git.iko.soy/laop
on master help sign in
swap
eea623154 ahead, 0 behind
.ghc.environment.x86_64-linux-8.6.5deleted−17
since splitsnapshots
unifiedsplit
@@ -1,17 +0,0 @@
1−-- This is a GHC environment file written by cabal. This means you can
2−-- run ghc or ghci and get the environment of the project as a whole.
3−-- But you still need to use cabal repl $target to get the environment
4−-- of specific components (libs, exes, tests etc) because each one can
5−-- have its own source dirs, cpp flags etc.
6−--
7−clear-package-db
8−global-package-db
9−package-db /home/bolt/.cabal/store/ghc-8.6.5/package.db
10−package-db packagedb/ghc-8.6.5
11−package-id laop-0.1.0.0-inplace
12−package-id base-4.12.0.0
13−package-id ghc-prim-0.5.3
14−package-id rts
15−package-id integer-gmp-1.0.2.0
16−package-id deepseq-1.4.4.0
17−package-id array-0.5.3.0
.github/workflows/ci.ymlnew file+43
@@ -0,0 +1,43 @@
1+name: CI
2+
3+# Trigger the workflow on push or pull request, but only for the master branch
4+on:
5+ pull_request:
6+ push:
7+ branches: [master]
8+
9+jobs:
10+ build:
11+ name: ghc ${{ matrix.ghc }}
12+ runs-on: ubuntu-16.04
13+ strategy:
14+ matrix:
15+ cabal: ["3.0"]
16+ ghc:
17+ - "8.6.5"
18+ - "8.8.3"
19+
20+ steps:
21+ - uses: actions/checkout@v2
22+ if: github.event.action == 'opened' || github.event.action == 'synchronize' || github.event.ref == 'refs/heads/master'
23+
24+ - uses: actions/setup-haskell@v1
25+ name: Setup Haskell
26+ with:
27+ ghc-version: ${{ matrix.ghc }}
28+ cabal-version: ${{ matrix.cabal }}
29+
30+ - uses: actions/cache@v1
31+ name: Cache ~/.cabal/store
32+ with:
33+ path: ~/.cabal/store
34+ key: ${{ runner.os }}-${{ matrix.ghc }}-cabal
35+
36+ - name: Build
37+ run: |
38+ cabal v2-update
39+ cabal v2-build --enable-tests --enable-benchmarks
40+
41+ - name: Test
42+ run: |
43+ cabal v2-test --enable-tests
.gitignore+54−2
@@ -1,3 +1,55 @@
1+### Haskell
2+dist
3+dist-*
4+cabal-dev
5+*.o
6+*.hi
7+*.chi
8+*.chs.h
9+*.dyn_o
10+*.dyn_hi
11+*.prof
12+*.aux
13+*.hp
14+*.eventlog
15+.virtualenv
16+.hsenv
17+.hpc
18+.cabal-sandbox/
19+cabal.sandbox.config
20+cabal.config
21+cabal.project.local
22+.ghc.environment.*
23+.HTF/
24+# Stack
1.stack-work/
25.stack-work/
2−laop.cabal
26+stack.yaml.lock
27+
28+### IDE/support
29+# Vim
30+[._]*.s[a-v][a-z]
31+[._]*.sw[a-p]
32+[._]s[a-v][a-z]
33+[._]sw[a-p]
3−*~
34+*~
35+tags
36+
37+# IntellijIDEA
38+.idea/
39+.ideaHaskellLib/
40+*.iml
41+
42+# Atom
43+.haskell-ghc-mod.json
44+
45+# VS
46+.vscode/
47+
48+# Emacs
49+*#
50+.dir-locals.el
51+TAGS
52+
53+# other
54+.DS_Store
55+hackage.sh
.travis.ymlnew file+47
@@ -0,0 +1,47 @@
1+sudo: true
2+language: haskell
3+
4+git:
5+ depth: 5
6+
7+cabal: "3.0"
8+
9+cache:
10+ directories:
11+ - "$HOME/.cabal/store"
12+ - "$HOME/.stack"
13+ - "$TRAVIS_BUILD_DIR/.stack-work"
14+
15+matrix:
16+ include:
17+ - ghc: 8.6.5
18+ - ghc: 8.8.3
19+
20+ - ghc: 8.8.3
21+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack-8.8.3.yaml"
22+
23+ - ghc: 8.6.5
24+ env: STACK_YAML="$TRAVIS_BUILD_DIR/stack.yaml"
25+
26+
27+install:
28+ - |
29+ if [ -z "$STACK_YAML" ]; then
30+ cabal v2-update
31+ cabal v2-build --enable-tests --enable-benchmarks
32+ else
33+ curl -sSL https://get.haskellstack.org/ | sh
34+ stack --version
35+ stack build --system-ghc --test --bench --no-run-tests --no-run-benchmarks
36+ fi
37+
38+script:
39+ - |
40+ if [ -z "$STACK_YAML" ]; then
41+ cabal v2-test --enable-tests
42+ else
43+ stack test --system-ghc
44+ fi
45+
46+notifications:
47+ email: false
CHANGELOG.mdnew file+17
@@ -0,0 +1,17 @@
1+# Changelog
2+
3+`laop` uses [PVP Versioning][1].
4+The changelog is available [on GitHub][2].
5+
6+## 0.1.1.1
7+
8+* Bump base version to work with cabal
9+
10+## 0.1.1.0
11+
12+* Package is more organized and now has tests and benchmarks separated
13+* Repository is more organized and clean
14+* Added CI
15+
16+[1]: https://pvp.haskell.org
17+[2]: https://github.com/bolt12/laop2/releases
ChangeLog.mddeleted−3
@@ -1,3 +0,0 @@
1−# Changelog for laop
2−
3−## Unreleased changes
LICENSE+17−26
@@ -1,30 +1,21 @@
1−Copyright Armando Santos (c) 2019
1+MIT License
2
2
3−All rights reserved.
3+Copyright (c) 2020 Armando Santos
4
4
5−Redistribution and use in source and binary forms, with or without
5+Permission is hereby granted, free of charge, to any person obtaining a copy
6+of this software and associated documentation files (the "Software"), to deal
7+in the Software without restriction, including without limitation the rights
8+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
9+copies of the Software, and to permit persons to whom the Software is
6−modification, are permitted provided that the following conditions are met:
10+furnished to do so, subject to the following conditions:
7
11
8− * Redistributions of source code must retain the above copyright
12+The above copyright notice and this permission notice shall be included in all
9− notice, this list of conditions and the following disclaimer.
13+copies or substantial portions of the Software.
10
14
11− * Redistributions in binary form must reproduce the above
15+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
12− copyright notice, this list of conditions and the following
13− disclaimer in the documentation and/or other materials provided
14− with the distribution.
15−
16− * Neither the name of Armando Santos nor the names of other
17− contributors may be used to endorse or promote products derived
18− from this software without specific prior written permission.
19−
20−THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
21−"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
22−LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
16+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
23−A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
17+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
24−OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
18+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
25−SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
26−LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
27−DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
28−THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
19+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
29−(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
20+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
30−OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
21+SOFTWARE.
README.md+35−27
@@ -1,5 +1,11 @@
1# laop - Linear Algebra of Programming library
1# laop - Linear Algebra of Programming library
2
2
3+[![GitHub CI](https://github.com/bolt12/laop/workflows/CI/badge.svg)](https://github.com/bolt12/laop/actions)
4+[![Build status](https://img.shields.io/travis/bolt12/laop.svg?logo=travis)](https://travis-ci.org/bolt12/laop)
5+[![Hackage](https://img.shields.io/hackage/v/laop.svg?logo=haskell)](https://hackage.haskell.org/package/laop)
6+[![Stackage Lts](http://stackage.org/package/laop/badge/lts)](http://stackage.org/lts/package/laop)
7+[![Stackage Nightly](http://stackage.org/package/laop/badge/nightly)](http://stackage.org/nightly/package/laop)
8+
3The LAoP discipline generalises relations and functions treating them as
9The LAoP discipline generalises relations and functions treating them as
4Boolean matrices and in turn consider these as arrows.
10Boolean matrices and in turn consider these as arrows.
5
11
@@ -10,36 +16,38 @@ motivation behind the library, the underlying theory, and implementation details
10This module offers many of the combinators mentioned in the work of
16This module offers many of the combinators mentioned in the work of
11[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).
17[Macedo (2012)](https://repositorium.sdum.uminho.pt/handle/1822/22894) and [Oliveira (2012)](https://pdfs.semanticscholar.org/ccf5/27fa9179081223bffe8067edd81948644fc0.pdf).
12
18
13−See the candidate package in hackage [here](https://hackage.haskell.org/package/laop-0.1.0.0)
19+See the package in hackage [here](https://hackage.haskell.org/package/laop-0.1.1.0)
20+
21+A Functional Pearl has been written and can be regarded as the [reference document](https://github.com/bolt12/tymfgg-pearl) for this library.
14
22
15## Features
23## Features
16
24
17−This library offers:
25+This library offers 3 main matrix programming modules:
18−
26+
19−- An internal module representation that uses normal datatypes (`Void`, `()`, `Either`) for matrix dimensions and two
27+- One in which matrices are typed with type level natural numbers;
20− type families `FromNat` and `Count` to make it easier to work with these type of matrices.
28+- One in which matrices are typed with arbitrary generic data types;
21−- An interfacing module that uses generalised types for the dimensions and is just a newtype wrapper
29+- One in which matrices are regarded as relations, i.e. boolean matrices;
22− around the internal matrix data type.
30+
23−- An interfacing module that uses type level naturals for the dimensions and is just a newtype wrapper
31+There's also an experimental module that uses matrices to represent probability
24− around the internal matrix data type.
32+distributions.
25−- An interfacing module that restricts the matrix contents to being only 0's or 1's (True
33+
26− and False). This module is called 'Relation' and offers many of the combinators seen in
34+The most interesting feature is that matrices are represented as an inductive data type.
27− AoP and Relational Calculus/Algebra.
35+Given this formulation, matrix algorithms can be expressed in a much more elegant and
28−- A very simple Probabilistic Programming module and several functions and data types to
36+calculational style than the traditional vector of vectors representation. This data type
29− make it easier to deal with sample space.
37+guarantees that a matrix will always have valid dimensions. It can also express block matrix
30−
38+computations naturally, which leads to total, efficient and statically typed manipulation
31−Given this, this new matrix formulation compared to other libraries one has much more advantages, such as:
39+and transformation functions.
32−
40+
33−- Has an inductive definition enabling writing matrix manipulation functions in a much more elegant and calculational way;
41+Like matrix multiplication, other common operations, such as matrix transposition, benefit from a block-oriented structure that leads to a simple and natural divide-and-conquer algorithmic solution. Performance wise, this means that without much effort we can obtain optimal cache-oblivious algorithms.
34−- Statically typed dimensions;
42+
35−- Polymorphic data type dimensions;
43+Given this, this new matrix formulation compared to other libraries:
36−- Polymorphic matrix content;
44+
37−- Awesome dimension type inference;
45+- Is more compositional and polymorphic and does not have partial matrix manipulation functions (hence less chances for usage errors);
38−- Fast type natural conversion via `FromNat` type family;
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.
39−- Matrix `Junc` and `Split`-ing in O(1);
47+
40−- Matrix composition takes advantage of divide-and-conquer and fusion laws.
48+## Known issues
41−
49+
42−Unfortunately, this approach does not solve the issue of type dimensions being in some way constrainted making it impossible to write Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed.
50+Unfortunately, due to the use of type-level programming features, this approach sometimes requires type dimensions to be constrainted, in some way, impossible to write idiomatic Arrow instances, for example. Type inference isn't perfect, when it comes to infer the types of matrices which dimensions are computed using type level naturals multiplication, the compiler needs type annotations in order to succeed. And not all kinds of programs can be modeled using matrices, namely programs that deal with arbitrary infinite data types such as lists or integers (although there are some workarounds).
43
51
44## Notes
52## Notes
45
53
Setup.hsdeleted−2
@@ -1,2 +0,0 @@
1−import Distribution.Simple
2−main = defaultMain
appveyor.ymlnew file+34
@@ -0,0 +1,34 @@
1+clone_folder: "c:\\WORK"
2+clone_depth: 5
3+
4+# Do not build feature branch with open Pull Requests
5+skip_branch_with_pr: true
6+
7+platform:
8+ - x86_64
9+
10+cache:
11+ - "C:\\SR"
12+ - dist-newstyle
13+
14+environment:
15+ global:
16+ CABOPTS: --store-dir=C:\\SR
17+
18+ matrix:
19+ - GHCVER: 8.6.5
20+
21+install:
22+ - choco source add -n mistuke -s https://www.myget.org/F/mistuke/api/v2
23+ - choco install -y cabal --version 2.4.1.0
24+ - choco install -y ghc --version 8.6.3
25+ - refreshenv
26+
27+before_build:
28+ - cabal --version
29+ - ghc --version
30+ - cabal %CABOPTS% v2-update
31+
32+build_script:
33+ - cabal %CABOPTS% v2-build --enable-tests
34+ - cabal %CABOPTS% v2-test --enable-tests
benchmark/Main.hsnew file+175
@@ -0,0 +1,175 @@
1+{-# LANGUAGE KindSignatures #-}
2+{-# LANGUAGE GeneralizedNewtypeDeriving #-}
3+{-# LANGUAGE DerivingVia #-}
4+{-# LANGUAGE DataKinds #-}
5+{-# LANGUAGE BangPatterns #-}
6+{-# LANGUAGE TypeApplications #-}
7+{-# LANGUAGE FlexibleContexts #-}
8+{-# LANGUAGE AllowAmbiguousTypes #-}
9+{-# LANGUAGE ScopedTypeVariables #-}
10+
11+module Main (main) where
12+
13+import LAoP.Dist.Internal
14+import LAoP.Utils
15+import LAoP.Matrix.Type hiding (CountableN, Countable, FLN)
16+import qualified LAoP.Matrix.Type as T (FLN)
17+import Criterion.Main
18+import Control.DeepSeq
19+import Test.QuickCheck
20+import GHC.TypeLits
21+import Data.Proxy
22+import Prelude hiding (id, (.))
23+
24+selectM2 ::
25+ ( Num e,
26+ T.FLN b b,
27+ CountableN b
28+ ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
29+selectM2 m y = join y iden `comp` m
30+
31+selectD2 ::
32+ ( FLN b b,
33+ CountableN b
34+ ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
35+selectD2 (D d) m = D (join m iden `comp` d)
36+
37+-- Composition with iden
38+compId m = comp iden m
39+
40+randomDist :: forall a . (Countable a, FLN () a) => Gen (Dist a)
41+randomDist = do
42+ let size = fromInteger (natVal (Proxy :: Proxy (Count a)))
43+ l <- vectorOf size (arbitrary :: Gen Prob)
44+ let ln = normalize l
45+ lr = map (:[]) ln
46+ m = fromLists lr
47+ return (D m)
48+
49+randomMatrix :: forall a b . (CountableDims a b, FLN a b) => Gen (Matrix Prob a b)
50+randomMatrix = do
51+ let cols = fromInteger (natVal (Proxy :: Proxy (Count a)))
52+ rows = fromInteger (natVal (Proxy :: Proxy (Count b)))
53+ l <- vectorOf (cols * rows) arbitrary
54+ let lr = buildList l cols
55+ m = fromLists lr
56+ return m
57+
58+buildList [] _ = []
59+buildList l r = take r l : buildList (drop r l) r
60+
61+normalize :: [Prob] -> [Prob]
62+normalize l = let l' = map abs l
63+ s = sum l'
64+ in map (/ s) l'
65+
66+-- Probability
67+newtype Probability = P Double
68+ deriving (Num, Show, Fractional, NFData) via Double
69+newtype Dist' a = D' { unD :: [(a,Probability)] }
70+ deriving (Show, NFData)
71+
72+instance Functor Dist' where
73+ fmap f (D' d) = D' [(f x,p) | (x,p) <- d]
74+
75+instance Applicative Dist' where
76+ pure x = D' [(x, 1)]
77+ fm <*> m = D' [(f x,q*p) | (f,p) <- unD fm, (x,q) <- unD m]
78+
79+class Applicative f => Selective f where
80+ select :: f (Either a b) -> f (a -> b) -> f b
81+
82+instance Selective Dist' where
83+ select x y = (\e f -> either f id e) <$> x <*> y -- selectA
84+
85+instance Monad Dist' where
86+ return = pure
87+ (D' d) >>= f = D' [(y,q*p) | (x,p) <- d, (y,q) <- unD (f x)]
88+
89+normalize2 :: forall a . (Enum a, Bounded a) => [Probability] -> [(a, Probability)]
90+normalize2 l = let l' = map abs l
91+ as = [minBound .. maxBound]
92+ s = sum l'
93+ probs = map (/ s) l'
94+ in zip as probs
95+
96+randomDist2 :: forall a . (Countable a, Enum a, Bounded a) => Gen (Dist' a)
97+randomDist2 = do
98+ let size = fromInteger (natVal (Proxy :: Proxy (Count a)))
99+ l <- vectorOf size (arbitrary :: Gen Prob)
100+ let ln = normalize2 @a (map P l)
101+ return (D' ln)
102+
103+normalize3 :: forall a . [a] -> [Probability] -> [(a, Probability)]
104+normalize3 as l = let s = sum l
105+ probs = map (/ s) l
106+ in zip as probs
107+
108+randomDistF :: forall a b . (CountableDims a b, CoArbitrary a, Arbitrary b) => Gen (Dist' (a -> b))
109+randomDistF = do
110+ let a = fromInteger (natVal (Proxy :: Proxy (Count a)))
111+ b = fromInteger (natVal (Proxy :: Proxy (Count b)))
112+ size = a * b
113+ l <- vectorOf size (arbitrary :: Gen Prob)
114+ l2 <- vectorOf size (arbitrary :: Gen (a -> b))
115+ let !ln = normalize3 l2 (map P l)
116+ return (D' ln)
117+
118+instance CoArbitrary (Natural a b)
119+
120+instance forall (a :: Nat) (b :: Nat). (KnownNat a, KnownNat b) => Arbitrary (Natural a b)
121+ where
122+ arbitrary =
123+ let bottom = fromInteger (natVal (Proxy :: Proxy a))
124+ top = fromInteger (natVal (Proxy :: Proxy b))
125+ in do
126+ x <- arbitrary `suchThat` (\n -> n >= bottom && n <= top)
127+ return (nat x)
128+
129+setupEnv = do
130+ m11 <- generate (resize 1 (randomMatrix @(Natural 0 10) @(Natural 0 10) ))
131+ m12 <- generate (resize 1 (randomMatrix @(Natural 0 10) @(Natural 0 10) ))
132+ m21 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
133+ m22 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
134+ m31 <- generate (resize 1 (randomMatrix @(Natural 0 200) @(Natural 0 200) ))
135+ m32 <- generate (resize 1 (randomMatrix @(Natural 0 200) @(Natural 0 200) ))
136+ return (m11, m12, m21, m22, m31, m32)
137+
138+setupEnv2 = do
139+ m21 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
140+ m40 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Either (Natural 0 100) (Natural 0 100) )))
141+ dist <- generate (resize 1 (randomDist @(Either (Natural 0 100) (Natural 0 100)) ))
142+ dist2 <- generate (resize 1 (randomMatrix @(Natural 0 100) @(Natural 0 100) ))
143+ distList1 <- generate (resize 1 (randomDist2 @(Either (Natural 0 100) (Natural 0 100))))
144+ distList2 <- generate (resize 1 (randomDistF @(Natural 0 100) @(Natural 0 100)))
145+ return (m21, m40, dist, dist2, distList1, distList2)
146+
147+main :: IO ()
148+main = defaultMain [
149+ env setupEnv $ \ ~(m11, m12, m21, m22, m31, m32) -> bgroup "Matrix Composition" [
150+ bgroup "10x10" [
151+ bench "WHNF - 10x10" $ whnf (comp m11) m12
152+ , bench "NF - 10x10" $ nf (comp m11) m12
153+ ],
154+ bgroup "100x100" [
155+ bench "WHNF - 100x100" $ whnf (comp m21) m22
156+ , bench "NF - 100x100" $ nf (comp m21) m22
157+ ],
158+ bgroup "200x200" [
159+ bench "WHNF - 200x200" $ whnf (comp m31) m32
160+ , bench "NF - 200x200" $ nf (comp m31) m32
161+ ],
162+ bgroup "200x200 - RULES" [
163+ bench "No rules - 200x200" $ nf (comp m21) iden
164+ , bench "Rules - 200x200" $ nf compId m21
165+ ] ],
166+ env setupEnv2 $ \ ~(m21, m40, dist, dist2, dl1, dl2) -> bgroup "Matrix vs List - `select`" [
167+ bgroup "Distribution `select` - 100+100 / 100x100" [
168+ bench "List Distribution - Applicative version" $ nf (select dl1) dl2
169+ , bench "Matrix Distribution - Applicative version" $ nf (selectD2 dist) dist2
170+ , bench "Matrix Distribution - Selective version" $ nf (selectD dist) dist2
171+ ],
172+ bgroup "Matrix `select` - 100x(100+100) / 100x100" [
173+ bench "Applicative version" $ nf (selectM2 m40) m21
174+ , bench "Selective version" $ nf (selectM m40) m21
175+ ] ] ]
hackage.shdeleted−10
@@ -1,10 +0,0 @@
1−#!/bin/sh
2−set -e
3−
4−dir=$(mktemp -d dist-docs.XXXXXX)
5−trap 'rm -r "$dir"' EXIT
6−
7−# assumes cabal 2.4 or later
8−cabal v2-haddock --builddir="$dir" --haddock-for-hackage --enable-doc
9−
10−cabal upload -d $dir/*-docs.tar.gz --publish
laop-0.1.0.0-docs.tar.gzdeleted

Binary file not shown.

laop.cabalnew file+89
@@ -0,0 +1,89 @@
1+cabal-version: 2.4
2+name: laop
3+version: 0.1.1.1
4+synopsis: Matrix programming library
5+description: Matrix programming library that uses a type-safe inductive definition
6+ inspired by the Linear Algebra of Programming
7+homepage: https://github.com/bolt12/laop
8+bug-reports: https://github.com/bolt12/laop/issues
9+license: MIT
10+license-file: LICENSE
11+author: Armando Santos
12+maintainer: Armando Santos <armandoifsantos@gmail.com>
13+copyright: 2020 Armando Santos
14+category: Data
15+build-type: Simple
16+extra-doc-files: README.md
17+ CHANGELOG.md
18+tested-with: GHC == 8.6.5
19+ GHC == 8.8.3
20+
21+source-repository head
22+ type: git
23+ location: https://github.com/bolt12/laop.git
24+
25+common common-options
26+ build-depends: base >= 4.12.0.0 && < 5,
27+ deepseq
28+
29+ ghc-options: -Wall
30+ -Wcompat
31+ -Widentities
32+ -Wincomplete-uni-patterns
33+ -Wincomplete-record-updates
34+ if impl(ghc >= 8.0)
35+ ghc-options: -Wredundant-constraints
36+ if impl(ghc >= 8.2)
37+ ghc-options: -fhide-source-paths
38+ if impl(ghc >= 8.4)
39+ ghc-options: -Wmissing-export-lists
40+ -Wpartial-fields
41+ if impl(ghc >= 8.8)
42+ ghc-options: -Wmissing-deriving-strategies
43+
44+ default-language: Haskell2010
45+
46+library
47+ import: common-options
48+ hs-source-dirs: src
49+ exposed-modules: LAoP.Dist
50+ LAoP.Dist.Internal
51+ LAoP.Matrix.Internal
52+ LAoP.Matrix.Nat
53+ LAoP.Matrix.Type
54+ LAoP.Relation
55+ LAoP.Relation.Internal
56+ LAoP.Utils
57+ LAoP.Utils.Internal
58+
59+test-suite laop-test
60+ import: common-options
61+ type: exitcode-stdio-1.0
62+ hs-source-dirs: test
63+ other-modules: Examples.Readme
64+ main-is: Spec.hs
65+ build-depends: laop
66+ ghc-options: -threaded
67+ -rtsopts
68+ -with-rtsopts=-N
69+ -funfolding-use-threshold=16
70+ -fexcess-precision
71+ -optc-O3
72+ -optc-ffast-math
73+
74+benchmark laop-benchmark
75+ import: common-options
76+ type: exitcode-stdio-1.0
77+ hs-source-dirs: benchmark
78+ main-is: Main.hs
79+ build-depends: criterion
80+ , QuickCheck
81+ , deepseq
82+ , laop
83+ ghc-options: -threaded
84+ -rtsopts
85+ -with-rtsopts=-N
86+ -funfolding-use-threshold=16
87+ -fexcess-precision
88+ -optc-O3
89+ -optc-ffast-math
package.yamldeleted−48
@@ -1,48 +0,0 @@
1−name: laop
2−version: 0.1.0.0
3−github: "bolt12/laop"
4−license: BSD3
5−author: "Armando Santos"
6−maintainer: "armandoifsantos@gmail.com"
7−copyright: "MIT"
8−
9−extra-source-files:
10−- README.md
11−- ChangeLog.md
12−
13−# Metadata used when publishing your package
14−# synopsis: Short description of your package
15−# category: Web
16−
17−# To avoid duplicated efforts in documentation and dealing with the
18−# complications of embedding Haddock markup inside cabal files, it is
19−# common to point users to the README.md file.
20−description: Please see the README on GitHub at <https://github.com/bolt12/laop#readme>
21−
22−dependencies:
23−- base >= 4.7 && < 5
24−
25−library:
26− source-dirs: src
27−
28−#executables:
29−# laop-exe:
30−# main: Main.hs
31−# source-dirs: app
32−# ghc-options:
33−# - -threaded
34−# - -rtsopts
35−# - -with-rtsopts=-N
36−# dependencies:
37−# - laop
38−
39−tests:
40− laop-test:
41− main: Spec.hs
42− source-dirs: test
43− ghc-options:
44− - -threaded
45− - -rtsopts
46− - -with-rtsopts=-N
47− dependencies:
48− - laop
src/LAoP/Dist.hs+4−3
@@ -39,11 +39,10 @@ module LAoP.Dist
39 -- * Constraint type synonyms
39 -- * Constraint type synonyms
40 Countable,
40 Countable,
41 CountableN,
41 CountableN,
42− CountableDimensionsN,
42+ CountableDimsN,
43− FromListsN,
43+ FLN,
44 Liftable,
44 Liftable,
45 TrivialP,
45 TrivialP,
46− TrivialE,
47
46
48 -- * Functor instance equivalent functions
47 -- * Functor instance equivalent functions
49 fmapD,
48 fmapD,
@@ -54,6 +53,8 @@ module LAoP.Dist
54
53
55 -- * Selective equivalent functions
54 -- * Selective equivalent functions
56 selectD,
55 selectD,
56+ branchD,
57+ ifD,
57
58
58 -- * Monad equivalent functions
59 -- * Monad equivalent functions
59 returnD,
60 returnD,
src/LAoP/Dist/Internal.hs+71−33
@@ -1,3 +1,4 @@
1+{-# LANGUAGE DerivingVia #-}
1{-# LANGUAGE ConstraintKinds #-}
2{-# LANGUAGE ConstraintKinds #-}
2{-# LANGUAGE GADTs #-}
3{-# LANGUAGE GADTs #-}
3{-# LANGUAGE AllowAmbiguousTypes #-}
4{-# LANGUAGE AllowAmbiguousTypes #-}
@@ -15,16 +16,17 @@ module LAoP.Dist.Internal
15
16
16 Countable,
17 Countable,
17 CountableN,
18 CountableN,
18− CountableDimensionsN,
19+ CountableDimsN,
19− FromListsN,
20+ FLN,
20 Liftable,
21 Liftable,
21 TrivialP,
22 TrivialP,
22− TrivialE,
23
23
24 fmapD,
24 fmapD,
25 unitD,
25 unitD,
26 multD,
26 multD,
27 selectD,
27 selectD,
28+ branchD,
29+ ifD,
28 returnD,
30 returnD,
29 bindD,
31 bindD,
30 (??),
32 (??),
@@ -41,12 +43,15 @@ module LAoP.Dist.Internal
41 )
43 )
42 where
44 where
43
45
44−import LAoP.Matrix.Type hiding (TrivialP, TrivialE, Countable, CountableDimensions, CountableN, CountableDimensionsN, Liftable, FromListsN)
46+import LAoP.Matrix.Type hiding (TrivialP, Countable, CountableDims, CountableN, CountableDimsN, Liftable, FLN)
47+import Prelude hiding (id, (.))
