R a) = I.pretty a
1005
406
1006-- | Relation pretty printing
407-- | Relation pretty printing
1007−prettyPrint :: (CountableDimsN a b) => Relation a b -> IO ()
408+prettyPrint :: (KnownNat (I.Count (I.Normalize a))) => Relation a b -> IO ()
1008prettyPrint (R a) = I.prettyPrint a
409prettyPrint (R a) = I.prettyPrint a
src/LAoP/Utils.hs−6
@@ -94,12 +94,6 @@ 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(..)
103 )
97 )
104where
98where
105
99
src/LAoP/Utils/Internal.hs+74−94
@@ -1,15 +1,84 @@
1−{-# LANGUAGE TypeFamilies #-}
2−{-# LANGUAGE DeriveGeneric #-}
3{-# LANGUAGE TypeApplications #-}
1{-# LANGUAGE TypeApplications #-}
4{-# LANGUAGE DataKinds #-}
2{-# LANGUAGE DataKinds #-}
5{-# LANGUAGE FlexibleInstances #-}
3{-# LANGUAGE FlexibleInstances #-}
6{-# LANGUAGE GeneralizedNewtypeDeriving #-}
4{-# LANGUAGE GeneralizedNewtypeDeriving #-}
5+{-# LANGUAGE KindSignatures #-}
7{-# LANGUAGE ScopedTypeVariables #-}
6{-# LANGUAGE ScopedTypeVariables #-}
8
7
9module LAoP.Utils.Internal
8module LAoP.Utils.Internal
10− (
9+ ( -- | This is an internal module and it is not meant to be imported.
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+ --
11 -- * 'Natural' data type
80 -- * 'Natural' data type
12− Natural(..),
81+ Natural,
13 nat,
82 nat,
14
83
15 -- * Coerce auxiliar functions to help promote 'Int' typed functions to
84 -- * Coerce auxiliar functions to help promote 'Int' typed functions to
@@ -17,31 +86,19 @@ module LAoP.Utils.Internal
17 coerceNat,
86 coerceNat,
18 coerceNat2,
87 coerceNat2,
19 coerceNat3,
88 coerceNat3,
20−
21− -- * 'List' data type
22− List (..),
23−
24− -- * Category type class
25− Category(..)
26 )
89 )
27where
90where
28
91
29import Data.Coerce
92import Data.Coerce
30import Data.Proxy
93import Data.Proxy
31−import Data.List
32−import Data.Maybe
33import GHC.TypeLits
94import GHC.TypeLits
34import Control.DeepSeq
95import Control.DeepSeq
35−import GHC.Generics
36−import Data.Kind
37−import Prelude hiding (id, (.))
38−import qualified Prelude
39
96
40-- | Wrapper around 'Int's that have a restrictive semantic associated.
97-- | Wrapper around 'Int's that have a restrictive semantic associated.
41-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'
98-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'
42-- @i@ that's @n <= i <= m@.
99-- @i@ that's @n <= i <= m@.
43newtype Natural (start :: Nat) (end :: Nat) = Nat Int
100newtype Natural (start :: Nat) (end :: Nat) = Nat Int
44− deriving (Show, Read, Eq, Ord, NFData, Generic)
101+ deriving (Show, Read, Eq, Ord, NFData)
45
102
46-- | Throws a runtime error if any of the operations overflows or
103-- | Throws a runtime error if any of the operations overflows or
47-- underflows.
104-- underflows.
@@ -117,80 +174,3 @@ instance
117 fstI = fromEnum a
174 fstI = fromEnum a
118 sndI = fromEnum b
175 sndI = fromEnum b
119 in fstI * lengthB + sndI
176 in fstI * lengthB + sndI
120−
121−instance
122− ( Bounded a,
123− Bounded b
124− ) => Bounded (Either a b)
125− where
126− minBound = Left (minBound :: a)
127− maxBound = Right (maxBound :: b)
128−
129−instance
130− ( Enum a,
131− Bounded a,
132− Enum b,
133− Bounded b
134− ) => Enum (Either a b)
135− where
136− toEnum i =
137− let la = fmap Left ([minBound..maxBound] :: [a])
138− lb = fmap Right ([minBound..maxBound] :: [b])
139− in (la ++ lb) !! i
140−
141− fromEnum (Left a) = fromEnum a
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.yamldeleted−1
@@ -1 +0,0 @@
1−resolver: lts-15.5
stack.yaml+66−1
@@ -1 +1,66 @@
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
1−resolver: lts-14.3
20+resolver: lts-14.18
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: 523878
9+ size: 524789
10− url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/3.yaml
10+ url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/18.yaml
11− sha256: 470c46c27746a48c7c50f829efc0cf00112787a7804ee4ac7a27754658f6d92c
11+ sha256: 646be71223e08234131c6989912e6011e01b9767bc447b6d466a35e14360bdf2
12− original: lts-14.3
12+ original: lts-14.18
test/Examples/Quantum.hsdeleted−31
@@ -1,31 +0,0 @@
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)
test/Examples/Readme.hsdeleted−184
@@ -1,184 +0,0 @@
1−{-# LANGUAGE GADTs #-}
2−{-# LANGUAGE DeriveGeneric #-}
3−{-# LANGUAGE FlexibleContexts #-}
4−{-# LANGUAGE TypeApplications #-}
5−{-# LANGUAGE DataKinds #-}
6−
7−module Examples.Readme
8− ( exec
9− )
10− where
11−
12−import LAoP.Matrix.Type
13−import qualified LAoP.Relation as R
14−import LAoP.Utils
15−import LAoP.Dist
16−import GHC.TypeLits
17−import Data.Coerce
18−import qualified GHC.Generics as G
19−import Prelude hiding (id, (.))