45import qualified LAoP.Matrix.Internal as I
48import qualified LAoP.Matrix.Internal as I
46import LAoP.Utils
49import LAoP.Utils
47import GHC.TypeLits
50import GHC.TypeLits
48import Data.Proxy
51import Data.Proxy
49import Data.List (sortBy)
52import Data.List (sortBy)
53+import Control.DeepSeq
54+import Data.Bool
50
55
51-- | Type synonym for probability value
56-- | Type synonym for probability value
52type Prob = Double
57type Prob = Double
@@ -54,21 +59,21 @@ type Prob = Double
54-- | Type synonym for column vector matrices. This represents a probability
59-- | Type synonym for column vector matrices. This represents a probability
55-- distribution.
60-- distribution.
56newtype Dist a = D (Matrix Prob () a)
61newtype Dist a = D (Matrix Prob () a)
62+ deriving (Show, Num, Eq, Ord, NFData) via (Matrix Prob () a)
57
63
58-- | Constraint type synonyms to keep the type signatures less convoluted
64-- | Constraint type synonyms to keep the type signatures less convoluted
59−type Countable a = KnownNat (I.Count a)
65+type Countable a = KnownNat (I.Count a)
60−type CountableN a = KnownNat (I.Count (I.Normalize a))
66+type CountableN a = KnownNat (I.Count (I.Normalize a))
61−type CountableDimensionsN a b = (CountableN a, CountableN b)
67+type CountableDimsN a b = (CountableN a, CountableN b)
62−type FromListsN a b = I.FromLists Prob (I.Normalize a) (I.Normalize b)
68+type FLN a b = I.FL (I.Normalize a) (I.Normalize b)
63−type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Prob, Ord Prob)
69+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Prob, Ord Prob)
64−type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)
70+type TrivialP a b = Normalize (a, b) ~ Normalize (Normalize a, Normalize b)
65−type TrivialE a b = Normalize (Either a b) ~ Either (Normalize a) (Normalize b)
66
71
67-- | Functor instance
72-- | Functor instance
68fmapD ::
73fmapD ::
69 ( Liftable a b,
74 ( Liftable a b,
70− CountableDimensionsN a b,
75+ CountableDimsN a b,
71− FromListsN b a
76+ FLN b a
72 )
77 )
73 =>
78 =>
74 (a -> b) -> Dist a -> Dist b
79 (a -> b) -> Dist a -> Dist b
@@ -80,25 +85,58 @@ unitD = D (one 1)
80
85
81-- | Applicative/Monoidal instance 'mult' function
86-- | Applicative/Monoidal instance 'mult' function
82multD ::
87multD ::
83− ( CountableDimensionsN a b,
88+ ( CountableDimsN a b,
84 CountableN (a, b),
89 CountableN (a, b),
85− FromListsN (a, b) a,
90+ FLN (a, b) a,
86− FromListsN (a, b) b,
91+ FLN (a, b) b,
87 TrivialP a b
92 TrivialP a b
88 ) => Dist a -> Dist b -> Dist (a, b)
93 ) => Dist a -> Dist b -> Dist (a, b)
89−multD (D a) (D b) = D (khatri a b)
94+multD (D a) (D b) = D (kr a b)
90
95
91-- | Selective instance function
96-- | Selective instance function
92selectD ::
97selectD ::
93− ( TrivialE a b,
98+ ( FLN b b,
94− FromListsN b b,
95 CountableN b
99 CountableN b
96 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
100 ) => Dist (Either a b) -> Matrix Prob a b -> Dist b
97−selectD (D d) m = D (junc m identity `comp` d)
101+selectD (D d) m = D (selectM d m)
98−
102+
103+-- | Chooses which of the two given effectful
104+-- functions to apply to a given argument;
105+branchD ::
106+ ( Num e,
107+ CountableDimsN a b,
108+ CountableDimsN c (Either b c),
109+ FLN c b,
110+ FLN a b,
111+ FLN a a,
112+ FLN b b,
113+ FLN c c,
114+ FLN b a,
115+ FLN b c,
116+ FLN (Either b c) b,
117+ FLN (Either b c) c
118+ )
119+ => Dist (Either a b) -> Matrix Prob a c -> Matrix Prob b c -> Dist c
120+branchD x l r = f x `selectD` g l `selectD` r
121+ where
122+ f (D m) = D (fork (tr i1) (i1 `comp` tr i2) `comp` m)
123+ g m = i2 `comp` m
124+
125+-- | Branch on a Boolean value, skipping unnecessary computations.
126+ifD ::
127+ ( CountableDimsN a (Either () a),
128+ FLN a a,
129+ FLN a (),
130+ FLN () a,
131+ FLN (Either () a) a
132+ )
133+ => Dist Bool -> Dist a -> Dist a -> Dist a
134+ifD x (D t) (D e) = branchD x' t e
135+ where
136+ x' = bool (Right ()) (Left ()) `fmapD` x
99
137
100-- | Monad instance 'return' function
138-- | Monad instance 'return' function
101−returnD :: forall a . (Enum a, FromListsN () a, Countable a) => a -> Dist a
139+returnD :: forall a . (Enum a, FLN () a, Countable a) => a -> Dist a
102returnD a = D (col l)
140returnD a = D (col l)
103 where
141 where
104 i = fromInteger $ natVal (Proxy :: Proxy (Count a))
142 i = fromInteger $ natVal (Proxy :: Proxy (Count a))
@@ -113,7 +151,7 @@ bindD (D d) m = D (m `comp` d)
113(??) ::
151(??) ::
114 ( Enum a,
152 ( Enum a,
115 Countable a,
153 Countable a,
116− FromListsN () a
154+ FLN () a
117 ) => (a -> Bool) -> Dist a -> Prob
155 ) => (a -> Bool) -> Dist a -> Prob
118(??) p d =
156(??) p d =
119 let l = toValues d
157 let l = toValues d
@@ -123,11 +161,11 @@ bindD (D d) m = D (m `comp` d)
123-- Distribution Construction
161-- Distribution Construction
124
162
125-- | Constructs a Bernoulli distribution
163-- | Constructs a Bernoulli distribution
126−choose :: (FromListsN () a) => Prob -> Dist a
164+choose :: (FLN () a) => Prob -> Dist a
127choose prob = D (col [prob, 1 - prob])
165choose prob = D (col [prob, 1 - prob])
128
166
129-- | Creates a distribution given a shape function
167-- | Creates a distribution given a shape function
130−shape :: (FromListsN () a) => (Prob -> Prob) -> [a] -> Dist a
168+shape :: (FLN () a) => (Prob -> Prob) -> [a] -> Dist a
131shape _ [] = error "Probability.shape: empty list"
169shape _ [] = error "Probability.shape: empty list"
132shape f xs =
170shape f xs =
133 let incr = 1 / fromIntegral (length xs - 1)
171 let incr = 1 / fromIntegral (length xs - 1)
@@ -135,23 +173,23 @@ shape f xs =
135 in fromFreqs (zip xs ps)
173 in fromFreqs (zip xs ps)
136
174
137-- | Constructs a Linear distribution
175-- | Constructs a Linear distribution
138−linear :: (FromListsN () a) => [a] -> Dist a
176+linear :: (FLN () a) => [a] -> Dist a
139linear = shape id
177linear = shape id
140
178
141-- | Constructs an Uniform distribution
179-- | Constructs an Uniform distribution
142−uniform :: (FromListsN () a) => [a] -> Dist a
180+uniform :: (FLN () a) => [a] -> Dist a
143uniform = shape (const 1)
181uniform = shape (const 1)
144
182
145-- | Constructs an Negative Exponential distribution
183-- | Constructs an Negative Exponential distribution
146−negExp :: (FromListsN () a) => [a] -> Dist a
184+negExp :: (FLN () a) => [a] -> Dist a
147negExp = shape (\x -> exp (-x))
185negExp = shape (\x -> exp (-x))
148
186
149-- | Constructs an Normal distribution
187-- | Constructs an Normal distribution
150−normal :: (FromListsN () a) => [a] -> Dist a
188+normal :: (FLN () a) => [a] -> Dist a
151normal = shape (normalCurve 0.5 0.5)
189normal = shape (normalCurve 0.5 0.5)
152
190
153-- | Transforms a 'Dist' into a list of pairs.
191-- | Transforms a 'Dist' into a list of pairs.
154−toValues :: forall a . (Enum a, Countable a, FromListsN () a) => Dist a -> [(a, Prob)]
192+toValues :: forall a . (Enum a, Countable a, FLN () a) => Dist a -> [(a, Prob)]
155toValues (D d) =
193toValues (D d) =
156 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))
194 let rows = fromInteger (natVal (Proxy :: Proxy (Count a)))
157 probs = toList d
195 probs = toList d
@@ -159,7 +197,7 @@ toValues (D d) =
159 in res
197 in res
160
198
161-- | Pretty a distribution
199-- | Pretty a distribution
162−prettyDist :: forall a. (Show a, Enum a, Countable a, FromListsN () a) => Dist a -> String
200+prettyDist :: forall a. (Show a, Enum a, Countable a, FLN () a) => Dist a -> String
163prettyDist d =
201prettyDist d =
164 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)
202 let values = sortBy (\(a, p1) (b, p2) -> compare p2 p1) (toValues @a d)
165 w = maximum (map (length . show . fst) values)
203 w = maximum (map (length . show . fst) values)
@@ -171,12 +209,12 @@ prettyDist d =
171 showR n x = show x ++ " "
209 showR n x = show x ++ " "
172
210
173-- | Pretty Print a distribution
211-- | Pretty Print a distribution
174−prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FromListsN () a) => Dist a -> IO ()
212+prettyPrintDist :: forall a . (Show a, Enum a, Countable a, FLN () a) => Dist a -> IO ()
175prettyPrintDist = putStrLn . prettyDist @a
213prettyPrintDist = putStrLn . prettyDist @a
176
214
177-- Auxiliary functions
215-- Auxiliary functions
178
216
179−fromFreqs :: (FromListsN () a) => [(a,Prob)] -> Dist a
217+fromFreqs :: (FLN () a) => [(a,Prob)] -> Dist a
180fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))
218fromFreqs xs = D (col (map (\(x,p) -> p/q) xs))
181 where q = sum $ map snd xs
219 where q = sum $ map snd xs
182
220
src/LAoP/Matrix/Internal.hs+414−379
@@ -1,3 +1,4 @@
1+{-# LANGUAGE StrictData #-}
1{-# LANGUAGE TypeFamilyDependencies #-}
2{-# LANGUAGE TypeFamilyDependencies #-}
2{-# LANGUAGE ConstraintKinds #-}
3{-# LANGUAGE ConstraintKinds #-}
3{-# LANGUAGE AllowAmbiguousTypes #-}
4{-# LANGUAGE AllowAmbiguousTypes #-}
@@ -29,7 +30,7 @@
29-- motivation behind the library, the underlying theory, and implementation details.
30-- motivation behind the library, the underlying theory, and implementation details.
30--
31--
31-- This module offers many of the combinators mentioned in the work of
32-- This module offers many of the combinators mentioned in the work of
32−-- Macedo (2012) and Oliveira (2012).
33+-- Macedo (2012) and Oliveira (2012).
33--
34--
34-- This is an Internal module and it is no supposed to be imported.
35-- This is an Internal module and it is no supposed to be imported.
35--
36--
@@ -42,17 +43,26 @@ module LAoP.Matrix.Internal
42 --
43 --
43 -- There exists two type families that make it easier to write
44 -- There exists two type families that make it easier to write
44 -- matrix dimensions: 'FromNat' and 'Count'. This approach
45 -- matrix dimensions: 'FromNat' and 'Count'. This approach
45− -- leads to a very straightforward implementation
46+ -- leads to a very straightforward implementation
46− -- of LAoP combinators.
47+ -- of LAoP combinators.
47
48
48 -- * Type safe matrix representation
49 -- * Type safe matrix representation
49 Matrix (..),
50 Matrix (..),
50
51
52+ -- * Constraint type aliases
53+ Countable,
54+ CountableDims,
55+ CountableN,
56+ CountableDimsN,
57+ FL,
58+ FLN,
59+ Liftable,
60+ Trivial,
61+
51 -- * Primitives
62 -- * Primitives
52− empty,
53 one,
63 one,
54− junc,
64+ join,
55− split,
65+ fork,
56
66
57 -- * Auxiliary type families
67 -- * Auxiliary type families
58 FromNat,
68 FromNat,
@@ -65,6 +75,7 @@ module LAoP.Matrix.Internal
65 toLists,
75 toLists,
66 toList,
76 toList,
67 matrixBuilder,
77 matrixBuilder,
78+ matrixBuilder',
68 row,
79 row,
69 col,
80 col,
70 zeros,
81 zeros,
@@ -75,28 +86,38 @@ module LAoP.Matrix.Internal
75 -- * Misc
86 -- * Misc
76 -- ** Get dimensions
87 -- ** Get dimensions
77 columns,
88 columns,
89+ columns',
78 rows,
90 rows,
91+ rows',
79
92
80 -- ** Matrix Transposition
93 -- ** Matrix Transposition
81 tr,
94 tr,
82
95
96+ -- ** Scalar multiplication/division of matrices
97+ (.|),
98+ (./),
99+
83 -- ** Selective operator
100 -- ** Selective operator
84− select,
101+ select,
102+ branch,
85
103
86 -- ** McCarthy's Conditional
104 -- ** McCarthy's Conditional
87 cond,
105 cond,
88
106
89 -- ** Matrix "abiding"
107 -- ** Matrix "abiding"
90− abideJS,
108+ abideJF,
91− abideSJ,
109+ abideFJ,
110+
111+ -- ** Zip matrices
112+ zipWithM,
92
113
93 -- * Biproduct approach
114 -- * Biproduct approach
94− -- ** Split
115+ -- ** Fork
95 (===),
116 (===),
96 -- *** Projections
117 -- *** Projections
97 p1,
118 p1,
98 p2,
119 p2,
99− -- ** Junc
120+ -- ** Join
100 (|||),
121 (|||),
101 -- *** Injections
122 -- *** Injections
102 i1,
123 i1,
@@ -109,12 +130,13 @@ module LAoP.Matrix.Internal
109
130
110 -- | Note that given the restrictions imposed it is not possible to
131 -- | Note that given the restrictions imposed it is not possible to
111 -- implement the standard type classes present in standard Haskell.
132 -- implement the standard type classes present in standard Haskell.
133+
112 -- *** Matrix pairing projections
134 -- *** Matrix pairing projections
113− kp1,
135+ fstM,
114− kp2,
136+ sndM,
115
137
116 -- *** Matrix pairing
138 -- *** Matrix pairing
117− khatri,
139+ kr,
118
140
119 -- * Matrix composition and lifting
141 -- * Matrix composition and lifting
120
142
@@ -122,10 +144,10 @@ module LAoP.Matrix.Internal
122
144
123 -- | Note that given the restrictions imposed it is not possible to
145 -- | Note that given the restrictions imposed it is not possible to
124 -- implement the standard type classes present in standard Haskell.
146 -- implement the standard type classes present in standard Haskell.
125− identity,
147+ iden,
126 comp,
148 comp,
127− fromF,
128 fromF',
149 fromF',
150+ fromF,
129
151
130 -- * Matrix printing
152 -- * Matrix printing
131 pretty,
153 pretty,
@@ -152,21 +174,22 @@ import LAoP.Utils.Internal
152import Data.Bool
174import Data.Bool
153import Data.Kind
175import Data.Kind
154import Data.List
176import Data.List
177+import Data.Maybe
155import Data.Proxy
178import Data.Proxy
179+import Data.Typeable
156import Data.Void
180import Data.Void
157import GHC.TypeLits
181import GHC.TypeLits
158import Data.Type.Equality
182import Data.Type.Equality
159import GHC.Generics
183import GHC.Generics
160import Control.DeepSeq
184import Control.DeepSeq
161−import Control.Category
185+import Prelude hiding (id, (.))
162−import Prelude hiding ((.))
186+import qualified Prelude (id, (.))
163
187
164-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
188-- | LAoP (Linear Algebra of Programming) Inductive Matrix definition.
165data Matrix e cols rows where
189data Matrix e cols rows where
166− Empty :: Matrix e Void Void
167 One :: e -> Matrix e () ()
190 One :: e -> Matrix e () ()
168− Junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
191+ Join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
169− Split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
192+ Fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
170
193
171deriving instance (Show e) => Show (Matrix e cols rows)
194deriving instance (Show e) => Show (Matrix e cols rows)
172
195
@@ -193,205 +216,197 @@ type family Count (d :: Type) :: Nat where
193--
216--
194-- Thanks to Li-Yao Xia this type family is super fast.
217-- Thanks to Li-Yao Xia this type family is super fast.
195type family FromNat (n :: Nat) :: Type where
218type family FromNat (n :: Nat) :: Type where
196− FromNat 0 = Void
197 FromNat 1 = ()
219 FromNat 1 = ()
198 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
220 FromNat n = FromNat' (Mod n 2 == 0) (FromNat (Div n 2))
199
221
200type family FromNat' (b :: Bool) (m :: Type) :: Type where
222type family FromNat' (b :: Bool) (m :: Type) :: Type where
201− FromNat' 'True m = Either m m
223+ FromNat' 'True m = Either m m
202 FromNat' 'False m = Either () (Either m m)
224 FromNat' 'False m = Either () (Either m m)
203
225
204-- | Type family that normalizes the representation of a given data
226-- | Type family that normalizes the representation of a given data
205-- structure
227-- structure
206type family Normalize (d :: Type) :: Type where
228type family Normalize (d :: Type) :: Type where
229+ Normalize (Either a b) = Either (Normalize a) (Normalize b)
207− Normalize d = FromNat (Count d)
230+ Normalize d = FromNat (Count d)
208
231
209-- | Constraint type synonyms to keep the type signatures less convoluted
232-- | Constraint type synonyms to keep the type signatures less convoluted
210type Countable a = KnownNat (Count a)
233type Countable a = KnownNat (Count a)
211type CountableN a = KnownNat (Count (Normalize a))
234type CountableN a = KnownNat (Count (Normalize a))
212−type CountableDimensions a b = (Countable a, Countable b)
235+type CountableDims a b = (Countable a, Countable b)
213−type CountableDimensionsN a b = (CountableN a, CountableN b)
236+type CountableDimsN a b = (CountableN a, CountableN b)
237+type FL a b = FromLists a b
214−type FromListsN e a b = FromLists e (Normalize a) (Normalize b)
238+type FLN a b = FromLists (Normalize a) (Normalize b)
215type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
239type 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
216
241
217−-- | It isn't possible to implement the 'id' function so it's
242+-- | It is possible to implement a constrained version of the category type
218−-- implementation is 'undefined'. However 'comp' can be and this partial
243+-- class.
219−-- class implementation exists just to make the code more readable.
220−--
221−-- Please use 'identity' instead.
222instance (Num e) => Category (Matrix e) where
244instance (Num e) => Category (Matrix e) where
223− id = undefined
245+ type Object (Matrix e) a = (FL a a, Countable a)
246+ id = iden
224− (.) = comp
247+ (.) = comp
225
248
226instance NFData e => NFData (Matrix e cols rows) where
249instance NFData e => NFData (Matrix e cols rows) where
227− rnf Empty = ()
228− rnf (One e) = rnf e
250+ rnf (One e) = rnf e
229− rnf (Junc a b) = rnf a `seq` rnf b
251+ rnf (Join a b) = rnf a `seq` rnf b
230− rnf (Split a b) = rnf a `seq` rnf b
252+ rnf (Fork a b) = rnf a `seq` rnf b
231
253
232instance Eq e => Eq (Matrix e cols rows) where
254instance Eq e => Eq (Matrix e cols rows) where
233− Empty == Empty = True
234− (One a) == (One b) = a == b
255+ (One a) == (One b) = a == b
235− (Junc a b) == (Junc c d) = a == c && b == d
256+ (Join a b) == (Join c d) = a == c && b == d
236− (Split a b) == (Split c d) = a == c && b == d
257+ (Fork a b) == (Fork c d) = a == c && b == d
237− x@(Split _ _) == y@(Junc _ _) = x == abideJS y
258+ x@(Fork _ _) == y@(Join _ _) = x == abideJF y
238− x@(Junc _ _) == y@(Split _ _) = abideJS x == y
259+ x@(Join _ _) == y@(Fork _ _) = abideJF x == y
239
260
240instance Num e => Num (Matrix e cols rows) where
261instance Num e => Num (Matrix e cols rows) where
262+ a + b = zipWithM (+) a b
263+
264+ a - b = zipWithM (-) a b
265+
266+ a * b = zipWithM (*) a b
241
267
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− Empty - Empty = Empty
250− (One a) - (One b) = One (a - b)
251− (Junc a b) - (Junc c d) = Junc (a - c) (b - d)
252− (Split a b) - (Split c d) = Split (a - c) (b - d)
253− x@(Split _ _) - y@(Junc _ _) = x - abideJS y
254− x@(Junc _ _) - y@(Split _ _) = abideJS x - y
255−
256− Empty * Empty = Empty
257− (One a) * (One b) = One (a * b)
258− (Junc a b) * (Junc c d) = Junc (a * c) (b * d)
259− (Split a b) * (Split c d) = Split (a * c) (b * d)
260− x@(Split _ _) * y@(Junc _ _) = x * abideJS y
261− x@(Junc _ _) * y@(Split _ _) = abideJS x * y
262−
263− abs Empty = Empty
264− abs (One a) = One (abs a)
268+ abs (One a) = One (abs a)
265− abs (Junc a b) = Junc (abs a) (abs b)
269+ abs (Join a b) = Join (abs a) (abs b)
266− abs (Split a b) = Split (abs a) (abs b)
270+ abs (Fork a b) = Fork (abs a) (abs b)
267−
271+
268− signum Empty = Empty
269− signum (One a) = One (signum a)
272+ signum (One a) = One (signum a)
270− signum (Junc a b) = Junc (signum a) (signum b)
273+ signum (Join a b) = Join (signum a) (signum b)
271− signum (Split a b) = Split (signum a) (signum b)
274+ signum (Fork a b) = Fork (signum a) (signum b)
272
275
273instance Ord e => Ord (Matrix e cols rows) where
276instance Ord e => Ord (Matrix e cols rows) where
274− Empty <= Empty = True
275− (One a) <= (One b) = a <= b
277+ (One a) <= (One b) = a <= b
276− (Junc a b) <= (Junc c d) = (a <= c) && (b <= d)
278+ (Join a b) <= (Join c d) = (a <= c) && (b <= d)
277− (Split a b) <= (Split c d) = (a <= c) && (b <= d)
279+ (Fork a b) <= (Fork c d) = (a <= c) && (b <= d)
278− x@(Split _ _) <= y@(Junc _ _) = x <= abideJS y
280+ x@(Fork _ _) <= y@(Join _ _) = x <= abideJF y
279− x@(Junc _ _) <= y@(Split _ _) = abideJS x <= y
281+ x@(Join _ _) <= y@(Fork _ _) = abideJF x <= y
280
282
281-- Primitives
283-- Primitives
282
284
283−-- | Empty matrix constructor
284−empty :: Matrix e Void Void
285−empty = Empty
286−
287-- | Unit matrix constructor
285-- | Unit matrix constructor
288one :: e -> Matrix e () ()
286one :: e -> Matrix e () ()
289one = One
287one = One
290
288
291−-- | Matrix 'Junc' constructor
289+-- | Matrix 'Join' constructor
292−junc :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
290+join :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
293−junc = Junc
291+join = Join
294
292
295infixl 3 |||
293infixl 3 |||
296
294
297−-- | Matrix 'Junc' constructor
295+-- | Matrix 'Join' constructor
298(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
296(|||) :: Matrix e a rows -> Matrix e b rows -> Matrix e (Either a b) rows
299−(|||) = Junc
297+(|||) = Join
300
298
301−-- | Matrix 'Split' constructor
299+-- | Matrix 'Fork' constructor
302−split :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
300+fork :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
303−split = Split
301+fork = Fork
304
302
305infixl 2 ===
303infixl 2 ===
306
304
307−-- | Matrix 'Split' constructor
305+-- | Matrix 'Fork' constructor
308(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
306(===) :: Matrix e cols a -> Matrix e cols b -> Matrix e cols (Either a b)
309−(===) = Split
307+(===) = Fork
310
308
311-- Construction
309-- Construction
312
310
313-- | Type class for defining the 'fromList' conversion function.
311-- | Type class for defining the 'fromList' conversion function.
314--
312--
315-- Given that it is not possible to branch on types at the term level type
313-- Given that it is not possible to branch on types at the term level type
316−-- classes are needed bery much like an inductive definition but on types.
314+-- classes are needed very much like an inductive definition but on types.
317−class FromLists e cols rows where
315+class FromLists cols rows where
318 -- | Build a matrix out of a list of list of elements. Throws a runtime
316 -- | Build a matrix out of a list of list of elements. Throws a runtime
319 -- error if the dimensions do not match.
317 -- error if the dimensions do not match.
320 fromLists :: [[e]] -> Matrix e cols rows
318 fromLists :: [[e]] -> Matrix e cols rows
321
319
322−instance FromLists e Void Void where
320+instance FromLists () () where
323− fromLists [] = Empty
324− fromLists _ = error "Wrong dimensions"
325−
326−instance {-# OVERLAPPING #-} FromLists e () () where
327 fromLists [[e]] = One e
321 fromLists [[e]] = One e
328 fromLists _ = error "Wrong dimensions"
322 fromLists _ = error "Wrong dimensions"
329
323
330−instance {-# OVERLAPPING #-} (FromLists e cols ()) => FromLists e (Either () cols) () where
324+instance (FromLists cols ()) => FromLists (Either () cols) () where
331− fromLists [h : t] = Junc (One h) (fromLists [t])
325+ fromLists [h : t] = Join (One h) (fromLists [t])
332 fromLists _ = error "Wrong dimensions"
326 fromLists _ = error "Wrong dimensions"
333
327
328+instance {-# OVERLAPPABLE #-}
334−instance {-# OVERLAPPABLE #-} (FromLists e a (), FromLists e b (), Countable a) => FromLists e (Either a b) () where
329+ (FromLists a (), FromLists b (), Countable a) => FromLists (Either a b) () where
335− fromLists [l] =
330+ fromLists [l] =
336 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
331 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
337− in Junc (fromLists [take rowsA l]) (fromLists [drop rowsA l])
332+ in Join (fromLists [take rowsA l]) (fromLists [drop rowsA l])
338 fromLists _ = error "Wrong dimensions"
333 fromLists _ = error "Wrong dimensions"
339
334
340−instance {-# OVERLAPPING #-} (FromLists e () rows) => FromLists e () (Either () rows) where
335+instance (FromLists () rows) => FromLists () (Either () rows) where
341− fromLists ([h] : t) = Split (One h) (fromLists t)
336+ fromLists ([h] : t) = Fork (One h) (fromLists t)
342 fromLists _ = error "Wrong dimensions"
337 fromLists _ = error "Wrong dimensions"
343
338
339+instance {-# OVERLAPPABLE #-}
344−instance {-# OVERLAPPABLE #-} (FromLists e () a, FromLists e () b, Countable a) => FromLists e () (Either a b) where
340+ (FromLists () a, FromLists () b, Countable a) => FromLists () (Either a b) where
345− fromLists l@([_] : _) =
341+ fromLists l@([_] : _) =
346 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
342 let rowsA = fromInteger (natVal (Proxy :: Proxy (Count a)))
347− in Split (fromLists (take rowsA l)) (fromLists (drop rowsA l))
343+ in Fork (fromLists (take rowsA l)) (fromLists (drop rowsA l))
348 fromLists _ = error "Wrong dimensions"
344 fromLists _ = error "Wrong dimensions"
349
345
350−instance {-# OVERLAPPABLE #-} (FromLists e (Either a b) c, FromLists e (Either a b) d, Countable c) => FromLists e (Either a b) (Either c d) where
346+instance (FromLists (Either a b) c, FromLists (Either a b) d, Countable c)
347+ => FromLists (Either a b) (Either c d) where
351 fromLists l@(h : t) =
348 fromLists l@(h : t) =
352 let lh = length h
349 let lh = length h
353 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
350 rowsC = fromInteger (natVal (Proxy :: Proxy (Count c)))
354− condition = all (== lh) (map length t)
351+ condition = all ((== lh) . length) t
355 in if lh > 0 && condition
352 in if lh > 0 && condition
356− then Split (fromLists (take rowsC l)) (fromLists (drop rowsC l))
353+ then Fork (fromLists (take rowsC l)) (fromLists (drop rowsC l))
357 else error "Not all rows have the same length"
354 else error "Not all rows have the same length"
358
355
359-- | Matrix builder function. Constructs a matrix provided with
356-- | Matrix builder function. Constructs a matrix provided with
360−-- a construction function.
357+-- a construction function that operates with indices.
361−matrixBuilder ::
358+matrixBuilder' ::
362 forall e cols rows.
359 forall e cols rows.
363− ( FromLists e cols rows,
360+ ( FL cols rows,
364− CountableDimensions cols rows
361+ CountableDims cols rows
365− ) =>
366− ((Int, Int) -> e) ->
367− Matrix e cols rows
362+ ) => ((Int, Int) -> e) -> Matrix e cols rows
368−matrixBuilder f =
363+matrixBuilder' f =
369 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
364 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
370 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
365 r = fromInteger $ natVal (Proxy :: Proxy (Count rows))
371 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
366 positions = [(a, b) | a <- [0 .. (r - 1)], b <- [0 .. (c - 1)]]
372 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
367 in fromLists . map (map f) . groupBy (\(x, _) (w, _) -> x == w) $ positions
373
368
369+-- | Matrix builder function. Constructs a matrix provided with
370+-- a construction function that operates with arbitrary types.
371+matrixBuilder ::
372+ forall e a b.
373+ ( FLN a b,
374+ Enum a,
375+ Enum b,
376+ Bounded a,
377+ Bounded b,
378+ Eq a,
379+ Countable b
380+ ) => ((a, b) -> e) -> Matrix e (Normalize a) (Normalize b)
381+matrixBuilder f =
382+ let r = fromInteger $ natVal (Proxy :: Proxy (Count b))
383+ positions = [(a, b) | a <- [minBound .. maxBound], b <- [minBound .. maxBound]]
384+ in fromLists . map (map f) . transpose . buildList r $ positions
385+ where
386+ buildList _ [] = []
387+ buildList r l = take r l : buildList r (drop r l)
388+
374-- | Constructs a column vector matrix
389-- | Constructs a column vector matrix
375−col :: (FromLists e () rows) => [e] -> Matrix e () rows
390+col :: (FL () rows) => [e] -> Matrix e () rows
376col = fromLists . map (: [])
391col = fromLists . map (: [])
377
392
378-- | Constructs a row vector matrix
393-- | Constructs a row vector matrix
379−row :: (FromLists e cols ()) => [e] -> Matrix e cols ()
394+row :: (FL cols ()) => [e] -> Matrix e cols ()
380row = fromLists . (: [])
395row = fromLists . (: [])
381
396
382-- | Lifts functions to matrices with arbitrary dimensions.
397-- | Lifts functions to matrices with arbitrary dimensions.
383--
398--
384-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
399-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
385-- types @a@ or @b@ allows.
400-- types @a@ or @b@ allows.
386−fromF ::
401+fromF' ::
387 forall a b cols rows e.
402 forall a b cols rows e.
388 ( Liftable e a b,
403 ( Liftable e a b,
389− CountableDimensions cols rows,
404+ CountableDims cols rows,
390− FromLists e rows cols
405+ FL rows cols
391 ) =>
406 ) =>
392 (a -> b) ->
407 (a -> b) ->
393 Matrix e cols rows
408 Matrix e cols rows
394−fromF f =
409+fromF' f =
395 let minA = minBound @a
410 let minA = minBound @a
396 maxA = maxBound @a
411 maxA = maxBound @a
397 minB = minBound @b
412 minB = minBound @b
@@ -401,8 +416,8 @@ fromF f =
401 elementsA = take ccols [minA .. maxA]
416 elementsA = take ccols [minA .. maxA]
402 elementsB = take rrows [minB .. maxB]
417 elementsB = take rrows [minB .. maxB]
403 combinations = (,) <$> elementsA <*> elementsB
418 combinations = (,) <$> elementsA <*> elementsB
404− combAp = map snd . sort . map (\(a, b) -> if f a == b
419+ combAp = map snd . sort . map (\(a, b) -> if f a == b
405− then ((fromEnum a, fromEnum b), 1)
420+ then ((fromEnum a, fromEnum b), 1)
406 else ((fromEnum a, fromEnum b), 0)) $ combinations
421 else ((fromEnum a, fromEnum b), 0)) $ combinations
407 mList = buildList combAp rrows
422 mList = buildList combAp rrows
408 in tr $ fromLists mList
423 in tr $ fromLists mList
@@ -412,41 +427,23 @@ fromF f =
412
427
413-- | Lifts functions to matrices with dimensions matching @a@ and @b@
428-- | Lifts functions to matrices with dimensions matching @a@ and @b@
414-- cardinality's.
429-- cardinality's.
415−fromF' ::
430+fromF ::
416 forall a b e.
431 forall a b e.
417 ( Liftable e a b,
432 ( Liftable e a b,
418− CountableDimensionsN a b,
433+ CountableDimsN a b,
419− FromListsN e b a
434+ FLN b a
420 ) =>
435 ) =>
421 (a -> b) ->
436 (a -> b) ->
422 Matrix e (Normalize a) (Normalize b)
437 Matrix e (Normalize a) (Normalize b)
423−fromF' f =
438+fromF = fromF'
424− let minA = minBound @a
425− maxA = maxBound @a
426− minB = minBound @b
427− maxB = maxBound @b
428− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
429− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
430− elementsA = take ccols [minA .. maxA]
431− elementsB = take rrows [minB .. maxB]
432− combinations = (,) <$> elementsA <*> elementsB
433− combAp = map snd . sort . map (\(a, b) -> if f a == b
434− then ((fromEnum a, fromEnum b), 1)
435− else ((fromEnum a, fromEnum b), 0)) $ combinations
436− mList = buildList combAp rrows
437− in tr $ fromLists mList
438− where
439− buildList [] _ = []
440− buildList l r = take r l : buildList (drop r l) r
441
439
442-- Conversion
440-- Conversion
443
441
444-- | Converts a matrix to a list of lists of elements.
442-- | Converts a matrix to a list of lists of elements.
445toLists :: Matrix e cols rows -> [[e]]
443toLists :: Matrix e cols rows -> [[e]]
446−toLists Empty = []
447toLists (One e) = [[e]]
444toLists (One e) = [[e]]
448−toLists (Split l r) = toLists l ++ toLists r
445+toLists (Fork l r) = toLists l ++ toLists r
449−toLists (Junc l r) = zipWith (++) (toLists l) (toLists r)
446+toLists (Join l r) = zipWith (++) (toLists l) (toLists r)
450
447
451-- | Converts a matrix to a list of elements.
448-- | Converts a matrix to a list of elements.
452toList :: Matrix e cols rows -> [e]
449toList :: Matrix e cols rows -> [e]
@@ -455,37 +452,38 @@ toList = concat . toLists
455-- Zeros Matrix
452-- Zeros Matrix
456
453
457-- | The zero matrix. A matrix wholly filled with zeros.
454-- | The zero matrix. A matrix wholly filled with zeros.
458−zeros :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows
455+zeros :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
459−zeros = matrixBuilder (const 0)
456+zeros = matrixBuilder' (const 0)
460
457
461-- Ones Matrix
458-- Ones Matrix
462
459
463-- | The ones matrix. A matrix wholly filled with ones.
460-- | The ones matrix. A matrix wholly filled with ones.
464--
461--
465-- Also known as T (Top) matrix.
462-- Also known as T (Top) matrix.
466−ones :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => Matrix e cols rows
463+ones :: (Num e, FL cols rows, CountableDims cols rows) => Matrix e cols rows
467−ones = matrixBuilder (const 1)
464+ones = matrixBuilder' (const 1)
468
465
469-- Const Matrix
466-- Const Matrix
470
467
471-- | The constant matrix constructor. A matrix wholly filled with a given
468-- | The constant matrix constructor. A matrix wholly filled with a given
472-- value.
469-- value.
473−constant :: (Num e, FromLists e cols rows, CountableDimensions cols rows) => e -> Matrix e cols rows
470+constant :: (Num e, FL cols rows, CountableDims cols rows) => e -> Matrix e cols rows
474−constant e = matrixBuilder (const e)
471+constant e = matrixBuilder' (const e)
475
472
476-- Bang Matrix
473-- Bang Matrix
477
474
478-- | The T (Top) row vector matrix.
475-- | The T (Top) row vector matrix.
479−bang :: forall e cols. (Num e, Enum e, FromLists e cols (), Countable cols) => Matrix e cols ()
476+bang :: forall e cols. (Num e, Enum e, FL cols (), Countable cols) => Matrix e cols ()
480bang =
477bang =
481 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
478 let c = fromInteger $ natVal (Proxy :: Proxy (Count cols))
482 in fromLists [take c [1, 1 ..]]
479 in fromLists [take c [1, 1 ..]]
483
480
484−-- Identity Matrix
481+-- iden Matrix
485
482
486−-- | Identity matrix.
483+-- | iden matrix.
487−identity :: (Num e, FromLists e cols cols, Countable cols) => Matrix e cols cols
484+iden :: forall e cols . (Num e, FL cols cols, Countable cols) => Matrix e cols cols
488−identity = matrixBuilder (bool 0 1 . uncurry (==))
485+iden = matrixBuilder' (bool 0 1 . uncurry (==))
486+{-# NOINLINE iden #-}
489
487
490-- Matrix composition (MMM)
488-- Matrix composition (MMM)
491
489
@@ -494,60 +492,90 @@ identity = matrixBuilder (bool 0 1 . uncurry (==))
494-- This definition takes advantage of divide-and-conquer and fusion laws
492-- This definition takes advantage of divide-and-conquer and fusion laws
495-- from LAoP.
493-- from LAoP.
496comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
494comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
497−comp Empty Empty = Empty
498−comp (One a) (One b) = One (a * b)
495+comp (One a) (One b) = One (a * b)
499−comp (Junc a b) (Split c d) = comp a c + comp b d -- Divide-and-conquer law
496+comp (Join a b) (Fork c d) = comp a c + comp b d -- Divide-and-conquer law
500−comp (Split a b) c = Split (comp a c) (comp b c) -- Split fusion law
497+comp (Fork a b) c = Fork (comp a c) (comp b c) -- Fork fusion law
501−comp c (Junc a b) = Junc (comp c a) (comp c b) -- Junc fusion law
498+comp c (Join a b) = Join (comp c a) (comp c b) -- Join fusion law
499+{-# NOINLINE comp #-}
500+{-# RULES
501+ "comp/iden1" forall m. comp m iden = m ;
502+ "comp/iden2" forall m. comp iden m = m
503+#-}
504+
505+-- Scalar multiplication of matrices
506+
507+infixl 7 .|
508+-- | Scalar multiplication of matrices.
509+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows
510+(.|) e (One a) = One (e * a)
511+(.|) e (Join a b) = Join (e .| a) (e .| b)
512+(.|) e (Fork a b) = Fork (e .| a) (e .| b)
513+
514+-- Scalar division of matrices
515+
516+infixl 7 ./
517+-- | Scalar multiplication of matrices.
518+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows
519+(./) (One a) e = One (a / e)
520+(./) (Join a b) e = Join (a ./ e) (b ./ e)
521+(./) (Fork a b) e = Fork (a ./ e) (b ./ e)
502
522
503-- Projections
523-- Projections
504
524
505-- | Biproduct first component projection
525-- | Biproduct first component projection
506−p1 :: forall e m n. (Num e, CountableDimensions n m, FromLists e n m, FromLists e m m) => Matrix e (Either m n) m
526+p1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e (Either m n) m
507−p1 =
527+p1 = join id zeros
508− let iden = identity :: Matrix e m m
509− zero = zeros :: Matrix e n m
510− in junc iden zero
511
528
512-- | Biproduct second component projection
529-- | Biproduct second component projection
513−p2 :: forall e m n. (Num e, CountableDimensions n m, FromLists e m n, FromLists e n n) => Matrix e (Either m n) n
530+p2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e (Either m n) n
514−p2 =
531+p2 = join zeros id
515− let iden = identity :: Matrix e n n
516− zero = zeros :: Matrix e m n
517− in junc zero iden
518
532
519-- Injections
533-- Injections
520
534
521-- | Biproduct first component injection
535-- | Biproduct first component injection
522−i1 :: (Num e, CountableDimensions n m, FromLists e n m, FromLists e m m) => Matrix e m (Either m n)
536+i1 :: (Num e, CountableDims n m, FL n m, FL m m) => Matrix e m (Either m n)
523i1 = tr p1
537i1 = tr p1
524
538
525-- | Biproduct second component injection
539-- | Biproduct second component injection
526−i2 :: (Num e, CountableDimensions n m, FromLists e m n, FromLists e n n) => Matrix e n (Either m n)
540+i2 :: (Num e, CountableDims n m, FL m n, FL n n) => Matrix e n (Either m n)
527i2 = tr p2
541i2 = tr p2
528
542
529-- Dimensions
543-- Dimensions
530
544
531-- | Obtain the number of rows.
545-- | Obtain the number of rows.
532--
546--
533−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
547+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
534−-- dimensions in constant time.
548+-- dimensions in constant time. For a version that doesn't require the
535−--
549+-- constraint see 'rows''.
536−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
537−-- exchange for performance.
538rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int
550rows :: forall e cols rows. (Countable rows) => Matrix e cols rows -> Int
539rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
551rows _ = fromInteger $ natVal (Proxy :: Proxy (Count rows))
540
552
553+-- | Obtain the number of rows in an inefficient manner, but without any
554+-- constraints.
555+--
556+-- For a more efficient version see 'rows'.
557+rows' :: Matrix e cols rows -> Int
558+rows' (One _) = 1
559+rows' (Join lhs _) = rows' lhs
560+rows' (Fork top bottom) = rows' top + rows' bottom
561+
541-- | Obtain the number of columns.
562-- | Obtain the number of columns.
542−--
543−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
544−-- dimensions in constant time.
545--
563--
546−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
564+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
547−-- exchange for performance.
565+-- dimensions in constant time. For a version that doesn't require the
566+-- constraint see 'columns''.
548columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int
567columns :: forall e cols rows. (Countable cols) => Matrix e cols rows -> Int
549columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
568columns _ = fromInteger $ natVal (Proxy :: Proxy (Count cols))
550
569
570+-- | Obtain the number of columns in an inefficient manner, but without any
571+-- constraints.
572+--
573+-- For a more efficient version see 'columns'.
574+columns' :: Matrix e cols rows -> Int
575+columns' (One _) = 1
576+columns' (Join lhs rhs) = columns' lhs + columns' rhs
577+columns' (Fork top _) = columns' top
578+
551-- Coproduct Bifunctor
579-- Coproduct Bifunctor
552
580
553infixl 5 -|-
581infixl 5 -|-
@@ -556,150 +584,172 @@ infixl 5 -|-
556(-|-) ::
584(-|-) ::
557 forall e n k m j.
585 forall e n k m j.
558 ( Num e,
586 ( Num e,
559− CountableDimensions j k,
587+ CountableDims j k,
560− FromLists e k k,
588+ FL k k,
561− FromLists e j k,
589+ FL j k,
562− FromLists e k j,
590+ FL k j,
563− FromLists e j j
591+ FL j j
564 ) =>
592 ) =>
565 Matrix e n k ->
593 Matrix e n k ->
566 Matrix e m j ->
594 Matrix e m j ->
567 Matrix e (Either n m) (Either k j)
595 Matrix e (Either n m) (Either k j)
568−(-|-) a b = Junc (i1 . a) (i2 . b)
596+(-|-) a b = Join (i1 . a) (i2 . b)
569
597
570-- Khatri Rao Product and projections
598-- Khatri Rao Product and projections
571
599
572-- | Khatri Rao product first component projection matrix.
600-- | Khatri Rao product first component projection matrix.
573−kp1 ::
601+fstM ::
574 forall e m k .
602 forall e m k .
575 ( Num e,
603 ( Num e,
576− CountableDimensions k m,
604+ CountableDims k m,
577− FromLists e (Normalize (m, k)) m,
605+ FL (Normalize (m, k)) m,
578 CountableN (m, k)
606 CountableN (m, k)
579 ) => Matrix e (Normalize (m, k)) m
607 ) => Matrix e (Normalize (m, k)) m
580−kp1 = matrixBuilder f
608+fstM = matrixBuilder' f
581 where
609 where
582 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
610 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
583 f (x, y)
611 f (x, y)
584 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
612 | y >= (x * offset) && y <= (x * offset + offset - 1) = 1
585− | otherwise = 0
613+ | otherwise = 0
586
614
587-- | Khatri Rao product second component projection matrix.
615-- | Khatri Rao product second component projection matrix.
588−kp2 ::
616+sndM ::
589 forall e m k .
617 forall e m k .
590 ( Num e,
618 ( Num e,
591− CountableDimensions k m,
619+ CountableDims k m,
592− FromLists e (Normalize (m, k)) k,
620+ FL (Normalize (m, k)) k,
593 CountableN (m, k)
621 CountableN (m, k)
594 ) => Matrix e (Normalize (m, k)) k
622 ) => Matrix e (Normalize (m, k)) k
595−kp2 = matrixBuilder f
623+sndM = matrixBuilder' f
596 where
624 where
597 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
625 offset = fromInteger (natVal (Proxy :: Proxy (Count k)))
598 f (x, y)
626 f (x, y)
599 | x == y || mod (y - x) offset == 0 = 1
627 | x == y || mod (y - x) offset == 0 = 1
600− | otherwise = 0
628+ | otherwise = 0
601
629
602-- | Khatri Rao Matrix product also known as matrix pairing.
630-- | Khatri Rao Matrix product also known as matrix pairing.
603--
631--
604-- NOTE: That this is not a true categorical product, see for instance:
632-- NOTE: That this is not a true categorical product, see for instance:
605−--
633+--
606-- @
634-- @
607−-- | kp1 . khatri a b == a
635+-- | fstM . kr a b == a
608−-- khatri a b ==> |
636+-- kr a b ==> |
609−-- | kp2 . khatri a b == b
637+-- | sndM . kr a b == b
610-- @
638-- @
611--
639--
612-- __Emphasis__ on the implication symbol.
640-- __Emphasis__ on the implication symbol.
613−khatri ::
641+kr ::
614− forall e cols a b.
642+ forall e cols a b.
615 ( Num e,
643 ( Num e,
616− CountableDimensions a b,
644+ CountableDims a b,
617 CountableN (a, b),
645 CountableN (a, b),
618− FromLists e (Normalize (a, b)) a,
646+ FL (Normalize (a, b)) a,
619− FromLists e (Normalize (a, b)) b
647+ FL (Normalize (a, b)) b
620 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
648 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (Normalize (a, b))
621−khatri a b =
649+kr a b =
622− let kp1' = kp1 @e @a @b
650+ let fstM' = fstM @e @a @b
623− kp2' = kp2 @e @a @b
651+ sndM' = sndM @e @a @b
624− in (tr kp1') . a * (tr kp2') . b
652+ in (tr fstM' . a) * (tr sndM' . b)
625
653
626-- Product Bifunctor (Kronecker)
654-- Product Bifunctor (Kronecker)
627
655
628infixl 4 ><
656infixl 4 ><
629
657
630-- | Matrix product functor also known as kronecker product
658-- | Matrix product functor also known as kronecker product
631−(><) ::
659+(><) ::
632− forall e m p n q.
660+ forall e m p n q.
633 ( Num e,
661 ( Num e,
634− CountableDimensions m n,
662+ CountableDims m n,
635− CountableDimensions p q,
663+ CountableDims p q,
636− CountableDimensionsN (m, n) (p, q),
664+ CountableDimsN (m, n) (p, q),
637− FromLists e (Normalize (m, n)) m,
665+ FL (Normalize (m, n)) m,
638− FromLists e (Normalize (m, n)) n,
666+ FL (Normalize (m, n)) n,
639− FromLists e (Normalize (p, q)) p,
667+ FL (Normalize (p, q)) p,
640− FromLists e (Normalize (p, q)) q
668+ FL (Normalize (p, q)) q
641− )
669+ )
642 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
670 => Matrix e m p -> Matrix e n q -> Matrix e (Normalize (m, n)) (Normalize (p, q))
643(><) a b =
671(><) a b =
644− let kp1' = kp1 @e @m @n
672+ let fstM' = fstM @e @m @n
645− kp2' = kp2 @e @m @n
673+ sndM' = sndM @e @m @n
646− in khatri (a . kp1') (b . kp2')
674+ in kr (a . fstM') (b . sndM')
647
675
648−-- Matrix abide Junc Split
676+-- Matrix abide Join Fork
649
677
650−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
678+-- | Matrix "abiding" following the 'Join'-'Fork' exchange law.
651−--
679+--
652-- Law:
680-- Law:
653--
681--
654-- @
682-- @
655−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)
683+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)
656-- @
684-- @
657−abideJS :: Matrix e cols rows -> Matrix e cols rows
685+abideJF :: Matrix e cols rows -> Matrix e cols rows
658−abideJS (Junc (Split a c) (Split b d)) = Split (Junc (abideJS a) (abideJS b)) (Junc (abideJS c) (abideJS d)) -- Junc-Split abide law
686+abideJF (Join (Fork a c) (Fork b d)) = Fork (Join (abideJF a) (abideJF b)) (Join (abideJF c) (abideJF d)) -- Join-Fork abide law
659−abideJS Empty = Empty
660−abideJS (One e) = One e
687+abideJF (One e) = One e
661−abideJS (Junc a b) = Junc (abideJS a) (abideJS b)
688+abideJF (Join a b) = Join (abideJF a) (abideJF b)
662−abideJS (Split a b) = Split (abideJS a) (abideJS b)
689+abideJF (Fork a b) = Fork (abideJF a) (abideJF b)
663
690
664−-- Matrix abide Split Junc
691+-- Matrix abide Fork Join
665
692
666−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
693+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
667−--
694+--
668-- @
695-- @
669−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)
696+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)
670-- @
697-- @
671−abideSJ :: Matrix e cols rows -> Matrix e cols rows
698+abideFJ :: Matrix e cols rows -> Matrix e cols rows
672−abideSJ (Split (Junc a b) (Junc c d)) = Junc (Split (abideSJ a) (abideSJ c)) (Split (abideSJ b) (abideSJ d)) -- Split-Junc abide law
699+abideFJ (Fork (Join a b) (Join c d)) = Join (Fork (abideFJ a) (abideFJ c)) (Fork (abideFJ b) (abideFJ d)) -- Fork-Join abide law
673−abideSJ Empty = Empty
674−abideSJ (One e) = One e
700+abideFJ (One e) = One e
675−abideSJ (Junc a b) = Junc (abideSJ a) (abideSJ b)
701+abideFJ (Join a b) = Join (abideFJ a) (abideFJ b)
676−abideSJ (Split a b) = Split (abideSJ a) (abideSJ b)
702+abideFJ (Fork a b) = Fork (abideFJ a) (abideFJ b)
677
703
678-- Matrix transposition
704-- Matrix transposition
679
705
680-- | Matrix transposition.
706-- | Matrix transposition.
681tr :: Matrix e cols rows -> Matrix e rows cols
707tr :: Matrix e cols rows -> Matrix e rows cols
682−tr Empty = Empty
683−tr (One e) = One e
708+tr (One e) = One e
684−tr (Junc a b) = Split (tr a) (tr b)
709+tr (Join a b) = Fork (tr a) (tr b)
685−tr (Split a b) = Junc (tr a) (tr b)
710+tr (Fork a b) = Join (tr a) (tr b)
686
711
687-- Selective 'select' operator
712-- Selective 'select' operator
688
713
689-- | Selective functors 'select' operator equivalent inspired by the
714-- | Selective functors 'select' operator equivalent inspired by the
690-- ArrowMonad solution presented in the paper.
715-- ArrowMonad solution presented in the paper.
691−select :: (Num e, FromLists e b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
716+select :: (Num e, FL b b, Countable b) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
692−select m y = junc y identity . m
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
736+branch x l r = f x `select` g l `select` r
737+ where
738+ f :: (Num e, Countable a, CountableDims b c, FL a b, FL c b, FL b b, FL b a, FL a a)
739+ => Matrix e cols (Either a b) -> Matrix e cols (Either a (Either b c))
740+ f m = fork (tr i1) (i1 . tr i2) . m
741+ g :: (Num e, CountableDims b c, FL b c, FL c c) => Matrix e a c -> Matrix e a (Either b c)
742+ g m = i2 . m
693
743
694-- McCarthy's Conditional
744-- McCarthy's Conditional
695
745
696-- | McCarthy's Conditional expresses probabilistic choice.
746-- | McCarthy's Conditional expresses probabilistic choice.
697cond ::
747cond ::
698− ( cols ~ Normalize cols,
748+ ( Trivial cols,
699 Countable cols,
749 Countable cols,
700− FromLists e () cols,
750+ FL () cols,
701− FromLists e cols (),
751+ FL cols (),
702− FromLists e cols cols,
752+ FL cols cols,
703 Bounded a,
753 Bounded a,
704 Enum a,
754 Enum a,
705 Num e,
755 Num e,
@@ -707,14 +757,14 @@ cond ::
707 )
757 )
708 =>
758 =>
709 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
759 (a -> Bool) -> Matrix e cols rows -> Matrix e cols rows -> Matrix e cols rows
710−cond p f g = junc f g . grd p
760+cond p f g = join f g . grd p
711
761
712−grd ::
762+grd ::
713− ( q ~ Normalize q,
763+ ( Trivial q,
714 Countable q,
764 Countable q,
715− FromLists e () q,
765+ FL () q,
716− FromLists e q (),
766+ FL q (),
717− FromLists e q q,
767+ FL q q,
718 Bounded a,
768 Bounded a,
719 Enum a,
769 Enum a,
720 Num e,
770 Num e,
@@ -722,60 +772,86 @@ grd ::
722 )
772 )
723 =>
773 =>
724 (a -> Bool) -> Matrix e q (Either q q)
774 (a -> Bool) -> Matrix e q (Either q q)
725−grd f = split (corr f) (corr (not . f))
775+grd f = fork (corr f) (corr (not . f))
726
776
727−corr ::
777+corr ::
728− forall e a q .
778+ forall e a q .