20−
21−-- Monty Hall Problem
22−data Outcome = Win | Lose
23− deriving (Bounded, Enum, Eq, Show, G.Generic)
24−
25−switch :: Outcome -> Outcome
26−switch Win = Lose
27−switch Lose = Win
28−
29−firstChoice :: Matrix Double () Outcome
30−firstChoice = col [1/3, 2/3]
31−
32−secondChoice :: Matrix Double Outcome Outcome
33−secondChoice = fromF switch
34−
35−-- Dice sum
36−
37−type SS = Natural 1 6 -- Sample Space
38−
39−sumSS :: SS -> SS -> Natural 2 12
40−sumSS = coerceNat (+)
41−
42−sumSSM = fromF (uncurry sumSS)
43−
44−condition :: (Int, Int) -> Int -> Int
45−condition (fst, snd) thrd = if fst == snd
46− then fst * 3
47− else fst + snd + thrd
48−
49−conditionSS :: (SS, SS) -> SS -> Natural 3 18
50−conditionSS = coerceNat2 condition
51−
52−conditionalThrows = fromF (uncurry conditionSS) . kr (kr die die) die
53−
54−die :: Matrix Double () SS
55−die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]
56−
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)
63−
64−data R = No | Yes
65− deriving (Bounded, Enum, Eq, Show, G.Generic)
66−
67−rain :: Matrix Double () R
68−rain = matrixBuilder gen
69− where
70− gen (_, No) = 0.8
71− gen (_, Yes) = 0.2
72−
73−sprinkler :: Matrix Double R S
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
92−
93−tag f = kr f id
94−
95−state g s r = tag g . tag s . r
96−
97−grass_wet :: Matrix Double () (G, (S, R)) -> Matrix Double One One
98−grass_wet state = row [0,1] . fstM . state
99−
100−rainning :: Matrix Double (G, (S, R)) One
101−rainning = row [0,1] . sndM . sndM
102−
103−-- Alcuin Puzzle
104−
105−data Being = Farmer | Fox | Goose | Beans
106− deriving (Bounded, Enum, Eq, Show, G.Generic)
107−
108−data Bank = LeftB | RightB
109− deriving (Bounded, Enum, Eq, Show, G.Generic)
110−
111−eats :: Being -> Being -> Bool
112−eats Fox Goose = True
113−eats Goose Beans = True
114−eats _ _ = False
115−
116−eatsR :: R.Relation Being Being
117−eatsR = R.toRel eats
118−
119−cross :: Bank -> Bank
120−cross LeftB = RightB
121−cross RightB = LeftB
122−
123−crossR :: R.Relation Bank Bank
124−crossR = R.fromF cross
125−
126−-- | Initial state, everyone in the left bank
127−locationLeft :: Being -> Bank
128−locationLeft _ = LeftB
129−
130−locationLeftR :: R.Relation Being Bank
131−locationLeftR = R.fromF locationLeft
132−
133−-- | Initial state, everyone in the right bank
134−locationRight :: Being -> Bank
135−locationRight _ = RightB
136−
137−locationRightR :: R.Relation Being Bank
138−locationRightR = R.fromF locationRight
139−
140−-- Properties
141−
142−-- Being at the same bank
143−sameBank :: R.Relation Being Bank -> R.Relation Being Being
144−sameBank = R.ker
145−
146−-- Risk of somebody eating somebody else
147−canEat :: R.Relation Being Bank -> R.Relation Being Being
148−canEat w = sameBank w `R.intersection` eatsR
149−
150−-- "Starvation" property.
151−inv :: R.Relation Being Bank -> Bool
152−inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)
153− where
154− farmer :: R.Relation Being Being
155− farmer = R.fromF (const Farmer)
156−
157−-- Arbitrary state
158−bankState :: Being -> Bank -> Bool
159−bankState Farmer LeftB = True
160−bankState Fox LeftB = True
161−bankState Goose RightB = True
162−bankState Beans RightB = True
163−bankState _ _ = False
164−
165−bankStateR :: R.Relation Being Bank
166−bankStateR = R.toRel bankState
167−
168−-- Main
169−exec :: IO ()
170−exec = do
171− putStrLn "Monty Hall Problem solution:"
172− prettyPrint (secondChoice . firstChoice)
173− putStrLn "\n Sum of dices probability:"
174− prettyPrint (sumSSM . kr die die)
175− putStrLn "\n Conditional dice throw:"
176− prettyPrint conditionalThrows
177− putStrLn "\n Checking that the last result is indeed a distribution: "
178− prettyPrint (bang . sumSSM . kr die die)
179− putStrLn "\n Probability of grass being wet:"
180− prettyPrint (grass_wet (state grass sprinkler rain))
181− putStrLn "\n Probability of rain:"
182− prettyPrint (rainning . state grass sprinkler rain)
183− putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
184− print (inv bankStateR)
test/Examples/Spreadsheets.hsdeleted−47
@@ -1,47 +0,0 @@
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+1−7
@@ -1,8 +1,2 @@
1−module Main (main) where
2−
3−import qualified Examples.Readme as RD
4−
5main :: IO ()
1main :: IO ()
6−main = do
2+main = putStrLn "Test suite not yet implemented"
7− putStrLn ("Testing if README works" :: String)
8− RD.exec