729− ( q ~ Normalize q,
779+ ( Trivial q,
730 Countable q,
780 Countable q,
731− FromLists e () q,
781+ FL () q,
732− FromLists e q (),
782+ FL q (),
733− FromLists e q q,
783+ FL q q,
734 Liftable e a Bool
784 Liftable e a Bool
735− )
785+ )
736 => (a -> Bool) -> Matrix e q q
786 => (a -> Bool) -> Matrix e q q
737−corr p = let f = fromF p :: Matrix e q ()
787+corr p = let f = fromF' p :: Matrix e q ()
738− in khatri f (identity :: Matrix e q q)
788+ in kr f (id :: Matrix e q q)
739
789
740-- Pretty print
790-- Pretty print
741
791
742prettyAux :: Show e => [[e]] -> [[e]] -> String
792prettyAux :: Show e => [[e]] -> [[e]] -> String
743prettyAux [] _ = ""
793prettyAux [] _ = ""
744−prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
794+prettyAux [[e]] m = "│ " ++ fill (show e) ++ " │\n"
745 where
795 where
746 v = fmap show m
796 v = fmap show m
747 widest = maximum $ fmap length v
797 widest = maximum $ fmap length v
748 fill str = replicate (widest - length str - 2) ' ' ++ str
798 fill str = replicate (widest - length str - 2) ' ' ++ str
749−prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
799+prettyAux [h] m = "│ " ++ fill (unwords $ map show h) ++ " │\n"
750 where
800 where
751− v = fmap show m
801+ v = fmap show m
752− widest = maximum $ fmap length v
802+ widest = maximum $ fmap length v
753 fill str = replicate (widest - length str - 2) ' ' ++ str
803 fill str = replicate (widest - length str - 2) ' ' ++ str
754−prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
804+prettyAux (h : t) l = "│ " ++ fill (unwords $ map show h) ++ " │\n" ++
755 prettyAux t l
805 prettyAux t l
756 where
806 where
757− v = fmap show l
807+ v = fmap show l
758− widest = maximum $ fmap length v
808+ widest = maximum $ fmap length v
759 fill str = replicate (widest - length str - 2) ' ' ++ str
809 fill str = replicate (widest - length str - 2) ' ' ++ str
760
810
761-- | Matrix pretty printer
811-- | Matrix pretty printer
762−pretty :: (Countable cols, Show e) => Matrix e cols rows -> String
812+pretty :: (CountableDims cols rows, Show e) => Matrix e cols rows -> String
813+pretty m = concat
763−pretty m = "┌ " ++ unwords (replicate (columns m) blank) ++ " ┐\n" ++
814+ [ "┌ ", unwords (replicate (columns m) blank), " ┐\n"
764− prettyAux (toLists m) (toLists m) ++
815+ , unlines
816+ [ "│ " ++ unwords (fmap (\j -> fill $ show $ getElem i j m) [1..columns m]) ++ " │" | i <- [1..rows m] ]
765− "└ " ++ unwords (replicate (columns m) blank) ++ " ┘"
817+ , "└ ", unwords (replicate (columns m) blank), " ┘"
818+ ]
766− where
819+ where
767− v = fmap show (toList m)
820+ strings = map show (toList m)
768− widest = maximum $ fmap length v
821+ widest = maximum $ map length strings
769 fill str = replicate (widest - length str) ' ' ++ str
822 fill str = replicate (widest - length str) ' ' ++ str
770− blank = fill ""
823+ blank = fill ""
824+ safeGet i j m
825+ | i > rows m || j > columns m || i < 1 || j < 1 = Nothing
826+ | otherwise = Just $ unsafeGet i j m (toList m)
827+ unsafeGet i j m l = l !! encode (columns m) (i,j)
828+ encode m (i,j) = (i-1)*m + j - 1
829+ getElem i j m =
830+ fromMaybe
831+ (error $
832+ "getElem: Trying to get the "
833+ ++ show (i, j)
834+ ++ " element from a "
835+ ++ show (rows m) ++ "x" ++ show (columns m)
836+ ++ " matrix."
837+ )
838+ (safeGet i j m)
771
839
772-- | Matrix pretty printer
840-- | Matrix pretty printer
773−prettyPrint :: (Countable cols, Show e) => Matrix e cols rows -> IO ()
841+prettyPrint :: (CountableDims cols rows, Show e) => Matrix e cols rows -> IO ()
774prettyPrint = putStrLn . pretty
842prettyPrint = putStrLn . pretty
775
843
844+-- | Zip two matrices with a given binary function
845+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows
846+zipWithM f (One a) (One b) = One (f a b)
847+zipWithM f (Join a b) (Join c d) = Join (zipWithM f a c) (zipWithM f b d)
848+zipWithM f (Fork a b) (Fork c d) = Fork (zipWithM f a c) (zipWithM f b d)
849+zipWithM f x@(Fork _ _) y@(Join _ _) = zipWithM f x (abideJF y)
850+zipWithM f x@(Join _ _) y@(Fork _ _) = zipWithM f (abideJF x) y
851+
776-- Relational operators functions
852-- Relational operators functions
777
853
778−type Boolean = Natural 0 1
854+type Boolean = Natural 0 1
779type Relation a b = Matrix Boolean a b
855type Relation a b = Matrix Boolean a b
780
856
781-- | Helper conversion function
857-- | Helper conversion function
@@ -791,54 +867,47 @@ fromBool False = nat 0
791
867
792-- | Relational negation
868-- | Relational negation
793negateM :: Relation cols rows -> Relation cols rows
869negateM :: Relation cols rows -> Relation cols rows
794−negateM Empty = Empty
795−negateM (One (Nat 0)) = One (Nat 1)
870+negateM (One (Nat p)) = One (Nat (negate p))
796−negateM (One (Nat 1)) = One (Nat 0)
797−negateM (Junc a b) = Junc (negateM a) (negateM b)
871+negateM (Join a b) = Join (negateM a) (negateM b)
798−negateM (Split a b) = Split (negateM a) (negateM b)
872+negateM (Fork a b) = Fork (negateM a) (negateM b)
799
873
800-- | Relational addition
874-- | Relational addition
801orM :: Relation cols rows -> Relation cols rows -> Relation cols rows
875orM :: Relation cols rows -> Relation cols rows -> Relation cols rows
802−orM Empty Empty = Empty
803−orM (One a) (One b) = One (fromBool (toBool a || toBool b))
876+orM (One a) (One b) = One (fromBool (toBool a || toBool b))
804−orM (Junc a b) (Junc c d) = Junc (orM a c) (orM b d)
877+orM (Join a b) (Join c d) = Join (orM a c) (orM b d)
805−orM (Split a b) (Split c d) = Split (orM a c) (orM b d)
878+orM (Fork a b) (Fork c d) = Fork (orM a c) (orM b d)
806−orM x@(Split _ _) y@(Junc _ _) = orM x (abideJS y)
879+orM x@(Fork _ _) y@(Join _ _) = orM x (abideJF y)
807−orM x@(Junc _ _) y@(Split _ _) = orM (abideJS x) y
880+orM x@(Join _ _) y@(Fork _ _) = orM (abideJF x) y
808
881
809-- | Relational multiplication
882-- | Relational multiplication
810andM :: Relation cols rows -> Relation cols rows -> Relation cols rows
883andM :: Relation cols rows -> Relation cols rows -> Relation cols rows
811−andM Empty Empty = Empty
812−andM (One a) (One b) = One (fromBool (toBool a && toBool b))
884+andM (One a) (One b) = One (fromBool (toBool a && toBool b))
813−andM (Junc a b) (Junc c d) = Junc (andM a c) (andM b d)
885+andM (Join a b) (Join c d) = Join (andM a c) (andM b d)
814−andM (Split a b) (Split c d) = Split (andM a c) (andM b d)
886+andM (Fork a b) (Fork c d) = Fork (andM a c) (andM b d)
815−andM x@(Split _ _) y@(Junc _ _) = andM x (abideJS y)
887+andM x@(Fork _ _) y@(Join _ _) = andM x (abideJF y)
816−andM x@(Junc _ _) y@(Split _ _) = andM (abideJS x) y
888+andM x@(Join _ _) y@(Fork _ _) = andM (abideJF x) y
817
889
818-- | Relational subtraction
890-- | Relational subtraction
819subM :: Relation cols rows -> Relation cols rows -> Relation cols rows
891subM :: Relation cols rows -> Relation cols rows -> Relation cols rows
820−subM Empty Empty = Empty
821−subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
892+subM (One a) (One b) = if a - b < nat 0 then One (nat 0) else One (a - b)
822−subM (Junc a b) (Junc c d) = Junc (subM a c) (subM b d)
893+subM (Join a b) (Join c d) = Join (subM a c) (subM b d)
823−subM (Split a b) (Split c d) = Split (subM a c) (subM b d)
894+subM (Fork a b) (Fork c d) = Fork (subM a c) (subM b d)
824−subM x@(Split _ _) y@(Junc _ _) = subM x (abideJS y)
895+subM x@(Fork _ _) y@(Join _ _) = subM x (abideJF y)
825−subM x@(Junc _ _) y@(Split _ _) = subM (abideJS x) y
896+subM x@(Join _ _) y@(Fork _ _) = subM (abideJF x) y
826
897
827-- | Matrix relational composition.
898-- | Matrix relational composition.
828compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows
899compRel :: Relation cr rows -> Relation cols cr -> Relation cols rows
829−compRel Empty Empty = Empty
830−compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
900+compRel (One a) (One b) = One (fromBool (toBool a && toBool b))
831−compRel (Junc a b) (Split c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
901+compRel (Join a b) (Fork c d) = orM (compRel a c) (compRel b d) -- Divide-and-conquer law
832−compRel (Split a b) c = Split (compRel a c) (compRel b c) -- Split fusion law
902+compRel (Fork a b) c = Fork (compRel a c) (compRel b c) -- Fork fusion law
833−compRel c (Junc a b) = Junc (compRel c a) (compRel c b) -- Junc fusion law
903+compRel c (Join a b) = Join (compRel c a) (compRel c b) -- Join fusion law
834
904
835-- | Matrix relational right division
905-- | Matrix relational right division
836divR :: Relation b c -> Relation b a -> Relation a c
906divR :: Relation b c -> Relation b a -> Relation a c
837−divR Empty Empty = Empty
838divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a
907divR (One a) (One b) = One (fromBool (not (toBool b) || toBool a)) -- b implies a
839−divR (Junc a b) (Junc c d) = andM (divR a c) (divR b d)
908+divR (Join a b) (Join c d) = andM (divR a c) (divR b d)
840−divR (Split a b) c = Split (divR a c) (divR b c)
909+divR (Fork a b) c = Fork (divR a c) (divR b c)
841−divR c (Split a b) = Junc (divR c a) (divR c b)
910+divR c (Fork a b) = Join (divR c a) (divR c b)
842
911
843-- | Matrix relational left division
912-- | Matrix relational left division
844divL :: Relation c b -> Relation a b -> Relation a c
913divL :: Relation c b -> Relation a b -> Relation a c
@@ -854,86 +923,52 @@ divS s r = divL r s `intersection` divR (tr r) (tr s)
854--
923--
855-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
924-- NOTE: Be careful to not ask for a relation bigger than the cardinality of
856-- types @a@ or @b@ allows.
925-- types @a@ or @b@ allows.
857−fromFRel ::
926+fromFRel' ::
858 forall a b cols rows.
927 forall a b cols rows.
859 ( Liftable Boolean a b,
928 ( Liftable Boolean a b,
860− CountableDimensions cols rows,
929+ CountableDims cols rows,
861− FromLists Boolean rows cols
930+ FL rows cols
862 ) =>
931 ) =>
863 (a -> b) ->
932 (a -> b) ->
864 Relation cols rows
933 Relation cols rows
865−fromFRel f =
934+fromFRel' = fromF'
866− let minA = minBound @a
867− maxA = maxBound @a
868− minB = minBound @b
869− maxB = maxBound @b
870− ccols = fromInteger $ natVal (Proxy :: Proxy (Count cols))
871− rrows = fromInteger $ natVal (Proxy :: Proxy (Count rows))
872− elementsA = take ccols [minA .. maxA]
873− elementsB = take rrows [minB .. maxB]
874− combinations = (,) <$> elementsA <*> elementsB
875− combAp = map snd . sort . map (\(a, b) -> if f a == b
876− then ((fromEnum a, fromEnum b), nat 1)
877− else ((fromEnum a, fromEnum b), nat 0)) $ combinations
878− mList = buildList combAp rrows
879− in tr $ fromLists mList
880− where
881− buildList [] _ = []
882− buildList l r = take r l : buildList (drop r l) r
883
935
884-- | Lifts functions to relations with dimensions matching @a@ and @b@
936-- | Lifts functions to relations with dimensions matching @a@ and @b@
885-- cardinality's.
937-- cardinality's.
886−fromFRel' ::
938+fromFRel ::
887 forall a b.
939 forall a b.
888 ( Liftable Boolean a b,
940 ( Liftable Boolean a b,
889− CountableDimensionsN a b,
941+ CountableDimsN a b,
890− FromLists Boolean (Normalize b) (Normalize a)
942+ FLN b a
891 ) =>
943 ) =>
892 (a -> b) ->
944 (a -> b) ->
893 Relation (Normalize a) (Normalize b)
945 Relation (Normalize a) (Normalize b)
894−fromFRel' f =
946+fromFRel = fromFRel'
895− let minA = minBound @a
896− maxA = maxBound @a
897− minB = minBound @b
898− maxB = maxBound @b
899− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
900− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
901− elementsA = take ccols [minA .. maxA]
902− elementsB = take rrows [minB .. maxB]
903− combinations = (,) <$> elementsA <*> elementsB
904− combAp = map snd . sort . map (\(a, b) -> if f a == b
905− then ((fromEnum a, fromEnum b), nat 1)
906− else ((fromEnum a, fromEnum b), nat 0)) $ combinations
907− mList = buildList combAp rrows
908− in tr $ fromLists mList
909− where
910− buildList [] _ = []
911− buildList l r = take r l : buildList (drop r l) r
912
947
913-- | Lifts a relation function to a Boolean Matrix
948-- | Lifts a relation function to a Boolean Matrix
914−toRel ::
949+toRel ::
915 forall a b.
950 forall a b.
916 ( Bounded a,
951 ( Bounded a,
917 Bounded b,
952 Bounded b,
918 Enum a,
953 Enum a,
919 Enum b,
954 Enum b,
920 Eq b,
955 Eq b,
921− CountableDimensionsN a b,
956+ CountableDims a b,
922− FromListsN Boolean b a
957+ FLN b a
923− )
958+ )
924 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)
959 => (a -> b -> Bool) -> Relation (Normalize a) (Normalize b)
925toRel f =
960toRel f =
926 let minA = minBound @a
961 let minA = minBound @a
927 maxA = maxBound @a
962 maxA = maxBound @a
928 minB = minBound @b
963 minB = minBound @b
929 maxB = maxBound @b
964 maxB = maxBound @b
930− ccols = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize a)))
965+ ccols = fromInteger $ natVal (Proxy :: Proxy (Count a))
931− rrows = fromInteger $ natVal (Proxy :: Proxy (Count (Normalize b)))
966+ rrows = fromInteger $ natVal (Proxy :: Proxy (Count b))
932 elementsA = take ccols [minA .. maxA]
967 elementsA = take ccols [minA .. maxA]
933 elementsB = take rrows [minB .. maxB]
968 elementsB = take rrows [minB .. maxB]
934 combinations = (,) <$> elementsA <*> elementsB
969 combinations = (,) <$> elementsA <*> elementsB
935− combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
970+ combAp = map snd . sort . map (\(a, b) -> if uncurry f (a, b)
936− then ((fromEnum a, fromEnum b), nat 1)
971+ then ((fromEnum a, fromEnum b), nat 1)
937 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
972 else ((fromEnum a, fromEnum b), nat 0)) $ combinations
938 mList = buildList combAp rrows
973 mList = buildList combAp rrows
939 in tr $ fromLists mList
974 in tr $ fromLists mList
src/LAoP/Matrix/Nat.hs+139−124
@@ -56,21 +56,20 @@ module LAoP.Matrix.Nat
56
56
57 -- * Constraint type synonyms
57 -- * Constraint type synonyms
58 Countable,
58 Countable,
59− CountableDimensions,
59+ CountableDims,
60 CountableN,
60 CountableN,
61 CountableNz,
61 CountableNz,
62− CountableDimensionsN,
62+ CountableDimsN,
63− FromListsN,
63+ FLN,
64− FromListsNz,
64+ FLNz,
65 Liftable,
65 Liftable,
66 TrivialE,
66 TrivialE,
67 TrivialP,
67 TrivialP,
68
68
69 -- * Primitives
69 -- * Primitives
70− empty,
71 one,
70 one,
72− junc,
71+ join,
73− split,
72+ fork,
74
73
75 -- * Auxiliary type families
74 -- * Auxiliary type families
76 I.FromNat,
75 I.FromNat,
@@ -78,11 +77,11 @@ module LAoP.Matrix.Nat
78 I.Normalize,
77 I.Normalize,
79
78
80 -- * Matrix construction and conversion
79 -- * Matrix construction and conversion
81− I.FromLists,
80+ I.FL,
82 fromLists,
81 fromLists,
83 toLists,
82 toLists,
84 toList,
83 toList,
85− matrixBuilder,
84+ matrixBuilder',
86 row,
85 row,
87 col,
86 col,
88 zeros,
87 zeros,
@@ -98,6 +97,10 @@ module LAoP.Matrix.Nat
98 -- ** Matrix Transposition
97 -- ** Matrix Transposition
99 tr,
98 tr,
100
99
100+ -- ** Scalar multiplication/division of matrices
101+ (.|),
102+ (./),
103+
101 -- ** Selective operator
104 -- ** Selective operator
102 select,
105 select,
103
106
@@ -105,16 +108,19 @@ module LAoP.Matrix.Nat
105 cond,
108 cond,
106
109
107 -- ** Matrix "abiding"
110 -- ** Matrix "abiding"
108− abideJS,
111+ abideJF,
109− abideSJ,
112+ abideFJ,
113+
114+ -- ** Zip Matrices
115+ zipWithM,
110
116
111 -- * Biproduct approach
117 -- * Biproduct approach
112− -- ** Split
118+ -- ** Fork
113 (===),
119 (===),
114 -- *** Projections
120 -- *** Projections
115 p1,
121 p1,
116 p2,
122 p2,
117− -- ** Junc
123+ -- ** Join
118 (|||),
124 (|||),
119 -- *** Injections
125 -- *** Injections
120 i1,
126 i1,
@@ -128,11 +134,11 @@ module LAoP.Matrix.Nat
128 -- | Note that given the restrictions imposed it is not possible to
134 -- | Note that given the restrictions imposed it is not possible to
129 -- implement the standard type classes present in standard Haskell.
135 -- implement the standard type classes present in standard Haskell.
130 -- *** Matrix pairing projections
136 -- *** Matrix pairing projections
131− kp1,
137+ fstM,
132− kp2,
138+ sndM,
133
139
134 -- *** Matrix pairing
140 -- *** Matrix pairing
135− khatri,
141+ kr,
136
142
137 -- * Matrix composition and lifting
143 -- * Matrix composition and lifting
138
144
@@ -140,10 +146,10 @@ module LAoP.Matrix.Nat
140
146
141 -- | Note that given the restrictions imposed it is not possible to
147 -- | Note that given the restrictions imposed it is not possible to
142 -- implement the standard type classes present in standard Haskell.
148 -- implement the standard type classes present in standard Haskell.
143− identity,
149+ iden,
144 comp,
150 comp,
145− fromF,
146 fromF',
151 fromF',
152+ fromF,
147
153
148 -- * Matrix printing
154 -- * Matrix printing
149 pretty,
155 pretty,
@@ -154,47 +160,34 @@ where
154import Data.Proxy
160import Data.Proxy
155import GHC.TypeLits
161import GHC.TypeLits
156import Control.DeepSeq
162import Control.DeepSeq
157−import qualified Control.Category as C
158import qualified LAoP.Matrix.Internal as I
163import qualified LAoP.Matrix.Internal as I
159
164
160newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))
165newtype Matrix e (cols :: Nat) (rows :: Nat) = M (I.Matrix e (I.FromNat cols) (I.FromNat rows))
161 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))
166 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.FromNat cols) (I.FromNat rows))
162
167
163-- | Constraint type synonyms to keep the type signatures less convoluted
168-- | Constraint type synonyms to keep the type signatures less convoluted
164−type Countable a = KnownNat (I.Count a)
169+type Countable a = KnownNat (I.Count a)
165−type CountableDimensions a b = (Countable a, Countable b)
170+type CountableDims a b = (Countable a, Countable b)
166−type CountableN a = KnownNat (I.Count (I.FromNat a))
171+type CountableN a = KnownNat (I.Count (I.FromNat a))
167−type CountableNz a = KnownNat (I.Count (I.Normalize a))
172+type CountableNz a = KnownNat (I.Count (I.Normalize a))
168−type CountableDimensionsN a b = (CountableN a, CountableN b)
173+type CountableDimsN a b = (CountableN a, CountableN b)
169−type FromListsN e a b = I.FromLists e (I.FromNat a) (I.FromNat b)
174+type FLN e a b = I.FL (I.FromNat a) (I.FromNat b)
170−type FromListsNz e a b = I.FromLists e (I.Normalize a) (I.Normalize b)
175+type FLNz e a b = I.FL (I.Normalize a) (I.Normalize b)
171−type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
176+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
172−type TrivialE a b = I.FromNat (a + b) ~ Either (I.FromNat a) (I.FromNat b)
177+type TrivialE a b = I.FromNat (a + b) ~ Either (I.FromNat a) (I.FromNat b)
173−type TrivialP a b = I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))
178+type TrivialP a b = I.FromNat (a * b) ~ I.FromNat (I.Count (I.FromNat a) * I.Count (I.FromNat b))
174−
175−-- | It isn't possible to implement the 'id' function so it's
176−-- implementation is 'undefined'. However 'comp' can be and this partial
177−-- class implementation exists just to make the code more readable.
178−--
179−-- Please use 'identity' instead.
180−instance (Num e) => C.Category (Matrix e) where
181− id = undefined
182− (.) = comp
183
179
184-- Primitives
180-- Primitives
185
181
186−empty :: Matrix e 0 0
187−empty = M I.Empty
188−
189one :: e -> Matrix e 1 1
182one :: e -> Matrix e 1 1
190one = M . I.One
183one = M . I.One
191
184
192−junc ::
185+join ::
193 (TrivialE a b) =>
186 (TrivialE a b) =>
194 Matrix e a rows ->
187 Matrix e a rows ->
195 Matrix e b rows ->
188 Matrix e b rows ->
196 Matrix e (a + b) rows
189 Matrix e (a + b) rows
197−junc (M a) (M b) = M (I.Junc a b)
190+join (M a) (M b) = M (I.Join a b)
198
191
199infixl 3 |||
192infixl 3 |||
200(|||) ::
193(|||) ::
@@ -202,14 +195,14 @@ infixl 3 |||
202 Matrix e a rows ->
195 Matrix e a rows ->
203 Matrix e b rows ->
196 Matrix e b rows ->
204 Matrix e (a + b) rows
197 Matrix e (a + b) rows
205−(|||) = junc
198+(|||) = join
206
199
207−split ::
200+fork ::
208 (TrivialE a b) =>
201 (TrivialE a b) =>
209 Matrix e cols a ->
202 Matrix e cols a ->
210 Matrix e cols b ->
203 Matrix e cols b ->
211 Matrix e cols (a + b)
204 Matrix e cols (a + b)
212−split (M a) (M b) = M (I.Split a b)
205+fork (M a) (M b) = M (I.Fork a b)
213
206
214infixl 2 ===
207infixl 2 ===
215(===) ::
208(===) ::
@@ -217,43 +210,45 @@ infixl 2 ===
217 Matrix e cols a ->
210 Matrix e cols a ->
218 Matrix e cols b ->
211 Matrix e cols b ->
219 Matrix e cols (a + b)
212 Matrix e cols (a + b)
220−(===) = split
213+(===) = fork
221
214
222-- Construction
215-- Construction
223
216
224−fromLists :: (FromListsN e cols rows) => [[e]] -> Matrix e cols rows
217+fromLists :: (FLN e cols rows) => [[e]] -> Matrix e cols rows
225fromLists = M . I.fromLists
218fromLists = M . I.fromLists
226
219
220+-- | Matrix builder function. Constructs a matrix provided with
221+-- a construction function that operates with indices.
227−matrixBuilder ::
222+matrixBuilder' ::
228− (FromListsN e cols rows, CountableN cols, CountableN rows)
223+ (FLN e cols rows, CountableN cols, CountableN rows)
229 => ((Int, Int) -> e) -> Matrix e cols rows
224 => ((Int, Int) -> e) -> Matrix e cols rows
230−matrixBuilder = M . I.matrixBuilder
225+matrixBuilder' = M . I.matrixBuilder'
231
226
232−col :: (I.FromLists e () (I.FromNat rows)) => [e] -> Matrix e 1 rows
227+col :: (I.FL () (I.FromNat rows)) => [e] -> Matrix e 1 rows
233col = M . I.col
228col = M . I.col
234
229
235−row :: (I.FromLists e (I.FromNat cols) ()) => [e] -> Matrix e cols 1
230+row :: (I.FL (I.FromNat cols) ()) => [e] -> Matrix e cols 1
236row = M . I.row
231row = M . I.row
237
232
238−fromF ::
233+fromF' ::
239 ( Liftable e a b,
234 ( Liftable e a b,
240 CountableN cols,
235 CountableN cols,
241 CountableN rows,
236 CountableN rows,
242− FromListsN e rows cols
237+ FLN e rows cols
243 ) =>
238 ) =>
244 (a -> b) ->
239 (a -> b) ->
245 Matrix e cols rows
240 Matrix e cols rows
246−fromF = M . I.fromF
241+fromF' = M . I.fromF'
247
242
248−fromF' ::
243+fromF ::
249 ( Liftable e a b,
244 ( Liftable e a b,
250 CountableNz a,
245 CountableNz a,
251 CountableNz b,
246 CountableNz b,
252− FromListsNz e b a
247+ FLNz e b a
253 ) =>
248 ) =>
254 (a -> b) ->
249 (a -> b) ->
255 Matrix e (I.Count a) (I.Count b)
250 Matrix e (I.Count a) (I.Count b)
256−fromF' = M . I.fromF'
251+fromF = undefined -- M . I.fromF
257
252
258-- Conversion
253-- Conversion
259
254
@@ -266,21 +261,21 @@ toList (M m) = I.toList m
266-- Zeros Matrix
261-- Zeros Matrix
267
262
268zeros ::
263zeros ::
269− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>
264+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
270 Matrix e cols rows
265 Matrix e cols rows
271zeros = M I.zeros
266zeros = M I.zeros
272
267
273-- Ones Matrix
268-- Ones Matrix
274
269
275ones ::
270ones ::
276− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>
271+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
277 Matrix e cols rows
272 Matrix e cols rows
278ones = M I.ones
273ones = M I.ones
279
274
280-- Const Matrix
275-- Const Matrix
281
276
282constant ::
277constant ::
283− (Num e, FromListsN e cols rows, CountableN cols, CountableN rows) =>
278+ (Num e, FLN e cols rows, CountableN cols, CountableN rows) =>
284 e ->
279 e ->
285 Matrix e cols rows
280 Matrix e cols rows
286constant = M . I.constant
281constant = M . I.constant
@@ -289,27 +284,41 @@ constant = M . I.constant
289
284
290bang ::
285bang ::
291 forall e cols.
286 forall e cols.
292− (Num e, Enum e, I.FromLists e (I.FromNat cols) (), CountableN cols) =>
287+ (Num e, Enum e, I.FL (I.FromNat cols) (), CountableN cols) =>
293 Matrix e cols 1
288 Matrix e cols 1
294bang = M I.bang
289bang = M I.bang
295
290
296−-- Identity Matrix
291+-- iden Matrix
297
292
298−identity ::
293+iden ::
299− (Num e, FromListsN e cols cols, CountableN cols) =>
294+ (Num e, FLN e cols cols, CountableN cols) =>
300 Matrix e cols cols
295 Matrix e cols cols
301−identity = M I.identity
296+iden = M I.iden
302
297
303-- Matrix composition (MMM)
298-- Matrix composition (MMM)
304
299
305comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
300comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
306comp (M a) (M b) = M (I.comp a b)
301comp (M a) (M b) = M (I.comp a b)
307
302
303+-- Scalar multiplication of matrices
304+
305+infixl 7 .|
306+-- | Scalar multiplication of matrices.
307+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows
308+(.|) e (M m) = M (e I..| m)
309+
310+-- Scalar division of matrices
311+
312+infixl 7 ./
313+-- | Scalar multiplication of matrices.
314+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows
315+(./) (M m) e = M (m I../ e)
316+
308p1 ::
317p1 ::
309 ( Num e,
318 ( Num e,
310− CountableDimensionsN n m,
319+ CountableDimsN n m,
311− FromListsN e n m,
320+ FLN e n m,
312− FromListsN e m m,
321+ FLN e m m,
313 TrivialE m n
322 TrivialE m n
314 ) =>
323 ) =>
315 Matrix e (m + n) m
324 Matrix e (m + n) m
@@ -317,9 +326,9 @@ p1 = M I.p1
317
326
318p2 ::
327p2 ::
319 ( Num e,
328 ( Num e,
320− CountableDimensionsN n m,
329+ CountableDimsN n m,
321− FromListsN e m n,
330+ FLN e m n,
322− FromListsN e n n,
331+ FLN e n n,
323 TrivialE m n
332 TrivialE m n
324 ) =>
333 ) =>
325 Matrix e (m + n) n
334 Matrix e (m + n) n
@@ -329,9 +338,9 @@ p2 = M I.p2
329
338
330i1 ::
339i1 ::
331 ( Num e,
340 ( Num e,
332− CountableDimensionsN n rows,
341+ CountableDimsN n rows,
333− FromListsN e n rows,
342+ FLN e n rows,
334− FromListsN e rows rows,
343+ FLN e rows rows,
335 TrivialE rows n
344 TrivialE rows n
336 ) =>
345 ) =>
337 Matrix e rows (rows + n)
346 Matrix e rows (rows + n)
@@ -339,9 +348,9 @@ i1 = tr p1
339
348
340i2 ::
349i2 ::
341 ( Num e,
350 ( Num e,
342− CountableDimensionsN rows m,
351+ CountableDimsN rows m,
343− FromListsN e m rows,
352+ FLN e m rows,
344− FromListsN e rows rows,
353+ FLN e rows rows,
345 TrivialE m rows
354 TrivialE m rows
346 ) =>
355 ) =>
347 Matrix e rows (m + rows)
356 Matrix e rows (m + rows)
@@ -355,16 +364,16 @@ rows (M m) = I.rows m
355columns :: (CountableN cols) => Matrix e cols rows -> Int
364columns :: (CountableN cols) => Matrix e cols rows -> Int
356columns (M m) = I.columns m
365columns (M m) = I.columns m
357
366
358−-- | Coproduct Bifunctor
359infixl 5 -|-
367infixl 5 -|-
360
368
369+-- | Coproduct Bifunctor (Direct sum)
361(-|-) ::
370(-|-) ::
362 ( Num e,
371 ( Num e,
363− CountableDimensionsN j k,
372+ CountableDimsN j k,
364− FromListsN e k k,
373+ FLN e k k,
365− FromListsN e j k,
374+ FLN e j k,
366− FromListsN e k j,
375+ FLN e k j,
367− FromListsN e j j,
376+ FLN e j j,
368 TrivialE n m,
377 TrivialE n m,
369 TrivialE k j
378 TrivialE k j
370 ) =>
379 ) =>
@@ -373,40 +382,42 @@ infixl 5 -|-
373 Matrix e (n + m) (k + j)
382 Matrix e (n + m) (k + j)
374(-|-) (M a) (M b) = M ((I.-|-) a b)
383(-|-) (M a) (M b) = M ((I.-|-) a b)
375
384
376−-- | Khatri Rao Product and projections
385+-- | Khatri Rao Product first projection
377−kp1 ::
386+fstM ::
378 forall e m k .
387 forall e m k .
379 ( Num e,
388 ( Num e,
380− CountableDimensionsN m k,
389+ CountableDimsN m k,
381 CountableN (m * k),
390 CountableN (m * k),
382− FromListsN e (m * k) m,
391+ FLN e (m * k) m,
383 TrivialP m k
392 TrivialP m k
384 ) => Matrix e (m * k) m
393 ) => Matrix e (m * k) m
385−kp1 = M (I.kp1 @e @(I.FromNat m) @(I.FromNat k))
394+fstM = M (I.fstM @e @(I.FromNat m) @(I.FromNat k))
386
395
387−kp2 ::
396+-- | Khatri Rao Product second projection
397+sndM ::
388 forall e m k.
398 forall e m k.
389 ( Num e,
399 ( Num e,
390− CountableDimensionsN k m,
400+ CountableDimsN k m,
391− FromListsN e (m * k) k,
401+ FLN e (m * k) k,
392 CountableN (m * k),
402 CountableN (m * k),
393 TrivialP m k
403 TrivialP m k
394 ) => Matrix e (m * k) k
404 ) => Matrix e (m * k) k
395−kp2 = M (I.kp2 @e @(I.FromNat m) @(I.FromNat k))
405+sndM = M (I.sndM @e @(I.FromNat m) @(I.FromNat k))
396
406
397−khatri ::
407+-- | Khatri Rao Product
408+kr ::
398 forall e cols a b.
409 forall e cols a b.
399 ( Num e,
410 ( Num e,
400− CountableDimensionsN a b,
411+ CountableDimsN a b,
401 CountableN (a * b),
412 CountableN (a * b),
402− FromListsN e (a * b) a,
413+ FLN e (a * b) a,
403− FromListsN e (a * b) b,
414+ FLN e (a * b) b,
404 TrivialP a b
415 TrivialP a b
405 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)
416 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a * b)
406−khatri a b =
417+kr a b =
407− let kp1' = kp1 @e @a @b
418+ let fstM' = fstM @e @a @b
408− kp2' = kp2 @e @a @b
419+ sndM' = sndM @e @a @b
409− in comp (tr kp1') a * comp (tr kp2') b
420+ in comp (tr fstM') a * comp (tr sndM') b
410
421
411-- | Product Bifunctor (Kronecker)
422-- | Product Bifunctor (Kronecker)
412infixl 4 ><
423infixl 4 ><
@@ -414,28 +425,28 @@ infixl 4 ><
414(><) ::
425(><) ::
415 forall e m p n q.
426 forall e m p n q.
416 ( Num e,
427 ( Num e,
417− CountableDimensionsN m n,
428+ CountableDimsN m n,
418− CountableDimensionsN p q,
429+ CountableDimsN p q,
419− CountableDimensionsN (m * n) (p * q),
430+ CountableDimsN (m * n) (p * q),
420− FromListsN e (m * n) m,
431+ FLN e (m * n) m,
421− FromListsN e (m * n) n,
432+ FLN e (m * n) n,
422− FromListsN e (p * q) p,
433+ FLN e (p * q) p,
423− FromListsN e (p * q) q,
434+ FLN e (p * q) q,
424 TrivialP m n,
435 TrivialP m n,
425 TrivialP p q
436 TrivialP p q
426 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)
437 ) => Matrix e m p -> Matrix e n q -> Matrix e (m * n) (p * q)
427(><) a b =
438(><) a b =
428− let kp1' = kp1 @e @m @n
439+ let fstM' = fstM @e @m @n
429− kp2' = kp2 @e @m @n
440+ sndM' = sndM @e @m @n
430− in khatri (comp a kp1') (comp b kp2')
441+ in kr (comp a fstM') (comp b sndM')
431
442
432−-- | Matrix abide Junc Split
443+-- | Matrix abide Join Fork
433−abideJS :: Matrix e cols rows -> Matrix e cols rows
444+abideJF :: Matrix e cols rows -> Matrix e cols rows
434−abideJS (M m) = M (I.abideJS m)
445+abideJF (M m) = M (I.abideJF m)
435
446
436−-- | Matrix abide Split Junc
447+-- | Matrix abide Fork Join
437−abideSJ :: Matrix e cols rows -> Matrix e cols rows
448+abideFJ :: Matrix e cols rows -> Matrix e cols rows
438−abideSJ (M m) = M (I.abideSJ m)
449+abideFJ (M m) = M (I.abideFJ m)
439
450
440-- | Matrix transposition
451-- | Matrix transposition
441tr :: Matrix e cols rows -> Matrix e rows cols
452tr :: Matrix e cols rows -> Matrix e rows cols
@@ -444,7 +455,7 @@ tr (M m) = M (I.tr m)
444-- Selective 'select' operator
455-- Selective 'select' operator
445select ::
456select ::
446 ( Num e,
457 ( Num e,
447− FromListsN e rows1 rows1,
458+ FLN e rows1 rows1,
448 CountableN rows1,
459 CountableN rows1,
449 I.FromNat rows2 ~ I.FromNat rows1,
460 I.FromNat rows2 ~ I.FromNat rows1,
450 I.FromNat cols1 ~ I.FromNat cols2,
461 I.FromNat cols1 ~ I.FromNat cols2,
@@ -457,9 +468,9 @@ select (M m) (M y) = M (I.select m y)
457cond ::
468cond ::
458 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,
469 ( I.FromNat (I.Count (I.FromNat cols)) ~ I.FromNat cols,
459 CountableN cols,
470 CountableN cols,
460− I.FromLists e () (I.FromNat cols),
471+ I.FL () (I.FromNat cols),
461− I.FromLists e (I.FromNat cols) (),
472+ I.FL (I.FromNat cols) (),
462− FromListsN e cols cols,
473+ FLN e cols cols,
463 Liftable e a Bool
474 Liftable e a Bool
464 )
475 )
465 =>
476 =>
@@ -468,8 +479,12 @@ cond p (M a) (M b) = M (I.cond p a b)
468
479
469-- Pretty print
480-- Pretty print
470
481
471−pretty :: (CountableN cols, Show e) => Matrix e cols rows -> String
482+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String
472pretty (M m) = I.pretty m
483pretty (M m) = I.pretty m
473
484
474−prettyPrint :: (CountableN cols, Show e) => Matrix e cols rows -> IO ()
485+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()
475prettyPrint (M m) = I.prettyPrint m
486prettyPrint (M m) = I.prettyPrint m
487+
488+-- | Zip two matrices with a given binary function
489+zipWithM :: (e -> f -> g) -> Matrix e cols rows -> Matrix f cols rows -> Matrix g cols rows
490+zipWithM f (M a) (M b) = M (I.zipWithM f a b)
src/LAoP/Matrix/Type.hs+246−180
@@ -1,11 +1,11 @@
1{-# LANGUAGE ConstraintKinds #-}
1{-# LANGUAGE ConstraintKinds #-}
2+{-# LANGUAGE TypeFamilies #-}
2{-# LANGUAGE AllowAmbiguousTypes #-}
3{-# LANGUAGE AllowAmbiguousTypes #-}
3{-# LANGUAGE DataKinds #-}
4{-# LANGUAGE DataKinds #-}
4{-# LANGUAGE DerivingVia #-}
5{-# LANGUAGE DerivingVia #-}
5{-# LANGUAGE FlexibleContexts #-}
6{-# LANGUAGE FlexibleContexts #-}
6{-# LANGUAGE FlexibleInstances #-}
7{-# LANGUAGE FlexibleInstances #-}
7{-# LANGUAGE GADTs #-}
8{-# LANGUAGE GADTs #-}
8−{-# LANGUAGE KindSignatures #-}
9{-# LANGUAGE MultiParamTypeClasses #-}
9{-# LANGUAGE MultiParamTypeClasses #-}
10{-# LANGUAGE NoStarIsType #-}
10{-# LANGUAGE NoStarIsType #-}
11{-# LANGUAGE RankNTypes #-}
11{-# LANGUAGE RankNTypes #-}
@@ -58,13 +58,12 @@ module LAoP.Matrix.Type
58
58
59 -- * Constraint type aliases
59 -- * Constraint type aliases
60 Countable,
60 Countable,
61− CountableDimensions,
61+ CountableDims,
62 CountableN,
62 CountableN,
63− CountableDimensionsN,
63+ CountableDimsN,
64− FromListsN,
64+ FLN,
65 Liftable,
65 Liftable,
66 Trivial,
66 Trivial,
67− TrivialE,
68 TrivialP,
67 TrivialP,
69
68
70 -- * Type aliases
69 -- * Type aliases
@@ -72,10 +71,9 @@ module LAoP.Matrix.Type
72 One,
71 One,
73
72
74 -- * Primitives
73 -- * Primitives
75− empty,
76 one,
74 one,
77− junc,
75+ join,
78− split,
76+ fork,
79
77
80 -- * Auxiliary type families
78 -- * Auxiliary type families
81 I.FromNat,
79 I.FromNat,
@@ -87,24 +85,27 @@ module LAoP.Matrix.Type
87 fromLists,
85 fromLists,
88 toLists,
86 toLists,
89 toList,
87 toList,
88+ matrixBuilder',
90 matrixBuilder,
89 matrixBuilder,
91 row,
90 row,
92 col,
91 col,
93 zeros,
92 zeros,
94 ones,
93 ones,
95 bang,
94 bang,
95+ point,
96 constant,
96 constant,
97
97
98 -- * Functor instance equivalent function
98 -- * Functor instance equivalent function
99 fmapM,
99 fmapM,
100+ bimapM,
100
101
101 -- * Applicative/Monoidal instance equivalent functions
102 -- * Applicative/Monoidal instance equivalent functions
102 unitM,
103 unitM,
103 multM,
104 multM,
104
105
105 -- * Selective equivalent instance function
106 -- * Selective equivalent instance function
106− selectM,
107+ selectM,
107−
108+
108 -- * Monad equivalent instance function
109 -- * Monad equivalent instance function
109 returnM,
110 returnM,
110 bindM,
111 bindM,
@@ -112,25 +113,34 @@ module LAoP.Matrix.Type
112 -- * Misc
113 -- * Misc
113 -- ** Get dimensions
114 -- ** Get dimensions
114 columns,
115 columns,
116+ columns',
115 rows,
117 rows,
118+ rows',
116
119
117 -- ** Matrix Transposition
120 -- ** Matrix Transposition
118 tr,
121 tr,
119
122
123+ -- ** Scalar multiplication/division of matrices
124+ (.|),
125+ (./),
126+
120 -- ** McCarthy's Conditional
127 -- ** McCarthy's Conditional
121 cond,
128 cond,
122
129
123 -- ** Matrix "abiding"
130 -- ** Matrix "abiding"
124− abideJS,
131+ abideJF,
125− abideSJ,
132+ abideFJ,
133+
134+ -- ** Zip Matrices
135+ zipWithM,
126
136
127 -- * Biproduct approach
137 -- * Biproduct approach
128− -- ** Split
138+ -- ** Fork
129 (===),
139 (===),
130 -- *** Projections
140 -- *** Projections
131 p1,
141 p1,
132 p2,
142 p2,
133− -- ** Junc
143+ -- ** Join
134 (|||),
144 (|||),
135 -- *** Injections
145 -- *** Injections
136 i1,
146 i1,
@@ -144,11 +154,11 @@ module LAoP.Matrix.Type
144 -- | Note that given the restrictions imposed it is not possible to
154 -- | Note that given the restrictions imposed it is not possible to
145 -- implement the standard type classes present in standard Haskell.
155 -- implement the standard type classes present in standard Haskell.
146 -- *** Matrix pairing projections
156 -- *** Matrix pairing projections
147− kp1,
157+ fstM,
148− kp2,
158+ sndM,
149
159
150 -- *** Matrix pairing
160 -- *** Matrix pairing
151− khatri,
161+ kr,
152
162
153 -- * Matrix composition and lifting
163 -- * Matrix composition and lifting
154
164
@@ -156,10 +166,10 @@ module LAoP.Matrix.Type
156
166
157 -- | Note that given the restrictions imposed it is not possible to
167 -- | Note that given the restrictions imposed it is not possible to
158 -- implement the standard type classes present in standard Haskell.
168 -- implement the standard type classes present in standard Haskell.
159− identity,
169+ iden,
160 comp,
170 comp,
161− fromF,
162 fromF',
171 fromF',
172+ fromF,
163
173
164 -- * Relation
174 -- * Relation
165 toRel,
175 toRel,
@@ -173,34 +183,44 @@ where
173import Data.Void
183import Data.Void
174import Data.Proxy
184import Data.Proxy
175import Data.Kind
185import Data.Kind
176−import GHC.TypeLits
186+import GHC.TypeLits
177import Control.DeepSeq
187import Control.DeepSeq
178import LAoP.Utils
188import LAoP.Utils
179−import qualified Control.Category as C
180import qualified LAoP.Matrix.Internal as I
189import qualified LAoP.Matrix.Internal as I
190+import Prelude hiding (id, (.))
181
191
182newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))
192newtype Matrix e (cols :: Type) (rows :: Type) = M (I.Matrix e (I.Normalize cols) (I.Normalize rows))
183 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))
193 deriving (Show, Num, Eq, Ord, NFData) via (I.Matrix e (I.Normalize cols) (I.Normalize rows))
184
194
185-- | Constraint type synonyms to keep the type signatures less convoluted
195-- | Constraint type synonyms to keep the type signatures less convoluted
186−type Countable a = KnownNat (I.Count a)
196+type Countable a = KnownNat (I.Count a)
187−type CountableDimensions a b = (Countable a, Countable b)
197+type CountableDims a b = (Countable a, Countable b)
188−type CountableN a = KnownNat (I.Count (I.Normalize a))
198+type CountableN a = KnownNat (I.Count (I.Normalize a))
189−type CountableDimensionsN a b = (CountableN a, CountableN b)
199+type CountableDimsN a b = (CountableN a, CountableN b)
190−type FromListsN e a b = I.FromLists e (I.Normalize a) (I.Normalize b)
200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)
191−type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
201+type Liftable e a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num e, Ord e)
192−type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)
202+type Trivial a = I.Normalize (I.Normalize a) ~ I.Normalize (I.Normalize (I.Normalize a))
193−type TrivialE a b = I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)
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)
194−type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
205+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
195−
206+
196−-- | It isn't possible to implement the 'id' function so it's
207+-- | It is possible to implement a constrained version of the category type
197−-- implementation is 'undefined'. However 'comp' can be and this partial
208+-- class.
198−-- class implementation exists just to make the code more readable.
199−--
200−-- Please use 'identity' instead.
201−instance (Num e) => C.Category (Matrix e) where
209+instance (Num e) => Category (Matrix e) where
202− id = undefined
210+ type Object (Matrix e) a = (FLN a a, CountableN a)
211+ id = iden
203− (.) = comp
212+ (.) = comp
213+
214+-- | Bifunctor equivalent function
215+bimapM ::
216+ ( Liftable e a b,
217+ Liftable e c d,
218+ CountableDimsN a c,
219+ CountableDimsN b d,
220+ FLN d c,
221+ FLN b a
222+ ) => (a -> b) -> (c -> d) -> Matrix e a c -> Matrix e b d
223+bimapM f g m = fromF g . m . tr (fromF f)
204
224
205-- | Zero type alias
225-- | Zero type alias
206type Zero = Void
226type Zero = Void
@@ -210,82 +230,74 @@ type One = ()
210
230
211-- Primitives
231-- Primitives
212
232
213−-- | Empty matrix constructor
214−empty :: Matrix e Zero Zero
215−empty = M I.Empty
216−
217-- | Unit matrix constructor
233-- | Unit matrix constructor
218one :: e -> Matrix e One One
234one :: e -> Matrix e One One
219one = M . I.One
235one = M . I.One
220
236
221−-- | Matrix 'Junc' constructor
237+-- | Matrix 'Join' constructor
222−junc ::
238+join ::
223− (TrivialE a b) =>
224 Matrix e a rows ->
239 Matrix e a rows ->
225 Matrix e b rows ->
240 Matrix e b rows ->
226 Matrix e (Either a b) rows
241 Matrix e (Either a b) rows
227−junc (M a) (M b) = M (I.Junc a b)
242+join (M a) (M b) = M (I.Join a b)
228
243
229infixl 3 |||
244infixl 3 |||
230−-- | Matrix 'Junc' constructor
245+-- | Matrix 'Join' constructor
231(|||) ::
246(|||) ::
232− (TrivialE a b) =>
233 Matrix e a rows ->
247 Matrix e a rows ->
234 Matrix e b rows ->
248 Matrix e b rows ->
235 Matrix e (Either a b) rows
249 Matrix e (Either a b) rows
236−(|||) = junc
250+(|||) = join
237
251
238−-- | Matrix 'Split' constructor
252+-- | Matrix 'Fork' constructor
239−split ::
253+fork ::
240− (TrivialE a b) =>
241 Matrix e cols a ->
254 Matrix e cols a ->
242 Matrix e cols b ->
255 Matrix e cols b ->
243 Matrix e cols (Either a b)
256 Matrix e cols (Either a b)
244−split (M a) (M b) = M (I.Split a b)
257+fork (M a) (M b) = M (I.Fork a b)
245
258
246infixl 2 ===
259infixl 2 ===
247−-- | Matrix 'Split' constructor
260+-- | Matrix 'Fork' constructor
248(===) ::
261(===) ::
249− (TrivialE a b) =>
250 Matrix e cols a ->
262 Matrix e cols a ->
251 Matrix e cols b ->
263 Matrix e cols b ->
252 Matrix e cols (Either a b)
264 Matrix e cols (Either a b)
253−(===) = split
265+(===) = fork
254
266
255-- Functor hierarchy
267-- Functor hierarchy
256
268
257-- | Functor instance equivalent function
269-- | Functor instance equivalent function
258−fmapM ::
270+fmapM ::
259 ( Liftable e a b,
271 ( Liftable e a b,
260− CountableDimensionsN a b,
272+ CountableDimsN a b,
261− FromListsN e b a
273+ FLN b a
262 )
274 )
263 =>
275 =>
264 (a -> b) -> Matrix e c a -> Matrix e c b
276 (a -> b) -> Matrix e c a -> Matrix e c b
265−fmapM f m = fromF' f `comp` m
277+fmapM f m = fromF f . m
266
278
267-- | Applicative instance equivalent 'unit' function,
279-- | Applicative instance equivalent 'unit' function,
268unitM :: (Num e) => Matrix e () ()
280unitM :: (Num e) => Matrix e () ()
269unitM = one 1
281unitM = one 1
270
282
271-- | Applicative instance equivalent 'unit' function,
283-- | Applicative instance equivalent 'unit' function,
272−multM ::
284+multM ::
273− ( CountableDimensionsN a b,
285+ ( CountableDimsN a b,
274 CountableN (a, b),
286 CountableN (a, b),
275 Num e,
287 Num e,
276− FromListsN e (a, b) a,
288+ FLN (a, b) a,
277− FromListsN e (a, b) b,
289+ FLN (a, b) b,
278 TrivialP a b
290 TrivialP a b
279 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)
291 ) => Matrix e c a -> Matrix e c b -> Matrix e c (a, b)
280−multM = khatri
292+multM = kr
281
293
282-- | Monad instance equivalent 'return' function,
294-- | Monad instance equivalent 'return' function,
283−returnM ::
295+returnM ::
284− forall e a .
296+ forall e a .
285 ( Num e,
297 ( Num e,
286 Enum e,
298 Enum e,
287 Enum a,
299 Enum a,
288− FromListsN e () a,
300+ FLN () a,
289 Countable a
301 Countable a
290 ) => a -> Matrix e One a
302 ) => a -> Matrix e One a
291returnM a = col l
303returnM a = col l
@@ -302,51 +314,64 @@ bindM = flip comp
302
314
303-- | Build a matrix out of a list of list of elements. Throws a runtime
315-- | Build a matrix out of a list of list of elements. Throws a runtime
304-- error if the dimensions do not match.
316-- error if the dimensions do not match.
305−fromLists :: (FromListsN e cols rows) => [[e]] -> Matrix e cols rows
317+fromLists :: (FLN cols rows) => [[e]] -> Matrix e cols rows
306fromLists = M . I.fromLists
318fromLists = M . I.fromLists
307
319
308-- | Matrix builder function. Constructs a matrix provided with
320-- | Matrix builder function. Constructs a matrix provided with
309−-- a construction function.
321+-- a construction function that operates with indices.
310−matrixBuilder ::
322+matrixBuilder' ::
311− (FromListsN e cols rows, CountableDimensionsN cols rows )
323+ (FLN cols rows, CountableDimsN cols rows )
312 => ((Int, Int) -> e) -> Matrix e cols rows
324 => ((Int, Int) -> e) -> Matrix e cols rows
313−matrixBuilder = M . I.matrixBuilder
325+matrixBuilder' = M . I.matrixBuilder'
326+
327+-- | Matrix builder function. Constructs a matrix provided with
328+-- a construction function that operates with arbitrary types.
329+matrixBuilder ::
330+ ( FLN a b,
331+ Enum a,
332+ Enum b,
333+ Bounded a,
334+ Bounded b,
335+ Eq a,
336+ Countable b
337+ ) => ((a, b) -> e) -> Matrix e a b
338+matrixBuilder f = M (I.matrixBuilder f)
314
339
315-- | Constructs a column vector matrix
340-- | Constructs a column vector matrix
316−col :: (FromListsN e () rows) => [e] -> Matrix e One rows
341+col :: (FLN () rows) => [e] -> Matrix e One rows
317col = M . I.col
342col = M . I.col
318
343
319-- | Constructs a row vector matrix
344-- | Constructs a row vector matrix
320−row :: (FromListsN e cols ()) => [e] -> Matrix e cols One
345+row :: (FLN cols ()) => [e] -> Matrix e cols One
321row = M . I.row
346row = M . I.row
322
347
323-- | Lifts functions to matrices with arbitrary dimensions.
348-- | Lifts functions to matrices with arbitrary dimensions.
324--
349--
325-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
350-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
326-- types @a@ or @b@ allows.
351-- types @a@ or @b@ allows.
327−fromF ::
352+fromF' ::
328 ( Liftable e a b,
353 ( Liftable e a b,
329− CountableDimensionsN cols rows,
354+ CountableDimsN cols rows,
330− FromListsN e rows cols
355+ FLN rows cols
331 ) =>
356 ) =>
332 (a -> b) -> Matrix e cols rows
357 (a -> b) -> Matrix e cols rows
333−fromF = M . I.fromF
358+fromF' = M . I.fromF'
334
359
335-- | Lifts functions to matrices with dimensions matching @a@ and @b@
360-- | Lifts functions to matrices with dimensions matching @a@ and @b@
336-- cardinality's.
361-- cardinality's.
337−fromF' ::
362+fromF ::
338 ( Liftable e a b,
363 ( Liftable e a b,
339− CountableDimensionsN a b,
364+ CountableDimsN a b,
340− FromListsN e b a
365+ FLN b a
341 ) =>
366 ) =>
342 (a -> b) -> Matrix e a b
367 (a -> b) -> Matrix e a b
343−fromF' = M . I.fromF'
368+fromF = M . I.fromF
344
369
345-- | Lifts relation functions to Boolean Matrix
370-- | Lifts relation functions to Boolean Matrix
346toRel ::
371toRel ::
347 ( Liftable (Natural 0 1) a b,
372 ( Liftable (Natural 0 1) a b,
348− CountableDimensionsN a b,
373+ CountableDims a b,
349− FromListsN (Natural 0 1) b a
374+ FLN b a
350 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b
375 ) => (a -> b -> Bool) -> Matrix (Natural 0 1) a b
351toRel = M . I.toRel
376toRel = M . I.toRel
352
377
@@ -364,7 +389,7 @@ toList (M m) = I.toList m
364
389
365-- | The zero matrix. A matrix wholly filled with zeros.
390-- | The zero matrix. A matrix wholly filled with zeros.
366zeros ::
391zeros ::
367− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)
392+ (Num e, FLN cols rows, CountableDimsN cols rows)
368 => Matrix e cols rows
393 => Matrix e cols rows
369zeros = M I.zeros
394zeros = M I.zeros
370
395
@@ -374,7 +399,7 @@ zeros = M I.zeros
374--
399--
375-- Also known as T (Top) matrix.
400-- Also known as T (Top) matrix.
376ones ::
401ones ::
377− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)
402+ (Num e, FLN cols rows, CountableDimsN cols rows)
378 => Matrix e cols rows
403 => Matrix e cols rows
379ones = M I.ones
404ones = M I.ones
380
405
@@ -383,7 +408,7 @@ ones = M I.ones
383-- | The constant matrix constructor. A matrix wholly filled with a given
408-- | The constant matrix constructor. A matrix wholly filled with a given
384-- value.
409-- value.
385constant ::
410constant ::
386− (Num e, FromListsN e cols rows, CountableDimensionsN cols rows)
411+ (Num e, FLN cols rows, CountableDimsN cols rows)
387 => e -> Matrix e cols rows
412 => e -> Matrix e cols rows
388constant = M . I.constant
413constant = M . I.constant
389
414
@@ -392,17 +417,30 @@ constant = M . I.constant
392-- | The T (Top) row vector matrix.
417-- | The T (Top) row vector matrix.
393bang ::
418bang ::
394 forall e cols.
419 forall e cols.
395− (Num e, Enum e, FromListsN e cols (), CountableN cols) =>
420+ (Num e, Enum e, FLN cols (), CountableN cols) =>
396 Matrix e cols One
421 Matrix e cols One
397bang = M I.bang
422bang = M I.bang
398
423
424+-- | Point constant relation
425+point ::
426+ ( Bounded a,
427+ Enum a,
428+ Eq a,
429+ Num e,
430+ Ord e,
431+ CountableN a,
432+ FLN a One
399−-- Identity Matrix
433+ ) => a -> Matrix e One a
434+point = fromF . const
400
435
436+-- iden Matrix
437+
401−-- | Identity matrix
438+-- | iden matrix
402−identity ::
439+iden ::
403− (Num e, FromListsN e a a, CountableN a) =>
440+ (Num e, FLN a a, CountableN a) =>
404 Matrix e a a
441 Matrix e a a
405−identity = M I.identity
442+iden = M I.iden
443+{-# NOINLINE iden #-}
406
444
407-- Matrix composition (MMM)
445-- Matrix composition (MMM)
408
446
@@ -412,14 +450,32 @@ identity = M I.identity
412-- from LAoP.
450-- from LAoP.
413comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
451comp :: (Num e) => Matrix e cr rows -> Matrix e cols cr -> Matrix e cols rows
414comp (M a) (M b) = M (I.comp a b)
452comp (M a) (M b) = M (I.comp a b)
453+{-# NOINLINE comp #-}
454+{-# RULES
455+ "comp/iden1" forall m. comp m iden = m ;
456+ "comp/iden2" forall m. comp iden m = m
457+#-}
458+
459+-- Scalar multiplication of matrices
460+
461+infixl 7 .|
462+-- | Scalar multiplication of matrices.
463+(.|) :: Num e => e -> Matrix e cols rows -> Matrix e cols rows
464+(.|) e (M m) = M (e I..| m)
465+
466+-- Scalar division of matrices
467+
468+infixl 7 ./
469+-- | Scalar multiplication of matrices.
470+(./) :: Fractional e => Matrix e cols rows -> e -> Matrix e cols rows
471+(./) (M m) e = M (m I../ e)
415
472
416-- | Biproduct first component projection
473-- | Biproduct first component projection
417p1 ::
474p1 ::
418 ( Num e,
475 ( Num e,
419− CountableDimensionsN n m,
476+ CountableDimsN n m,
420− FromListsN e n m,
477+ FLN n m,
421− FromListsN e m m,
478+ FLN m m
422− TrivialE m n
423 ) =>
479 ) =>
424 Matrix e (Either m n) m
480 Matrix e (Either m n) m
425p1 = M I.p1
481p1 = M I.p1
@@ -427,10 +483,9 @@ p1 = M I.p1
427-- | Biproduct second component projection
483-- | Biproduct second component projection
428p2 ::
484p2 ::
429 ( Num e,
485 ( Num e,
430− CountableDimensionsN n m,
486+ CountableDimsN n m,
431− FromListsN e m n,
487+ FLN m n,
432− FromListsN e n n,
488+ FLN n n
433− TrivialE m n
434 ) =>
489 ) =>
435 Matrix e (Either m n) n
490 Matrix e (Either m n) n
436p2 = M I.p2
491p2 = M I.p2
@@ -440,10 +495,9 @@ p2 = M I.p2
440-- | Biproduct first component injection
495-- | Biproduct first component injection
441i1 ::
496i1 ::
442 ( Num e,
497 ( Num e,
443− CountableDimensionsN n m,
498+ CountableDimsN n m,
444− FromListsN e n m,
499+ FLN n m,
445− FromListsN e m m,
500+ FLN m m
446− TrivialE m n
447 ) =>
501 ) =>
448 Matrix e m (Either m n)
502 Matrix e m (Either m n)
449i1 = tr p1
503i1 = tr p1
@@ -451,10 +505,9 @@ i1 = tr p1
451-- | Biproduct second component injection
505-- | Biproduct second component injection
452i2 ::
506i2 ::
453 ( Num e,
507 ( Num e,
454− CountableDimensionsN n m,
508+ CountableDimsN n m,
455− FromListsN e m n,
509+ FLN m n,
456− FromListsN e n n,
510+ FLN n n
457− TrivialE m n
458 ) =>
511 ) =>
459 Matrix e n (Either m n)
512 Matrix e n (Either m n)
460i2 = tr p2
513i2 = tr p2
@@ -463,24 +516,34 @@ i2 = tr p2
463
516
464-- | Obtain the number of rows.
517-- | Obtain the number of rows.
465--
518--
466−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
519+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
467−-- dimensions in constant time.
520+-- dimensions in constant time. For a version that doesn't require the
468−--
521+-- constraint see 'rows''.
469−-- TODO: A 'rows' function that does not need the 'KnownNat' constraint in
470−-- exchange for performance.
471rows :: (CountableN rows) => Matrix e cols rows -> Int
522rows :: (CountableN rows) => Matrix e cols rows -> Int
472rows (M m) = I.rows m
523rows (M m) = I.rows m
473
524
525+-- | Obtain the number of rows in an inefficient manner, but without any
526+-- constraints.
527+--
528+-- For a more efficient version see 'rows'.
529+rows' :: Matrix e cols rows -> Int
530+rows' (M m) = I.rows' m
531+
474-- | Obtain the number of columns.
532-- | Obtain the number of columns.
475−--
476−-- NOTE: The 'KnownNat' constaint is needed in order to obtain the
477−-- dimensions in constant time.
478--
533--
479−-- TODO: A 'columns' function that does not need the 'KnownNat' constraint in
534+-- NOTE: The 'KnownNat' constraint is needed in order to obtain the
480−-- exchange for performance.
535+-- dimensions in constant time. For a version that doesn't require the
536+-- constraint see 'columns''.
481columns :: (CountableN cols) => Matrix e cols rows -> Int
537columns :: (CountableN cols) => Matrix e cols rows -> Int
482columns (M m) = I.columns m
538columns (M m) = I.columns m
483
539
540+-- | Obtain the number of columns in an inefficient manner, but without any
541+-- constraints.
542+--
543+-- For a more efficient version see 'columns'.
544+columns' :: Matrix e cols rows -> Int
545+columns' (M m) = I.columns' m
546+
484-- Coproduct Bifunctor
547-- Coproduct Bifunctor
485
548
486infixl 5 -|-
549infixl 5 -|-
@@ -488,13 +551,11 @@ infixl 5 -|-
488-- | Matrix coproduct functor also known as matrix direct sum.
551-- | Matrix coproduct functor also known as matrix direct sum.
489(-|-) ::
552(-|-) ::
490 ( Num e,
553 ( Num e,
491− CountableDimensionsN j k,
554+ CountableDimsN j k,
492− FromListsN e k k,
555+ FLN k k,
493− FromListsN e j k,
556+ FLN j k,
494− FromListsN e k j,
557+ FLN k j,
495− FromListsN e j j,
558+ FLN j j
496− TrivialE n m,
497− TrivialE k j
498 ) =>
559 ) =>
499 Matrix e n k ->
560 Matrix e n k ->
500 Matrix e m j ->
561 Matrix e m j ->
@@ -504,98 +565,98 @@ infixl 5 -|-
504-- Khatri Rao Product and projections
565-- Khatri Rao Product and projections
505
566
506-- | Khatri Rao product first component projection matrix.
567-- | Khatri Rao product first component projection matrix.
507−kp1 ::
568+fstM ::
508 forall e m k .
569 forall e m k .
509 ( Num e,
570 ( Num e,
510− CountableDimensionsN m k,
571+ CountableDimsN m k,
511 CountableN (m, k),
572 CountableN (m, k),
512− FromListsN e (m, k) m,
573+ FLN (m, k) m,
513 TrivialP m k
574 TrivialP m k
514 ) => Matrix e (m, k) m
575 ) => Matrix e (m, k) m
515−kp1 = M (I.kp1 @e @(I.Normalize m) @(I.Normalize k))
576+fstM = M (I.fstM @e @(I.Normalize m) @(I.Normalize k))
516
577
517-- | Khatri Rao product second component projection matrix.
578-- | Khatri Rao product second component projection matrix.
518−kp2 ::
579+sndM ::
519 forall e m k.
580 forall e m k.
520 ( Num e,
581 ( Num e,
521− CountableDimensionsN k m,
582+ CountableDimsN k m,
522 CountableN (m, k),
583 CountableN (m, k),
523− FromListsN e (m, k) k,
584+ FLN (m, k) k,
524 TrivialP m k
585 TrivialP m k
525 ) => Matrix e (m, k) k
586 ) => Matrix e (m, k) k
526−kp2 = M (I.kp2 @e @(I.Normalize m) @(I.Normalize k))
587+sndM = M (I.sndM @e @(I.Normalize m) @(I.Normalize k))
527
588
528-- | Khatri Rao Matrix product also known as matrix pairing.
589-- | Khatri Rao Matrix product also known as matrix pairing.
529--
590--
530-- NOTE: That this is not a true categorical product, see for instance:
591-- NOTE: That this is not a true categorical product, see for instance:
531−--
592+--
532-- @
593-- @
533−-- | kp1 `comp` khatri a b == a
594+-- | fstM . kr a b == a
534−-- khatri a b ==> |
595+-- kr a b ==> |
535−-- | kp2 `comp` khatri a b == b
596+-- | sndM . kr a b == b
536-- @
597-- @
537--
598--
538-- __Emphasis__ on the implication symbol.
599-- __Emphasis__ on the implication symbol.
539−khatri ::
600+kr ::
540 forall e cols a b.
601 forall e cols a b.
541 ( Num e,
602 ( Num e,
542− CountableDimensionsN a b,
603+ CountableDimsN a b,
543 CountableN (a, b),
604 CountableN (a, b),
544− FromListsN e (a, b) a,
605+ FLN (a, b) a,
545− FromListsN e (a, b) b,
606+ FLN (a, b) b,
546 TrivialP a b
607 TrivialP a b
547 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)
608 ) => Matrix e cols a -> Matrix e cols b -> Matrix e cols (a, b)
548−khatri a b =
609+kr a b =
549− let kp1' = kp1 @e @a @b
610+ let fstM' = fstM @e @a @b
550− kp2' = kp2 @e @a @b
611+ sndM' = sndM @e @a @b
551− in comp (tr kp1') a * comp (tr kp2') b
612+ in comp (tr fstM') a * comp (tr sndM') b
552
613
553-- Product Bifunctor (Kronecker)
614-- Product Bifunctor (Kronecker)
554
615
555infixl 4 ><
616infixl 4 ><
556
617
557−-- | Matrix product functor also known as kronecker product
618+-- | Matrix product functor also known as Kronecker product
558(><) ::
619(><) ::
559 forall e m p n q.
620 forall e m p n q.
560 ( Num e,
621 ( Num e,
561− CountableDimensionsN m n,
622+ CountableDimsN m n,
562− CountableDimensionsN p q,
623+ CountableDimsN p q,
563− CountableDimensionsN (m, n) (p, q),
624+ CountableDimsN (m, n) (p, q),
564− FromListsN e (m, n) m,
625+ FLN (m, n) m,
565− FromListsN e (m, n) n,
626+ FLN (m, n) n,
566− FromListsN e (p, q) p,
627+ FLN (p, q) p,
567− FromListsN e (p, q) q,
628+ FLN (p, q) q,
568 TrivialP m n,
629 TrivialP m n,
569 TrivialP p q
630 TrivialP p q
570 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)
631 ) => Matrix e m p -> Matrix e n q -> Matrix e (m, n) (p, q)
571(><) a b =
632(><) a b =
572− let kp1' = kp1 @e @m @n
633+ let fstM' = fstM @e @m @n
573− kp2' = kp2 @e @m @n
634+ sndM' = sndM @e @m @n
574− in khatri (comp a kp1') (comp b kp2')
635+ in kr (comp a fstM') (comp b sndM')
575
636
576−-- Matrix abide Junc Split
637+-- Matrix abide Join Fork
577
638
578−-- | Matrix "abiding" followin the 'Junc'-'Split' abide law.
639+-- | Matrix "abiding" followin the 'Join'-'Fork' abide law.
579−--
640+--
580-- Law:
641-- Law:
581--
642--
582-- @
643-- @
583−-- 'Junc' ('Split' a c) ('Split' b d) == 'Split' ('Junc' a b) ('Junc' c d)
644+-- 'Join' ('Fork' a c) ('Fork' b d) == 'Fork' ('Join' a b) ('Join' c d)
584-- @
645-- @
585−abideJS :: Matrix e cols rows -> Matrix e cols rows
646+abideJF :: Matrix e cols rows -> Matrix e cols rows
586−abideJS (M m) = M (I.abideJS m)
647+abideJF (M m) = M (I.abideJF m)
587
648
588−-- Matrix abide Split Junc
649+-- Matrix abide Fork Join
589
650
590−-- | Matrix "abiding" followin the 'Split'-'Junc' abide law.
651+-- | Matrix "abiding" followin the 'Fork'-'Join' abide law.
591−--
652+--
592-- Law:
653-- Law:
593--
654--
594-- @
655-- @
595−-- 'Split' ('Junc' a b) ('Junc' c d) == 'Junc' ('Split' a c) ('Split' b d)
656+-- 'Fork' ('Join' a b) ('Join' c d) == 'Join' ('Fork' a c) ('Fork' b d)
596-- @
657-- @
597−abideSJ :: Matrix e cols rows -> Matrix e cols rows
658+abideFJ :: Matrix e cols rows -> Matrix e cols rows
598−abideSJ (M m) = M (I.abideSJ m)
659+abideFJ (M m) = M (I.abideFJ m)
599
660
600-- Matrix transposition
661-- Matrix transposition
601
662
@@ -607,11 +668,10 @@ tr (M m) = M (I.tr m)
607
668
608-- | Selective functors 'select' operator equivalent inspired by the
669-- | Selective functors 'select' operator equivalent inspired by the
609-- ArrowMonad solution presented in the paper.
670-- ArrowMonad solution presented in the paper.
610−selectM ::
671+selectM ::
611 ( Num e,
672 ( Num e,
612− FromListsN e b b,
673+ FLN b b,
613− CountableN b,
674+ CountableN b
614− TrivialE a b
615 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
675 ) => Matrix e cols (Either a b) -> Matrix e a b -> Matrix e cols b
616selectM (M m) (M y) = M (I.select m y)
676selectM (M m) (M y) = M (I.select m y)
617
677
@@ -620,10 +680,12 @@ selectM (M m) (M y) = M (I.select m y)
620-- | McCarthy's Conditional expresses probabilistic choice.
680-- | McCarthy's Conditional expresses probabilistic choice.
621cond ::
681cond ::
622 ( Trivial a,
682 ( Trivial a,
683+ Trivial2 a,
684+ Trivial3 a,
623 CountableN a,
685 CountableN a,
624− FromListsN e () a,
686+ FLN () a,
625− FromListsN e a (),
687+ FLN a (),
626− FromListsN e a a,
688+ FLN a a,
627 Liftable e a Bool
689 Liftable e a Bool
628 )
690 )
629 =>
691 =>
@@ -633,9 +695,13 @@ cond p (M a) (M b) = M (I.cond p a b)
633-- Pretty print
695-- Pretty print
634
696
635-- | Matrix pretty printer
697-- | Matrix pretty printer
636−pretty :: (CountableN cols, Show e) => Matrix e cols rows -> String
698+pretty :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> String
637pretty (M m) = I.pretty m
699pretty (M m) = I.pretty m
638
700
639-- | Matrix pretty printer
701-- | Matrix pretty printer
640−prettyPrint :: (CountableN cols, Show e) => Matrix e cols rows -> IO ()
702+prettyPrint :: (CountableDimsN cols rows, Show e) => Matrix e cols rows -> IO ()
641prettyPrint (M m) = I.prettyPrint m
703prettyPrint (M m) = I.prettyPrint m
704+
705+-- | Zip two matrices with a given binary function
706+zipWithM :: (e -> f -> g) -> Matrix e a b -> Matrix f a b -> Matrix g a b
707+zipWithM f (M a) (M b) = M (I.zipWithM f a b)
src/LAoP/Relation.hs+8−11
@@ -32,22 +32,19 @@ module LAoP.Relation
32
32
33 -- * Constraint type synonyms
33 -- * Constraint type synonyms
34 Countable,
34 Countable,
35− CountableDimensions,
35+ CountableDims,
36 CountableN,
36 CountableN,
37− CountableDimensionsN,
37+ CountableDimsN,
38− FromListsN,
38+ FLN,
39 Liftable,
39 Liftable,
40 Trivial,
40 Trivial,
41− TrivialE,
42 TrivialP,
41 TrivialP,
43− TrivialP2,
44
42
45 -- * Primitives
43 -- * Primitives
46− empty,
47 one,
44 one,
48− junc,
45+ join,
49 (|||),
46 (|||),
50− split,
47+ fork,
51 (===),
48 (===),
52
49
53 -- * Auxiliary type families
50 -- * Auxiliary type families
@@ -110,8 +107,8 @@ module LAoP.Relation
110 splitR,
107 splitR,
111
108
112 -- ** Projections
109 -- ** Projections
113− p1,
110+ fstR,
114− p2,
111+ sndR,
115 -- ** Bifunctor
112 -- ** Bifunctor
116 (><),
113 (><),
117 -- * Relational coproduct
114 -- * Relational coproduct
@@ -150,7 +147,7 @@ module LAoP.Relation
150 cond,
147 cond,
151
148
152 -- * Relational composition and lifting
149 -- * Relational composition and lifting
153− identity,
150+ iden,
154 comp,
151 comp,
155 fromF,
152 fromF,
156 fromF',
153 fromF',
src/LAoP/Relation/Internal.hs+250−254
@@ -1,3 +1,4 @@
1+{-# LANGUAGE TypeFamilies #-}
1{-# LANGUAGE ConstraintKinds #-}
2{-# LANGUAGE ConstraintKinds #-}
2{-# LANGUAGE GADTs #-}
3{-# LANGUAGE GADTs #-}
3{-# LANGUAGE UndecidableInstances #-}
4{-# LANGUAGE UndecidableInstances #-}
@@ -42,22 +43,19 @@ module LAoP.Relation.Internal
42
43
43 -- * Constraint type synonyms
44 -- * Constraint type synonyms
44 Countable,
45 Countable,
45− CountableDimensions,
46+ CountableDims,
46 CountableN,
47 CountableN,
47− CountableDimensionsN,
48+ CountableDimsN,
48− FromListsN,
49+ FLN,
49 Liftable,
50 Liftable,
50 Trivial,
51 Trivial,
51− TrivialE,
52 TrivialP,
52 TrivialP,
53− TrivialP2,
54
53
55 -- * Primitives
54 -- * Primitives
56− empty,
57 one,
55 one,
58− junc,
56+ join,
59 (|||),
57 (|||),
60− split,
58+ fork,
61 (===),
59 (===),
62
60
63 -- * Auxiliary type families
61 -- * Auxiliary type families
@@ -68,14 +66,15 @@ module LAoP.Relation.Internal
68 -- * Matrix construction and conversion
66 -- * Matrix construction and conversion
69 I.FromLists,
67 I.FromLists,
70 fromLists,
68 fromLists,
71− fromF,
72 fromF',
69 fromF',
70+ fromF,
73 toRel,
71 toRel,
74 toLists,
72 toLists,
75 toList,
73 toList,
76 toBool,
74 toBool,
77 pt,
75 pt,
78 belongs,
76 belongs,
77+ relationBuilder',
79 relationBuilder,
78 relationBuilder,
80 zeros,
79 zeros,
81 ones,
80 ones,
@@ -120,8 +119,8 @@ module LAoP.Relation.Internal
120 splitR,
119 splitR,
121
120
122 -- ** Projections
121 -- ** Projections
123− p1,
122+ fstR,
124− p2,
123+ sndR,
125 -- ** Bifunctor
124 -- ** Bifunctor
126 (><),
125 (><),
127 -- * Relational coproduct
126 -- * Relational coproduct
@@ -160,10 +159,10 @@ module LAoP.Relation.Internal
160 cond,
159 cond,
161
160
162 -- * Relational composition and lifting
161 -- * Relational composition and lifting
163− identity,
162+ iden,
164 comp,
163 comp,
165− fromF,
166 fromF',
164 fromF',
165+ fromF,
167
166
168 -- ** Relational application
167 -- ** Relational application
169 pointAp,
168 pointAp,
@@ -179,9 +178,9 @@ import Data.Void
179import qualified LAoP.Matrix.Internal as I
178import qualified LAoP.Matrix.Internal as I
180import LAoP.Utils.Internal
179import LAoP.Utils.Internal
181import Control.DeepSeq
180import Control.DeepSeq
182−import qualified Control.Category as C
183import Data.Bool
181import Data.Bool
184import GHC.TypeLits
182import GHC.TypeLits
183+import Prelude hiding (id, (.))
185
184
186-- | Boolean type synonym for working with boolean matrices
185-- | Boolean type synonym for working with boolean matrices
187type Boolean = Natural 0 1
186type Boolean = Natural 0 1
@@ -194,34 +193,30 @@ newtype Relation a b = R (I.Matrix Boolean (I.Normalize a) (I.Normalize b))
194deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)
193deriving instance (Read (I.Matrix Boolean (I.Normalize a) (I.Normalize b))) => Read (Relation a b)
195
194
196-- | Constraint type synonyms to keep the type signatures less convoluted
195-- | Constraint type synonyms to keep the type signatures less convoluted
197−type Countable a = KnownNat (I.Count a)
196+type Countable a = KnownNat (I.Count a)
198−type CountableDimensions a b = (Countable a, Countable b)
197+type CountableDims a b = (Countable a, Countable b)
199−type CountableN a = KnownNat (I.Count (I.Normalize a))
198+type CountableN a = KnownNat (I.Count (I.Normalize a))
200−type CountableDimensionsN a b = (CountableN a, CountableN b)
199+type CountableDimsN a b = (CountableN a, CountableN b)
201−type FromListsN a b = I.FromLists Boolean (I.Normalize a) (I.Normalize b)
200+type FLN a b = I.FromLists (I.Normalize a) (I.Normalize b)
202−type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Boolean, Ord Boolean)
201+type Liftable a b = (Bounded a, Bounded b, Enum a, Enum b, Eq b, Num Boolean, Ord Boolean)
203−type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)
202+type Trivial a = I.Normalize a ~ I.Normalize (I.Normalize a)
204−type TrivialE a b = I.Normalize (Either a b) ~ Either (I.Normalize a) (I.Normalize b)
205−type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
203+type TrivialP a b = I.Normalize (a, b) ~ I.Normalize (I.Normalize a, I.Normalize b)
206−type TrivialP2 a b = I.Normalize (I.Normalize a, I.Normalize b) ~ I.Normalize (I.Normalize (a, b))
204+
207−
208−-- | It isn't possible to implement the 'id' function so it's
205+-- | It is possible to implement a constrained version of the category type
209−-- implementation is 'undefined'. However 'comp' can be and this partial
206+-- class.
210−-- class implementation exists just to make the code more readable.
211−--
212−-- Please use 'identity' instead.
213−instance C.Category Relation where
207+instance Category Relation where
214− id = undefined
208+ type Object Relation a = (FLN a a, CountableN a)
209+ id = iden
215− (.) = comp
210+ (.) = comp
216
211
217instance Num (Relation a b) where
212instance Num (Relation a b) where
218− -- | Matrix addition becomes Boolean matrix disjunction
213+ -- | Matrix addition becomes Boolean matrix disjointion
219 (R a) + (R b) = R (I.orM a b)
214 (R a) + (R b) = R (I.orM a b)
220
215
221 -- | Matrix subtraction becomes Relational subtraction
216 -- | Matrix subtraction becomes Relational subtraction
222 (R a) - (R b) = R (I.subM a b)
217 (R a) - (R b) = R (I.subM a b)
223
218
224− -- | Matrix multiplication becomes Boolean matrix conjunction
219+ -- | Matrix multiplication becomes Boolean matrix conjointion
225 (R a) * (R b) = R (I.andM a b)
220 (R a) * (R b) = R (I.andM a b)
226
221
227 -- | Matrix negation becomes Boolean matrix negation
222 -- | Matrix negation becomes Boolean matrix negation
@@ -233,87 +228,92 @@ type One = ()
233
228
234-- Primitives
229-- Primitives
235
230
236−-- | Empty matrix constructor
237−empty :: Relation Zero Zero
238−empty = R I.Empty
239−
240-- | Unit matrix constructor
231-- | Unit matrix constructor
241one :: Boolean -> Relation One One
232one :: Boolean -> Relation One One
242one = R . I.One
233one = R . I.One
243
234
244−-- | Boolean Matrix 'Junc' constructor, also known as relational coproduct.
235+-- | Boolean Matrix 'Join' constructor, also known as relational coproduct.
245--
236--
246-- See 'eitherR'.
237-- See 'eitherR'.
247−junc :: (TrivialE a b)
248− => Relation a c -> Relation b c -> Relation (Either a b) c
238+join :: Relation a c -> Relation b c -> Relation (Either a b) c
249−junc (R a) (R b) = R (I.Junc a b)
239+join (R a) (R b) = R (I.Join a b)
250
240
251infixl 3 |||
241infixl 3 |||
252−-- | Boolean Matrix 'Junc' constructor
242+-- | Boolean Matrix 'Join' constructor
253--
243--
254-- See 'eitherR'.
244-- See 'eitherR'.
255(|||) ::
245(|||) ::
256− (TrivialE a b) =>
257 Relation a c ->
246 Relation a c ->
258 Relation b c ->
247 Relation b c ->
259 Relation (Either a b) c
248 Relation (Either a b) c
260−(|||) = junc
249+(|||) = join
261
250
262−-- | Boolean Matrix 'Split' constructor, also known as relational product.
251+-- | Boolean Matrix 'Fork' constructor, also known as relational product.
263−split :: (TrivialE a b)
264− => Relation c a -> Relation c b -> Relation c (Either a b)
252+fork :: Relation c a -> Relation c b -> Relation c (Either a b)
265−split (R a) (R b) = R (I.Split a b)
253+fork (R a) (R b) = R (I.Fork a b)
266
254
267infixl 2 ===
255infixl 2 ===
268−-- | Boolean Matrix 'Split' constructor
256+-- | Boolean Matrix 'Fork' constructor
269(===) ::
257(===) ::
270− (TrivialE a b) =>
271 Relation c a ->
258 Relation c a ->
272 Relation c b ->
259 Relation c b ->
273 Relation c (Either a b)
260 Relation c (Either a b)
274−(===) = split
261+(===) = fork
275
262
276-- Construction
263-- Construction
277
264
278-- | Build a matrix out of a list of list of elements. Throws a runtime
265-- | Build a matrix out of a list of list of elements. Throws a runtime
279-- error if the dimensions do not match.
266-- error if the dimensions do not match.
280−fromLists :: (FromListsN a b) => [[Boolean]] -> Relation a b
267+fromLists :: (FLN a b) => [[Boolean]] -> Relation a b
281fromLists = R . I.fromLists
268fromLists = R . I.fromLists
282
269
283−-- | Matrix builder function. Constructs a matrix provided with
270+-- | Relation builder function. Constructs a relation provided with
284−-- a construction function.
271+-- a construction function that operates with indices.
285−relationBuilder ::
272+relationBuilder' ::
286− (FromListsN a b, CountableDimensionsN a b) =>
273+ (FLN a b, CountableDimsN a b) =>
287 ((Int, Int) -> Boolean) -> Relation a b
274 ((Int, Int) -> Boolean) -> Relation a b
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
288relationBuilder = R . I.matrixBuilder
288relationBuilder = R . I.matrixBuilder
289
289
290-- | Lifts functions to matrices with arbitrary dimensions.
290-- | Lifts functions to matrices with arbitrary dimensions.
291--
291--
292-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
292-- NOTE: Be careful to not ask for a matrix bigger than the cardinality of
293-- types @a@ or @b@ allows.
293-- types @a@ or @b@ allows.
294−fromF ::
294+fromF' ::
295 ( Liftable a b,
295 ( Liftable a b,
296− CountableDimensionsN c d,
296+ CountableDimsN c d,
297− FromListsN d c
297+ FLN d c
298 )
298 )
299 => (a -> b) -> Relation c d
299 => (a -> b) -> Relation c d
300−fromF f = R (I.fromFRel f)
300+fromF' f = R (I.fromFRel' f)
301
301
302-- | Lifts functions to matrices with dimensions matching @a@ and @b@
302-- | Lifts functions to matrices with dimensions matching @a@ and @b@
303-- cardinality's.
303-- cardinality's.
304−fromF' ::
304+fromF ::
305 ( Liftable a b,
305 ( Liftable a b,
306− CountableDimensionsN a b,
306+ CountableDimsN a b,
307− FromListsN b a
307+ FLN b a
308 )
308 )
309 => (a -> b) -> Relation a b
309 => (a -> b) -> Relation a b
310−fromF' f = R (I.fromFRel' f)
310+fromF f = R (I.fromFRel f)
311
311
312-- | Lifts relation functions to 'Relation'
312-- | Lifts relation functions to 'Relation'
313toRel ::
313toRel ::
314 ( Liftable a b,
314 ( Liftable a b,
315− CountableDimensionsN a b,
315+ CountableDims a b,
316− FromListsN b a
316+ FLN b a
317 )
317 )
318 => (a -> b -> Bool) -> Relation a b
318 => (a -> b -> Bool) -> Relation a b
319toRel = R . I.toRel
319toRel = R . I.toRel
@@ -322,9 +322,9 @@ toRel = R . I.toRel
322fromRel ::
322fromRel ::
323 ( Liftable a b,
323 ( Liftable a b,
324 Eq a,
324 Eq a,
325− CountableDimensionsN a b,
325+ CountableDimsN a b,
326− FromListsN a One,
326+ FLN a One,
327− FromListsN b One
327+ FLN b One
328 )
328 )
329 => Relation a b -> (a -> b -> Bool)
329 => Relation a b -> (a -> b -> Bool)
330fromRel r a b = pointApBool a b r
330fromRel r a b = pointApBool a b r
@@ -348,12 +348,12 @@ toBool r = case toList r of
348-- | Power transpose.
348-- | Power transpose.
349--
349--
350-- Maps a relation to a set valued function.
350-- Maps a relation to a set valued function.
351−pt ::
351+pt ::
352 ( Liftable a b,
352 ( Liftable a b,
353 Eq a,
353 Eq a,
354− CountableDimensionsN a b,
354+ CountableDimsN a b,
355− FromListsN a One,
355+ FLN a One,
356− FromListsN b One
356+ FLN b One
357 )
357 )
358 => Relation a b -> (a -> List b)
358 => Relation a b -> (a -> List b)
359pt r a =
359pt r a =
@@ -365,8 +365,8 @@ belongs ::
365 ( Bounded a,
365 ( Bounded a,
366 Enum a,
366 Enum a,
367 Eq a,
367 Eq a,
368− CountableDimensionsN (List a) a,
368+ CountableDims (List a) a,
369− FromListsN a (List a)
369+ FLN a (List a)
370 )
370 )
371 => Relation (List a) a
371 => Relation (List a) a
372belongs = toRel elemR
372belongs = toRel elemR
@@ -378,16 +378,16 @@ belongs = toRel elemR
378-- | The zero relation. A relation where no element of type @a@ relates
378-- | The zero relation. A relation where no element of type @a@ relates
379-- with elements of type @b@.
379-- with elements of type @b@.
380--
380--
381−-- Also known as ⊥ (Bottom) Relation or empty Relation.
381+-- Also known as ⊥ (Bottom) Relation.
382--
382--
383-- @
383-- @
384−-- r ``comp`` ⊥ == ⊥ ``comp`` r == ⊥
384+-- r `.` ⊥ == ⊥ `.` r == ⊥
385-- ⊥ ``sse`` R && R ``sse`` T == True
385-- ⊥ ``sse`` R && R ``sse`` T == True
386-- @
386-- @
387zeros ::
387zeros ::
388− (FromListsN a b, CountableDimensionsN a b) =>
388+ (FLN a b, CountableDimsN a b) =>
389 Relation a b
389 Relation a b
390−zeros = relationBuilder (const (nat 0))
390+zeros = relationBuilder' (const (nat 0))
391
391
392-- Ones Matrix
392-- Ones Matrix
393
393
@@ -400,43 +400,43 @@ zeros = relationBuilder (const (nat 0))
400-- ⊥ ``sse`` R && R ``sse`` T == True
400-- ⊥ ``sse`` R && R ``sse`` T == True
401-- @
401-- @
402ones ::
402ones ::
403− (FromListsN a b, CountableDimensionsN a b) =>
403+ (FLN a b, CountableDimsN a b) =>
404 Relation a b
404 Relation a b
405−ones = relationBuilder (const (nat 1))
405+ones = relationBuilder' (const (nat 1))
406
406
407-- Bang Matrix
407-- Bang Matrix
408
408
409-- | The T (Top) row vector relation.
409-- | The T (Top) row vector relation.
410bang ::
410bang ::
411− (FromListsN a One, CountableN a) =>
411+ (FLN a One, CountableN a) =>
412 Relation a One
412 Relation a One
413bang = ones
413bang = ones
414
414
415-- | Point constant relation
415-- | Point constant relation
416−point ::
416+point ::
417 ( Bounded a,
417 ( Bounded a,
418 Enum a,
418 Enum a,
419 Eq a,
419 Eq a,
420 CountableN a,
420 CountableN a,
421− FromListsN a One
421+ FLN a One
422 ) => a -> Relation One a
422 ) => a -> Relation One a
423−point = fromF' . const
423+point = fromF . const
424
424
425−-- Identity Matrix
425+-- iden Matrix
426
426
427−-- | Identity matrix
427+-- | iden matrix
428--
428--
429-- @
429-- @
430−-- 'identity' ``comp`` r == r == r ``comp`` 'identity'
430+-- 'iden' `.` r == r == r `.` 'iden'
431-- @
431-- @
432−identity ::
432+iden ::
433− (FromListsN a a, CountableN a) => Relation a a
433+ (FLN a a, CountableN a) => Relation a a
434−identity = relationBuilder (bool (nat 0) (nat 1) . uncurry (==))
434+iden = relationBuilder' (bool (nat 0) (nat 1) . uncurry (==))
435
435
436-- | Relational composition
436-- | Relational composition
437--
437--
438-- @
438-- @
439−-- r ``comp`` (s ``comp`` p) = (r ``comp`` s) ``comp`` p
439+-- r `.` (s `.` p) = (r `.` s) `.` p
440-- @
440-- @
441comp :: Relation b c -> Relation a b -> Relation a c
441comp :: Relation b c -> Relation a b -> Relation a c
442comp (R a) (R b) = R (I.compRel a b)
442comp (R a) (R b) = R (I.compRel a b)
@@ -484,7 +484,7 @@ shrunkBy r s = r `intersection` divR s (conv r)
484-- r ``overriddenBy`` r == r
484-- r ``overriddenBy`` r == r
485-- @
485-- @
486overriddenBy ::
486overriddenBy ::
487− ( FromListsN b b,
487+ ( FLN b b,
488 CountableN b
488 CountableN b
489 ) => Relation a b -> Relation a b -> Relation a b
489 ) => Relation a b -> Relation a b -> Relation a b
490overriddenBy r s = s `union` r `intersection` divR zeros (conv s)
490overriddenBy r s = s `union` r `intersection` divR zeros (conv s)
@@ -496,11 +496,11 @@ overriddenBy r s = s `union` r `intersection` divR zeros (conv s)
496pointAp ::
496pointAp ::
497 ( Liftable a b,
497 ( Liftable a b,
498 Eq a,
498 Eq a,
499− CountableDimensionsN a b,
499+ CountableDimsN a b,
500− FromListsN a One,
500+ FLN a One,
501− FromListsN b One
501+ FLN b One
502 ) => a -> b -> Relation a b -> Relation One One
502 ) => a -> b -> Relation a b -> Relation One One
503−pointAp a b r = conv (point b) `comp` r `comp` point a
503+pointAp a b r = conv (point b) . r . point a
504
504
505-- | Relational application
505-- | Relational application
506--
506--
@@ -508,11 +508,11 @@ pointAp a b r = conv (point b) `comp` r `comp` point a
508pointApBool ::
508pointApBool ::
509 ( Liftable a b,
509 ( Liftable a b,
510 Eq a,
510 Eq a,
511− CountableDimensionsN a b,
511+ CountableDimsN a b,
512− FromListsN a One,
512+ FLN a One,
513− FromListsN b One
513+ FLN b One
514 ) => a -> b -> Relation a b -> Bool
514 ) => a -> b -> Relation a b -> Bool
515−pointApBool a b r = toBool $ conv (point b) `comp` r `comp` point a
515+pointApBool a b r = toBool $ conv (point b) . r . point a
516
516
517-- | Relational converse
517-- | Relational converse
518--
518--
@@ -538,7 +538,7 @@ iff r s = r == s
538
538
539-- | Relational intersection
539-- | Relational intersection
540--
540--
541−-- Lifts pointwise conjunction.
541+-- Lifts pointwise conjointion.
542--
542--
543-- @
543-- @
544-- (r ``intersection`` s) ``intersection`` t == r ``intersection`` (s ``intersection`` t)
544-- (r ``intersection`` s) ``intersection`` t == r ``intersection`` (s ``intersection`` t)
@@ -549,13 +549,13 @@ intersection a b = a * b
549
549
550-- | Relational union
550-- | Relational union
551--
551--
552−-- Lifts pointwise disjunction.
552+-- Lifts pointwise disjointion.
553--
553--
554-- @
554-- @
555-- (r ``union`` s) ``union`` t == r `'union' (s ``union`` t)
555-- (r ``union`` s) ``union`` t == r `'union' (s ``union`` t)
556-- r ``union`` s ``sse`` x == r ``sse`` x && s ``sse`` x
556-- r ``union`` s ``sse`` x == r ``sse`` x && s ``sse`` x
557−-- r ``comp`` (s ``union`` t) == (r ``comp`` s) ``union`` (r ``comp`` t)
557+-- r `.` (s ``union`` t) == (r `.` s) ``union`` (r `.` t)
558−-- (s ``union`` t) ``comp`` r == (s ``comp`` r) ``union`` (t ``comp`` r)
558+-- (s ``union`` t) `.` r == (s `.` r) ``union`` (t `.` r)
559-- @
559-- @
560union :: Relation a b -> Relation a b -> Relation a b
560union :: Relation a b -> Relation a b -> Relation a b
561union a b = a + b
561union a b = a + b
@@ -563,46 +563,46 @@ union a b = a + b
563-- | Relation Kernel
563-- | Relation Kernel
564--
564--
565-- @
565-- @
566−-- 'ker' r == 'conv' r ``comp`` r
566+-- 'ker' r == 'conv' r `.` r
567-- 'ker' r == 'img' ('conv' r)
567-- 'ker' r == 'img' ('conv' r)
568-- @
568-- @
569ker :: Relation a b -> Relation a a
569ker :: Relation a b -> Relation a a
570−ker r = conv r `comp` r
570+ker r = conv r . r
571
571
572-- | Relation Image
572-- | Relation Image
573--
573--
574-- @
574-- @
575−-- 'img' r == r ``comp`` conv r
575+-- 'img' r == r `.` conv r
576-- 'img' r == 'ker' ('conv' r)
576-- 'img' r == 'ker' ('conv' r)
577-- @
577-- @
578img :: Relation a b -> Relation b b
578img :: Relation a b -> Relation b b
579−img r = r `comp` conv r
579+img r = r . conv r
580
580
581-- | Function division. Special case of 'divS'.
581-- | Function division. Special case of 'divS'.
582--
582--
583-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and
583-- NOTE: _This is only valid_ if @f@ and @g@ are 'function's, i.e. 'simple' and
584-- 'entire'.
584-- 'entire'.
585--
585--
586−-- @'divisionF' f g == 'conv' g ``comp`` f@
586+-- @'divisionF' f g == 'conv' g `.` f@
587divisionF :: Relation a c -> Relation b c -> Relation a b
587divisionF :: Relation a c -> Relation b c -> Relation a b
588−divisionF f g = conv g `comp` f
588+divisionF f g = conv g . f
589
589
590-- Taxonomy of binary relations
590-- Taxonomy of binary relations
591
591
592-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@
592-- | A 'Relation' @r@ is 'simple' 'iff' @'coreflexive' ('img' r)@
593−simple :: (CountableN b, FromListsN b b) => Relation a b -> Bool
593+simple :: (CountableN b, FLN b b) => Relation a b -> Bool
594simple = coreflexive . img
594simple = coreflexive . img
595
595
596-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@
596-- | A 'Relation' @r@ is 'injective' 'iff' @'coreflexive' ('ker' r)@
597−injective :: (CountableN a, FromListsN a a) => Relation a b -> Bool
597+injective :: (CountableN a, FLN a a) => Relation a b -> Bool
598injective = coreflexive . ker
598injective = coreflexive . ker
599
599
600-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@
600-- | A 'Relation' @r@ is 'entire' 'iff' @'reflexive' ('ker' r)@
601−entire :: (CountableN a, FromListsN a a) => Relation a b -> Bool
601+entire :: (CountableN a, FLN a a) => Relation a b -> Bool
602entire = reflexive . ker
602entire = reflexive . ker
603
603
604-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@
604-- | A 'Relation' @r@ is 'surjective' 'iff' @'reflexive' ('img' r)@
605−surjective :: (CountableN b, FromListsN b b) => Relation a b -> Bool
605+surjective :: (CountableN b, FLN b b) => Relation a b -> Bool
606surjective = reflexive . img
606surjective = reflexive . img
607
607
608-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@
608-- | A 'Relation' @r@ is a 'function' 'iff' @'simple' r && 'entire' r@
@@ -611,13 +611,13 @@ surjective = reflexive . img
611-- relations:
611-- relations:
612--
612--
613-- @
613-- @
614−-- f ``comp`` r ``sse`` s == r ``sse`` f ``comp`` s
614+-- f `.` r ``sse`` s == r ``sse`` f `.` s
615−-- r ``comp`` f ``sse`` s == r ``sse`` s ``comp`` f
615+-- r `.` f ``sse`` s == r ``sse`` s `.` f
616-- @
616-- @
617function ::
617function ::
618− ( CountableDimensionsN a b,
618+ ( CountableDimsN a b,
619− FromListsN a a,
619+ FLN a a,
620− FromListsN b b
620+ FLN b b
621 )
621 )
622 => Relation a b -> Bool
622 => Relation a b -> Bool
623function r = simple r && entire r
623function r = simple r && entire r
@@ -625,7 +625,7 @@ function r = simple r && entire r
625-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@
625-- | A 'Relation' @r@ is a 'representation' 'iff' @'injective' r && 'entire' r@
626representation ::
626representation ::
627 ( CountableN a,
627 ( CountableN a,
628− FromListsN a a
628+ FLN a a
629 )
629 )
630 => Relation a b -> Bool
630 => Relation a b -> Bool
631representation r = injective r && entire r
631representation r = injective r && entire r
@@ -633,91 +633,91 @@ representation r = injective r && entire r
633-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@
633-- | A 'Relation' @r@ is an 'abstraction' 'iff' @'surjective' r && 'simple' r@
634abstraction ::
634abstraction ::
635 ( CountableN b,
635 ( CountableN b,
636− FromListsN b b
636+ FLN b b
637 )
637 )
638 => Relation a b -> Bool
638 => Relation a b -> Bool
639abstraction r = surjective r && simple r
639abstraction r = surjective r && simple r
640
640
641-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@
641-- | A 'Relation' @r@ is a 'surjection' 'iff' @'function' r && 'abstraction' r@
642surjection ::
642surjection ::
643− ( CountableDimensionsN a b,
643+ ( CountableDimsN a b,
644− FromListsN a a,
644+ FLN a a,
645− FromListsN b b
645+ FLN b b
646 )
646 )
647 => Relation a b -> Bool
647 => Relation a b -> Bool
648surjection r = function r && abstraction r
648surjection r = function r && abstraction r
649
649
650-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@
650-- | A 'Relation' @r@ is a 'injection' 'iff' @'function' r && 'representation' r@
651injection ::
651injection ::
652− ( CountableDimensionsN a b,
652+ ( CountableDimsN a b,
653− FromListsN a a,
653+ FLN a a,
654− FromListsN b b
654+ FLN b b
655 )
655 )
656 => Relation a b -> Bool
656 => Relation a b -> Bool
657injection r = function r && representation r
657injection r = function r && representation r
658
658
659-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@
659-- | A 'Relation' @r@ is an 'bijection' 'iff' @'injection' r && 'surjection' r@
660bijection ::
660bijection ::
661− ( CountableDimensionsN a b,
661+ ( CountableDimsN a b,
662− FromListsN b b,
662+ FLN b b,
663− FromListsN a a
663+ FLN a a
664 ) => Relation a b -> Bool
664 ) => Relation a b -> Bool
665bijection r = injection r && surjection r
665bijection r = injection r && surjection r
666
666
667-- Properties of relations
667-- Properties of relations
668
668
669−-- | A 'Relation' @r@ is 'reflexive' 'iff' @'identity' ``sse`` r@
669+-- | A 'Relation' @r@ is 'reflexive' 'iff' @'id' ``sse`` r@
670−reflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool
670+reflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
671−reflexive r = identity <= r
671+reflexive r = id <= r
672
672
673−-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'identity'@
673+-- | A 'Relation' @r@ is 'coreflexive' 'iff' @r ``sse`` 'id'@
674−coreflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool
674+coreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
675−coreflexive r = r <= identity
675+coreflexive r = r <= id
676
676
677−-- | A 'Relation' @r@ is 'transitive' 'iff' @(r ``comp`` r) ``sse`` r@
677+-- | A 'Relation' @r@ is 'transitive' 'iff' @(r `.` r) ``sse`` r@
678transitive :: Relation a a -> Bool
678transitive :: Relation a a -> Bool
679−transitive r = (r `comp` r) `sse` r
679+transitive r = (r . r) `sse` r
680
680
681-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@
681-- | A 'Relation' @r@ is 'symmetric' 'iff' @r == 'conv' r@
682symmetric :: Relation a a -> Bool
682symmetric :: Relation a a -> Bool
683symmetric r = r == conv r
683symmetric r = r == conv r
684
684
685−-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'identity'@
685+-- | A 'Relation' @r@ is anti-symmetric 'iff' @(r ``intersection`` 'conv' r) ``sse`` 'id'@
686−antiSymmetric :: (CountableN a, FromListsN a a) => Relation a a -> Bool
686+antiSymmetric :: (CountableN a, FLN a a) => Relation a a -> Bool
687−antiSymmetric r = (r `intersection` conv r) `sse` identity
687+antiSymmetric r = (r `intersection` conv r) `sse` id
688
688
689−-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'identity') == 'zeros'@
689+-- | A 'Relation' @r@ is 'irreflexive' 'iff' @(r ``intersection`` 'id') == 'zeros'@
690−irreflexive :: (CountableN a, FromListsN a a) => Relation a a -> Bool
690+irreflexive :: (CountableN a, FLN a a) => Relation a a -> Bool
691−irreflexive r = (r `intersection` identity) == zeros
691+irreflexive r = (r `intersection` id) == zeros
692
692
693-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@
693-- | A 'Relation' @r@ is 'connected' 'iff' @(r ``union`` 'conv' r) == 'ones'@
694−connected :: (CountableN a, FromListsN a a) => Relation a a -> Bool
694+connected :: (CountableN a, FLN a a) => Relation a a -> Bool
695connected r = (r `union` conv r) == ones
695connected r = (r `union` conv r) == ones
696
696
697-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@
697-- | A 'Relation' @r@ is a 'preorder' 'iff' @'reflexive' r && 'transitive' r@
698−preorder :: (CountableN a, FromListsN a a) => Relation a a -> Bool
698+preorder :: (CountableN a, FLN a a) => Relation a a -> Bool
699preorder r = reflexive r && transitive r
699preorder r = reflexive r && transitive r
700
700
701-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@
701-- | A 'Relation' @r@ is a partial-order 'iff' @'antiSymmetric' r && 'preorder' r@
702−partialOrder :: (CountableN a, FromListsN a a) => Relation a a -> Bool
702+partialOrder :: (CountableN a, FLN a a) => Relation a a -> Bool
703partialOrder r = antiSymmetric r && preorder r
703partialOrder r = antiSymmetric r && preorder r
704
704
705-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@
705-- | A 'Relation' @r@ is a linear-order 'iff' @'connected' r && 'partialOrder' r@
706−linearOrder :: (CountableN a, FromListsN a a) => Relation a a -> Bool
706+linearOrder :: (CountableN a, FLN a a) => Relation a a -> Bool
707linearOrder r = connected r && partialOrder r
707linearOrder r = connected r && partialOrder r
708
708
709-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@
709-- | A 'Relation' @r@ is an 'equivalence' 'iff' @'symmetric' r && 'preorder' r@
710−equivalence :: (CountableN a, FromListsN a a) => Relation a a -> Bool
710+equivalence :: (CountableN a, FLN a a) => Relation a a -> Bool
711equivalence r = symmetric r && preorder r
711equivalence r = symmetric r && preorder r
712
712
713-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@
713-- | A 'Relation' @r@ is a partial-equivalence 'iff' @'partialOrder' r && 'equivalence' r@
714−partialEquivalence :: (CountableN a, FromListsN a a) => Relation a a -> Bool
714+partialEquivalence :: (CountableN a, FLN a a) => Relation a a -> Bool
715partialEquivalence r = partialOrder r && equivalence r
715partialEquivalence r = partialOrder r && equivalence r
716
716
717-- | A 'Relation' @r@ is 'difunctional' or regular wherever
717-- | A 'Relation' @r@ is 'difunctional' or regular wherever
718−-- @r ``comp`` 'conv' r ``comp`` r == r@
718+-- @r `.` 'conv' r `.` r == r@
719difunctional :: Relation a b -> Bool
719difunctional :: Relation a b -> Bool
720−difunctional r = r `comp` conv r `comp` r == r
720+difunctional r = r . conv r . r == r
721
721
722-- Relational pairing
722-- Relational pairing
723
723
@@ -726,61 +726,61 @@ difunctional r = r `comp` conv r `comp` r == r
726-- NOTE: That this is not a true categorical product, see for instance:
726-- NOTE: That this is not a true categorical product, see for instance:
727--
727--
728-- @
728-- @
729−-- | 'p1' ``comp`` 'splitR' a b ``sse`` a
729+-- | 'fstR' `.` 'splitR' a b ``sse`` a
730-- 'splitR' a b <=> |
730-- 'splitR' a b <=> |
731−-- | 'p2' ``comp`` 'splitR' a b ``sse`` b
731+-- | 'sndR' `.` 'splitR' a b ``sse`` b
732-- @
732-- @
733--
733--
734-- __Emphasis__ on the 'sse'.
734-- __Emphasis__ on the 'sse'.
735--
735--
736-- @
736-- @
737−-- 'splitR' r s ``comp`` f == 'splitR' (r ``comp`` f) (s ``comp`` f)
737+-- 'splitR' r s `.` f == 'splitR' (r `.` f) (s `.` f)
738−-- (R '><' S) ``comp`` 'splitR' p q == 'splitR' (r ``comp`` p) (s ``comp`` q)
738+-- (R '><' S) `.` 'splitR' p q == 'splitR' (r `.` p) (s `.` q)
739−-- 'conv' ('splitR' r s) ``comp`` 'splitR' x y == ('conv' r ``comp`` x) ``intersection`` ('conv' s ``comp`` y)
739+-- 'conv' ('splitR' r s) `.` 'splitR' x y == ('conv' r `.` x) ``intersection`` ('conv' s `.` y)
740-- @
740-- @
741--
741--
742-- @
742-- @
743-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)
743-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)
744-- @
744-- @
745splitR ::
745splitR ::
746− ( CountableDimensionsN a b,
746+ ( CountableDimsN a b,
747 CountableN (a, b),
747 CountableN (a, b),
748− FromListsN (a, b) a,
748+ FLN (a, b) a,
749− FromListsN (a, b) b,
749+ FLN (a, b) b,
750 TrivialP a b
750 TrivialP a b
751 )
751 )
752 => Relation c a -> Relation c b -> Relation c (a, b)
752 => Relation c a -> Relation c b -> Relation c (a, b)
753−splitR (R f) (R g) = R (I.khatri f g)
753+splitR (R f) (R g) = R (I.kr f g)
754
754
755-- | Relational pairing first component projection
755-- | Relational pairing first component projection
756--
756--
757-- @
757-- @
758−-- 'p1' ``comp`` 'splitR' r s ``sse`` r
758+-- 'fstR' `.` 'splitR' r s ``sse`` r
759-- @
759-- @
760−p1 ::
760+fstR ::
761 forall a b .
761 forall a b .
762− ( CountableDimensionsN a b,
762+ ( CountableDimsN a b,
763 CountableN (a, b),
763 CountableN (a, b),
764− FromListsN (a, b) a,
764+ FLN (a, b) a,
765 TrivialP a b
765 TrivialP a b
766 )
766 )
767 => Relation (a, b) a
767 => Relation (a, b) a
768−p1 = R (I.kp1 @Boolean @(I.Normalize a) @(I.Normalize b))
768+fstR = R (I.fstM @Boolean @(I.Normalize a) @(I.Normalize b))
769
769
770-- | Relational pairing second component projection
770-- | Relational pairing second component projection
771--
771--
772-- @
772-- @
773−-- 'p2' ``comp`` 'splitR' r s ``sse`` s
773+-- 'sndR' `.` 'splitR' r s ``sse`` s
774-- @
774-- @
775−p2 ::
775+sndR ::
776 forall a b .
776 forall a b .
777− ( CountableDimensionsN a b,
777+ ( CountableDimsN a b,
778 CountableN (a, b),
778 CountableN (a, b),
779− FromListsN (a, b) b,
779+ FLN (a, b) b,
780 TrivialP a b
780 TrivialP a b
781 )
781 )
782 => Relation (a, b) b
782 => Relation (a, b) b
783−p2 = R (I.kp2 @Boolean @(I.Normalize a) @(I.Normalize b))
783+sndR = R (I.sndM @Boolean @(I.Normalize a) @(I.Normalize b))
784
784
785-- Relational pairing functor
785-- Relational pairing functor
786
786
@@ -788,17 +788,17 @@ infixl 4 ><
788-- | Relational pairing functor
788-- | Relational pairing functor
789--
789--
790-- @
790-- @
791−-- r '><' s == 'splitR' (r ``comp`` p1) (s ``comp`` p2)
791+-- r '><' s == 'splitR' (r `.` fstR) (s `.` sndR)
792−-- (r '><' s) ``comp`` (p '><' q) == (r ``comp`` p) '><' (s ``comp`` q)
792+-- (r '><' s) `.` (p '><' q) == (r `.` p) '><' (s `.` q)
793-- @
793-- @
794(><) ::
794(><) ::
795− ( CountableDimensionsN a b,
795+ ( CountableDimsN a b,
796− CountableDimensionsN c d,
796+ CountableDimsN c d,
797− CountableDimensionsN (a, c) (b, d),
797+ CountableDimsN (a, c) (b, d),
798− FromListsN (a, c) a,
798+ FLN (a, c) a,
799− FromListsN (a, c) c,
799+ FLN (a, c) c,
800− FromListsN (b, d) b,
800+ FLN (b, d) b,
801− FromListsN (b, d) d,
801+ FLN (b, d) d,
802 TrivialP a c,
802 TrivialP a c,
803 TrivialP b d
803 TrivialP b d
804 )
804 )
@@ -810,33 +810,31 @@ infixl 4 ><
810-- | Relational coproduct.
810-- | Relational coproduct.
811--
811--
812-- @
812-- @
813−-- | 'eitherR' a b ``comp`` 'i1' == a
813+-- | 'eitherR' a b `.` 'i1' == a
814-- 'eitherR' a b <=> |
814-- 'eitherR' a b <=> |
815−-- | 'eitherR' a b ``comp`` 'i2' == b
815+-- | 'eitherR' a b `.` 'i2' == b
816-- @
816-- @
817--
817--
818-- @
818-- @
819−-- 'eitherR' r s ``comp`` 'conv' ('eitherR' t u) == (r ``comp`` 'conv' t) ``union`` (s ``comp`` 'conv' u)
819+-- 'eitherR' r s `.` 'conv' ('eitherR' t u) == (r `.` 'conv' t) ``union`` (s `.` 'conv' u)
820-- @
820-- @
821--
821--
822-- @
822-- @
823-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)
823-- 'eitherR' ('splitR' r s) ('splitR' t v) == 'splitR' ('eitherR' r t) ('eitherR' s v)
824-- @
824-- @
825−eitherR :: (TrivialE a b)
826− => Relation a c -> Relation b c -> Relation (Either a b) c
825+eitherR :: Relation a c -> Relation b c -> Relation (Either a b) c
827−eitherR = junc
826+eitherR = join
828
827
829-- | Relational coproduct first component injection
828-- | Relational coproduct first component injection
830--
829--
831-- @
830-- @
832−-- 'img' 'i1' ``union`` 'img' 'i2' == 'identity'
831+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'
833−-- 'i1' ``comp`` 'i2' = 'zeros'
832+-- 'i1' `.` 'i2' = 'zeros'
834-- @
833-- @
835i1 ::
834i1 ::
836− ( CountableDimensionsN a b,
835+ ( CountableDimsN a b,
837− FromListsN b a,
836+ FLN b a,
838− FromListsN a a,
837+ FLN a a
839− TrivialE a b
840 )
838 )
841 => Relation a (Either a b)
839 => Relation a (Either a b)
842i1 = R I.i1
840i1 = R I.i1
@@ -844,14 +842,13 @@ i1 = R I.i1
844-- | Relational coproduct second component injection
842-- | Relational coproduct second component injection
845--
843--
846-- @
844-- @
847−-- 'img' 'i1' ``union`` 'img' 'i2' == 'identity'
845+-- 'img' 'i1' ``union`` 'img' 'i2' == 'id'
848−-- 'i1' ``comp`` 'i2' = 'zeros'
846+-- 'i1' `.` 'i2' = 'zeros'
849-- @
847-- @
850i2 ::
848i2 ::
851− ( CountableDimensionsN a b,
849+ ( CountableDimsN a b,
852− FromListsN a b,
850+ FLN a b,
853− FromListsN b b,
851+ FLN b b
854− TrivialE a b
855 )
852 )
856 => Relation b (Either a b)
853 => Relation b (Either a b)
857i2 = R I.i2
854i2 = R I.i2
@@ -861,16 +858,14 @@ infixl 5 -|-
861-- | Relational coproduct functor.
858-- | Relational coproduct functor.
862--
859--
863-- @
860-- @
864−-- r '-|-' s == 'eitherR' ('i1' ``comp`` r) ('i2' ``comp`` s)
861+-- r '-|-' s == 'eitherR' ('i1' `.` r) ('i2' `.` s)
865-- @
862-- @
866(-|-) ::
863(-|-) ::
867− ( CountableDimensionsN b d,
864+ ( CountableDimsN b d,
868− FromListsN b b,
865+ FLN b b,
869− FromListsN d b,
866+ FLN d b,
870− FromListsN b d,
867+ FLN b d,
871− FromListsN d d,
868+ FLN d d
872− TrivialE a c,
873− TrivialE b d
874 )
869 )
875 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)
870 => Relation a b -> Relation c d -> Relation (Either a c) (Either b d)
876(-|-) (R a) (R b) = R ((I.-|-) a b)
871(-|-) (R a) (R b) = R ((I.-|-) a b)
@@ -883,20 +878,20 @@ infixl 5 -|-
883-- 'trans'/'untrans';
878-- 'trans'/'untrans';
884-- more-over, where each particular attribute is placed (input/output) is irrelevant.
879-- more-over, where each particular attribute is placed (input/output) is irrelevant.
885trans ::
880trans ::
886− ( CountableDimensionsN a b,
881+ ( CountableDimsN a b,
887 CountableN c,
882 CountableN c,
888− CountableDimensionsN (a, b) (c, b),
883+ CountableDimsN (a, b) (c, b),
889− FromListsN (c, b) c,
884+ FLN (c, b) c,
890− FromListsN (c, b) b,
885+ FLN (c, b) b,
891− FromListsN (a, b) a,
886+ FLN (a, b) a,
892− FromListsN (a, b) b,
887+ FLN (a, b) b,
893 Trivial (a, b),
888 Trivial (a, b),
894 Trivial (c, b),
889 Trivial (c, b),
895− TrivialP2 a b,
890+ TrivialP a b,
896− TrivialP2 c b
891+ TrivialP c b
897 )
892 )
898 => Relation (a, b) c -> Relation a (c, b)
893 => Relation (a, b) c -> Relation a (c, b)
899−trans r = splitR r p2 `comp` conv p1
894+trans r = splitR r sndR . conv fstR
900
895
901-- | Relational 'untrans'
896-- | Relational 'untrans'
902--
897--
@@ -904,109 +899,110 @@ trans r = splitR r p2 `comp` conv p1
904-- 'trans'/'untrans';
899-- 'trans'/'untrans';
905-- more-over, where each particular attribute is placed (input/output) is irrelevant.
900-- more-over, where each particular attribute is placed (input/output) is irrelevant.
906untrans ::
901untrans ::
907− ( CountableDimensionsN a b,
902+ ( CountableDimsN a b,
908 CountableN c,
903 CountableN c,
909− CountableDimensionsN (a, b) (c, b),
904+ CountableDimsN (a, b) (c, b),
910− FromListsN (c, b) c,
905+ FLN (c, b) c,
911− FromListsN (c, b) b,
906+ FLN (c, b) b,
912− FromListsN (a, b) b,
907+ FLN (a, b) b,
913− FromListsN (a, b) a,
908+ FLN (a, b) a,
914 Trivial (a, b),
909 Trivial (a, b),
915 Trivial (c, b),
910 Trivial (c, b),
916− TrivialP2 a b,
911+ TrivialP a b,
917− TrivialP2 c b
912+ TrivialP c b
918 )
913 )
919 => Relation a (c, b) -> Relation (a, b) c
914 => Relation a (c, b) -> Relation (a, b) c
920−untrans s = p1 `comp` conv (splitR (conv s) p2)
915+untrans s = fstR . conv (splitR (conv s) sndR)
921
916
922-- | Transforms predicate @p@ into a correflexive relation.
917-- | Transforms predicate @p@ into a correflexive relation.
923--
918--
924-- @
919-- @
925−-- 'predR' ('const' True) == 'identity'
920+-- 'predR' ('const' True) == 'id'
926-- 'predR' ('const' False) == 'zeros'
921-- 'predR' ('const' False) == 'zeros'
927-- @
922-- @
928--
923--
929-- @
924-- @
930−-- 'predR' q ``comp`` 'predR' p == 'predR' q ``union`` 'predR' p
925+-- 'predR' q `.` 'predR' p == 'predR' q ``union`` 'predR' p
931-- @
926-- @
932predR ::
927predR ::
933 ( Bounded a,
928 ( Bounded a,
934 Enum a,
929 Enum a,
930+ Countable a,
935 CountableN a,
931 CountableN a,
936− FromListsN a a,
932+ FLN a a,
937− FromListsN Bool a
933+ FLN Bool a
938 )
934 )
939 => Relation a Bool -> Relation a a
935 => Relation a Bool -> Relation a a
940−predR p = identity `intersection` divisionF (fromF' (const True)) p
936+predR p = id `intersection` divisionF (fromF (const True)) p
941
937
942-- | Equalizes functions @f@ and @g@.
938-- | Equalizes functions @f@ and @g@.
943-- That is, @'equalizer' f g@ is the largest coreflexive
939-- That is, @'equalizer' f g@ is the largest coreflexive
944-- that restricts @g@ so that @f@ and @g@ yield the same outputs.
940-- that restricts @g@ so that @f@ and @g@ yield the same outputs.
945--
941--
946-- @
942-- @
947−-- 'equalizer' r r == 'identity'
943+-- 'equalizer' r r == 'id'
948-- 'equalizer' ('point' True) ('point' False) == 'zeros'
944-- 'equalizer' ('point' True) ('point' False) == 'zeros'
949-- @
945-- @
950equalizer ::
946equalizer ::
951 ( CountableN a,
947 ( CountableN a,
952− FromListsN a a
948+ FLN a a
953 )
949 )
954 => Relation a b -> Relation a b -> Relation a a
950 => Relation a b -> Relation a b -> Relation a a
955−equalizer f g = identity `intersection` divisionF f g
951+equalizer f g = id `intersection` divisionF f g
956
952
957-- | Relational conditional guard.
953-- | Relational conditional guard.
958--
954--
959-- @
955-- @
960−-- 'guard' p = 'i2' ``overriddenBy`` 'i1' ``comp`` 'predR' p
956+-- 'guard' p = 'i2' ``overriddenBy`` 'i1' `.` 'predR' p
961-- @
957-- @
962−guard ::
958+guard ::
963 ( Bounded b,
959 ( Bounded b,
964 Enum b,
960 Enum b,
965 CountableN b,
961 CountableN b,
966− FromListsN b b,
962+ Countable b,
963+ FLN b b,
967− FromListsN Bool b,
964+ FLN Bool b
968− TrivialE b b
969 ) => Relation b Bool -> Relation b (Either b b)
965 ) => Relation b Bool -> Relation b (Either b b)
970guard p = conv (eitherR (predR p) (predR (negate p)))
966guard p = conv (eitherR (predR p) (predR (negate p)))
971
967
972-- | Relational McCarthy's conditional.
968-- | Relational McCarthy's conditional.
973−cond ::
969+cond ::
974 ( Bounded b,
970 ( Bounded b,
975 Enum b,
971 Enum b,
972+ Countable b,
976 CountableN b,
973 CountableN b,
977− FromListsN b b,
974+ FLN b b,
978− FromListsN Bool b,
975+ FLN Bool b
979− TrivialE b b
980− )
976+ )
981 => Relation b Bool -> Relation b c -> Relation b c -> Relation b c
977 => Relation b Bool -> Relation b c -> Relation b c -> Relation b c
982−cond p r s = eitherR r s `comp` guard p
978+cond p r s = eitherR r s . guard p
983
979
984-- | Relational domain.
980-- | Relational domain.
985--
981--
986-- For injective relations, 'domain' and 'ker'nel coincide,
982-- For injective relations, 'domain' and 'ker'nel coincide,
987−-- since @'ker' r ``sse`` 'identity'@ in such situations.
983+-- since @'ker' r ``sse`` 'id'@ in such situations.
988domain ::
984domain ::
989 ( CountableN a,
985 ( CountableN a,
990− FromListsN a a
986+ FLN a a
991 ) => Relation a b -> Relation a a
987 ) => Relation a b -> Relation a a
992−domain r = ker r `intersection` identity
988+domain r = ker r `intersection` id
993
989
994-- | Relational range.
990-- | Relational range.
995--
991--
996-- For functions, 'range' and 'img' (image) coincide,
992-- For functions, 'range' and 'img' (image) coincide,
997−-- since @'img' f ``sse`` 'identity'@ for any @f@.
993+-- since @'img' f ``sse`` id@ for any @f@.
998range ::
994range ::
999 ( CountableN b,
995 ( CountableN b,
1000− FromListsN b b
996+ FLN b b
1001 ) => Relation a b -> Relation b b
997 ) => Relation a b -> Relation b b
1002−range r = img r `intersection` identity
998+range r = img r `intersection` id
1003
999
1004-- Relation pretty print
1000-- Relation pretty print
1005
1001
1006-- | Relation pretty printing
1002-- | Relation pretty printing
1007−pretty :: (CountableN a) => Relation a b -> String
1003+pretty :: (CountableDimsN a b) => Relation a b -> String
1008pretty (R a) = I.pretty a
1004pretty (R a) = I.pretty a
1009
1005
1010-- | Relation pretty printing
1006-- | Relation pretty printing
1011−prettyPrint :: (CountableN a) => Relation a b -> IO ()
1007+prettyPrint :: (CountableDimsN a b) => Relation a b -> IO ()
1012prettyPrint (R a) = I.prettyPrint a
1008prettyPrint (R a) = I.prettyPrint a
src/LAoP/Utils.hs+5−2
@@ -95,8 +95,11 @@ module LAoP.Utils
95 coerceNat2,
95 coerceNat2,
96 coerceNat3,
96 coerceNat3,
97
97
98− -- * Powerset data type
98+ -- * List data type
99− List (..)
99+ List (..),
100+
101+ -- * Category type-class
102+ Category(..)
100 )
103 )
101where
104where
102
105
src/LAoP/Utils/Internal.hs+33−75
@@ -1,82 +1,13 @@
1+{-# LANGUAGE TypeFamilies #-}
2+{-# LANGUAGE DeriveGeneric #-}
1{-# LANGUAGE TypeApplications #-}
3{-# LANGUAGE TypeApplications #-}
2{-# LANGUAGE DataKinds #-}
4{-# LANGUAGE DataKinds #-}
3{-# LANGUAGE FlexibleInstances #-}
5{-# LANGUAGE FlexibleInstances #-}
4{-# LANGUAGE GeneralizedNewtypeDeriving #-}
6{-# LANGUAGE GeneralizedNewtypeDeriving #-}
5−{-# LANGUAGE KindSignatures #-}
6{-# LANGUAGE ScopedTypeVariables #-}
7{-# LANGUAGE ScopedTypeVariables #-}
7
8
8module LAoP.Utils.Internal
9module LAoP.Utils.Internal
9− ( -- | This is an internal module and it is not meant to be imported.
10+ (
10− --
11− -- Utility module that provides the 'Natural' data type.
12− -- The semantic associated with this data type is that
13− -- it's meant to be a restricted 'Int' value. For example
14− -- the type @Natural 6@ can only be instanciated with @Nat n@
15− -- where @0 <= n <= 6@. Why, You might ask, because with normal
16− -- 'Int's it is not possible to have a decent @Enum (Int, Int)@
17− -- instance. See the following probabilistic programming model as and
18− -- example:
19− --
20− -- We want to calculate the probability of the sum of two dice throws.
21− -- To do this we start by defining the sample space:
22− --
23− -- @
24− -- type SampleSpace = Int -- We think 'Int' are enough
25− --
26− -- die :: Dist Int 6
27− -- die = unifrom [1..6]
28− --
29− -- -- Promote 'Int' addition to a matrix
30− -- addM = fromF (uncurry (+)) -- Impossible
31− -- @
32− --
33− -- The last line is impossible because @(Int, Int)@ does not have
34− -- a good 'Enum' instance: @[(0, 1), (0, 2), .. (0, maxBound), (1, 0),
35− -- ..]@. And we'd like the addition matrix to be of 36 columns by 12
36− -- rows but limited to integers up to @6@!
37− --
38− -- One way to solve this issue is by defining and auxilary data type to
39− -- represent the sample space:
40− --
41− -- @
42− -- data SampleSpace = S1 | S2 | S3 | S4 | S5 | S6
43− -- deriving (Show, Eq, Enum, Bounded) -- Enum and Bounded are
44− -- important
45− -- @
46− --
47− -- And write the sample space addition function:
48− --
49− -- @
50− -- ssAdd :: SampleSpace -> SampleSpace -> Int
51− -- ssAdd a b = (fromEnum a + 1) + (fromEnum b + 1)
52− -- @
53− --
54− -- And then promote that function to matrix and everything is alright:
55− --
56− -- @
57− -- ssAddM = fromF' (uncurry ssAdd)
58− --
59− -- dieSumProb = ssAddM `comp` (khatri die die)
60− -- @
61− --
62− -- This is a nice solution for small sample spaces. But for larger ones
63− -- it is not feasible to write a data type with hundreds of constructors
64− -- and then write manipulation functions that need to deal with them.
65− -- To mitigate this limitation the 'Natural' type comes a long way and
66− -- allows one to model the sample in an easier way. See for instance:
67− --
68− -- @
69− -- ssAdd :: Natural 6 -> Natural 6 -> Natural 12
70− -- ssAdd = coerceNat (+)
71− --
72− -- ssAddM = fromF' (uncurry sumSS)
73− --
74− -- die :: Dist (Natural 6) 6
75− -- die = uniform [nat @6 1 .. nat 6]
76− --
77− -- dieSumProb = ssAddM `comp` (khatri die die)
78− -- @
79− --
80 -- * 'Natural' data type
11 -- * 'Natural' data type
81 Natural(..),
12 Natural(..),
82 nat,
13 nat,
@@ -87,8 +18,11 @@ module LAoP.Utils.Internal
87 coerceNat2,
18 coerceNat2,
88 coerceNat3,
19 coerceNat3,
89
20
90− -- * Powerset data type
21+ -- * 'List' data type
91− List (..)
22+ List (..),
23+
24+ -- * Category type class
25+ Category(..)
92 )
26 )
93where
27where
94
28
@@ -98,12 +32,16 @@ import Data.List
98import Data.Maybe
32import Data.Maybe
99import GHC.TypeLits
33import GHC.TypeLits
100import Control.DeepSeq
34import Control.DeepSeq
35+import GHC.Generics
36+import Data.Kind
37+import Prelude hiding (id, (.))
38+import qualified Prelude
101
39
102-- | Wrapper around 'Int's that have a restrictive semantic associated.
40-- | Wrapper around 'Int's that have a restrictive semantic associated.
103-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'
41-- A value of type @'Natural' n m@ can only be instanciated with some 'Int'
104-- @i@ that's @n <= i <= m@.
42-- @i@ that's @n <= i <= m@.
105newtype Natural (start :: Nat) (end :: Nat) = Nat Int
43newtype Natural (start :: Nat) (end :: Nat) = Nat Int
106− deriving (Show, Read, Eq, Ord, NFData)
44+ deriving (Show, Read, Eq, Ord, NFData, Generic)
107
45
108-- | Throws a runtime error if any of the operations overflows or
46-- | Throws a runtime error if any of the operations overflows or
109-- underflows.
47-- underflows.
@@ -236,3 +174,23 @@ instance
236 fromEnum (L x) =
174 fromEnum (L x) =
237 let as = [minBound .. maxBound]
175 let as = [minBound .. maxBound]
238 in fromMaybe (error "Does not exist") $ elemIndex x (powerset as)
176 in fromMaybe (error "Does not exist") $ elemIndex x (powerset as)
177+
178+-- | Constrained category instance
179+class Category k where
180+ type Object k o :: Constraint
181+ type Object k o = ()
182+ id :: Object k a => k a a
183+ (.) :: k b c -> k a b -> k a c
184+
185+{-# RULES
186+"identity/left" forall p .
187+ id . p = p
188+"identity/right" forall p .
189+ p . id = p
190+"association" forall p q r .
191+ (p . q) . r = p . (q . r)
192+ #-}
193+
194+instance Category (->) where
195+ id = Prelude.id
196+ (.) = (Prelude..)
stack-8.8.3.yamlnew file+1
@@ -0,0 +1 @@
1+resolver: lts-15.5
stack.yaml+1−66
@@ -1,66 +1 @@
1−# This file was automatically generated by 'stack init'
2−#
3−# Some commonly used options have been documented as comments in this file.
4−# For advanced use and comprehensive documentation of the format, please see:
5−# https://docs.haskellstack.org/en/stable/yaml_configuration/
6−
7−# Resolver to choose a 'specific' stackage snapshot or a compiler version.
8−# A snapshot resolver dictates the compiler version and the set of packages
9−# to be used for project dependencies. For example:
10−#
11−# resolver: lts-3.5
12−# resolver: nightly-2015-09-21
13−# resolver: ghc-7.10.2
14−#
15−# The location of a snapshot can be provided as a file or url. Stack assumes
16−# a snapshot provided as a file might change, whereas a url resource does not.
17−#
18−# resolver: ./custom-snapshot.yaml
19−# resolver: https://example.com/snapshots/2018-01-01.yaml
20−resolver: lts-14.18
1+resolver: lts-14.3
21−
22−# User packages to be built.
23−# Various formats can be used as shown in the example below.
24−#
25−# packages:
26−# - some-directory
27−# - https://example.com/foo/bar/baz-0.0.2.tar.gz
28−# subdirs:
29−# - auto-update
30−# - wai
31−packages:
32−- .
33−# Dependency packages to be pulled from upstream that are not in the resolver.
34−# These entries can reference officially published versions as well as
35−# forks / in-progress versions pinned to a git hash. For example:
36−#
37−# extra-deps:
38−# - acme-missiles-0.3
39−# - git: https://github.com/commercialhaskell/stack.git
40−# commit: e7b331f14bcffb8367cd58fbfc8b40ec7642100a
41−#
42−# extra-deps: []
43−
44−# Override default flag values for local packages and extra-deps
45−# flags: {}
46−
47−# Extra package databases containing global packages
48−# extra-package-dbs: []
49−
50−# Control whether we use the GHC we find on the path
51−# system-ghc: true
52−#
53−# Require a specific version of stack, using version ranges
54−# require-stack-version: -any # Default
55−# require-stack-version: ">=2.1"
56−#
57−# Override the architecture used by stack, especially useful on Windows
58−# arch: i386
59−# arch: x86_64
60−#
61−# Extra directories used by stack for building
62−# extra-include-dirs: [/path/to/dir]
63−# extra-lib-dirs: [/path/to/dir]
64−#
65−# Allow a newer minor version of GHC than the snapshot specifies
66−# compiler-check: newer-minor
stack.yaml.lock+4−4
@@ -6,7 +6,7 @@
6packages: []
6packages: []
7snapshots:
7snapshots:
8- completed:
8- completed:
9− size: 524789
9+ size: 523878
10− url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/18.yaml
10+ url: https://raw.githubusercontent.com/commercialhaskell/stackage-snapshots/master/lts/14/3.yaml
11− sha256: 646be71223e08234131c6989912e6011e01b9767bc447b6d466a35e14360bdf2
11+ sha256: 470c46c27746a48c7c50f829efc0cf00112787a7804ee4ac7a27754658f6d92c
12− original: lts-14.18
12+ original: lts-14.3
test/Examples/Quantum.hsnew file+31
@@ -0,0 +1,31 @@
1+{-# LANGUAGE StandaloneDeriving #-}
2+module Examples.Quantum where
3+
4+import LAoP.Utils
5+import LAoP.Matrix.Type
6+import Data.Complex
7+import Prelude hiding (id, (.))
8+
9+deriving instance Ord a => Ord (Complex a)
10+
11+xor :: (Bool, Bool) -> Bool
12+xor (False, b) = b
13+xor (True, b) = not b
14+
15+ker :: Num e => Matrix e a b -> Matrix e a a
16+ker m = tr m . m
17+
18+cnot :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)
19+cnot = kr fstM (fromF xor)
20+
21+ccnot :: (Num e, Ord e) => Matrix e ((Bool, Bool), Bool) ((Bool, Bool), Bool)
22+ccnot = kr fstM (fromF f)
23+ where
24+ f = xor . tp (uncurry (&&)) id
25+ tp f g (a,b) = (f a, g b)
26+
27+had :: Matrix (Complex Double) Bool Bool
28+had = (1/sqrt 2) .| fromLists [[1, 1], [1, -1]]
29+
30+bell :: Matrix (Complex Double) (Bool, Bool) (Bool, Bool)
31+bell = cnot . (had >< iden)
app/Main.hs → test/Examples/Readme.hsrenamed+61−33
@@ -1,10 +1,13 @@
1+{-# LANGUAGE GADTs #-}
1{-# LANGUAGE DeriveGeneric #-}
2{-# LANGUAGE DeriveGeneric #-}
2{-# LANGUAGE FlexibleContexts #-}
3{-# LANGUAGE FlexibleContexts #-}
3−{-# LANGUAGE TypeOperators #-}
4{-# LANGUAGE TypeApplications #-}
4{-# LANGUAGE TypeApplications #-}
5{-# LANGUAGE DataKinds #-}
5{-# LANGUAGE DataKinds #-}
6
6
7+module Examples.Readme
8+ ( exec
9+ )
7−module Main where
10+ where
8
11
9import LAoP.Matrix.Type
12import LAoP.Matrix.Type
10import qualified LAoP.Relation as R
13import qualified LAoP.Relation as R
@@ -12,13 +15,12 @@ import LAoP.Utils
12import LAoP.Dist
15import LAoP.Dist
13import GHC.TypeLits
16import GHC.TypeLits
14import Data.Coerce
17import Data.Coerce
15−import GHC.Generics
18+import qualified GHC.Generics as G
16−import Control.Category hiding (id)
19+import Prelude hiding (id, (.))
17−import Prelude hiding ((.))
18
20
19-- Monty Hall Problem
21-- Monty Hall Problem
20data Outcome = Win | Lose
22data Outcome = Win | Lose
21− deriving (Bounded, Enum, Eq, Show, Generic)
23+ deriving (Bounded, Enum, Eq, Show, G.Generic)
22
24
23switch :: Outcome -> Outcome
25switch :: Outcome -> Outcome
24switch Win = Lose
26switch Win = Lose
@@ -28,7 +30,7 @@ firstChoice :: Matrix Double () Outcome
28firstChoice = col [1/3, 2/3]
30firstChoice = col [1/3, 2/3]
29
31
30secondChoice :: Matrix Double Outcome Outcome
32secondChoice :: Matrix Double Outcome Outcome
31−secondChoice = fromF' switch
33+secondChoice = fromF switch
32
34
33-- Dice sum
35-- Dice sum
34
36
@@ -37,7 +39,7 @@ type SS = Natural 1 6 -- Sample Space
37sumSS :: SS -> SS -> Natural 2 12
39sumSS :: SS -> SS -> Natural 2 12
38sumSS = coerceNat (+)
40sumSS = coerceNat (+)
39
41
40−sumSSM = fromF' (uncurry sumSS)
42+sumSSM = fromF (uncurry sumSS)
41
43
42condition :: (Int, Int) -> Int -> Int
44condition :: (Int, Int) -> Int -> Int
43condition (fst, snd) thrd = if fst == snd
45condition (fst, snd) thrd = if fst == snd
@@ -47,38 +49,64 @@ condition (fst, snd) thrd = if fst == snd
47conditionSS :: (SS, SS) -> SS -> Natural 3 18
49conditionSS :: (SS, SS) -> SS -> Natural 3 18
48conditionSS = coerceNat2 condition
50conditionSS = coerceNat2 condition
49
51
50−conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die
52+conditionalThrows = fromF (uncurry conditionSS) . kr (kr die die) die
51
53
52die :: Matrix Double () SS
54die :: Matrix Double () SS
53die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]
55die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]
54
56
55-- Sprinkler
57-- Sprinkler
58+data G = Dry | Wet
59+ deriving (Bounded, Enum, Eq, Show, G.Generic)
60+
61+data S = Off | On
62+ deriving (Bounded, Enum, Eq, Show, G.Generic)
56
63
64+data R = No | Yes
65+ deriving (Bounded, Enum, Eq, Show, G.Generic)
66+
57−rain :: Matrix Double () Bool
67+rain :: Matrix Double () R
58−rain = col [0.8, 0.2]
68+rain = matrixBuilder gen
69+ where
70+ gen (_, No) = 0.8
71+ gen (_, Yes) = 0.2
59
72
60−sprinkler :: Matrix Double Bool Bool
73+sprinkler :: Matrix Double R S
61−sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]
74+sprinkler = matrixBuilder gen
75+ where
76+ gen (No, Off) = 0.6
77+ gen (No, On) = 0.4
78+ gen (Yes, Off) = 0.99
79+ gen (Yes, On) = 0.01
80+
81+grass :: Matrix Double (S, R) G
82+grass = matrixBuilder gen
83+ where
84+ gen ((Off, No), Dry) = 1
85+ gen ((Off, Yes), Dry) = 0.2
86+ gen ((On, No), Dry) = 0.1
87+ gen ((On, Yes), Dry) = 0.01
88+ gen ((Off, No), Wet) = 0
89+ gen ((Off, Yes), Wet) = 0.8
90+ gen ((On, No), Wet) = 0.9
91+ gen ((On, Yes), Wet) = 0.99
62
92
63−grass :: Matrix Double (Bool, Bool) Bool
93+tag f = kr f id
64−grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]
65
94
66−state :: Matrix Double () (Bool, (Bool, Bool))
95+state g s r = tag g . tag s . r
67−state = khatri grass identity . khatri sprinkler identity . rain
68
96
69−grass_wet :: Matrix Double (Bool, (Bool, Bool)) One
97+grass_wet :: Matrix Double () (G, (S, R)) -> Matrix Double One One
70−grass_wet = row [0,1] . kp1
98+grass_wet state = row [0,1] . fstM . state
71
99
72−rainning :: Matrix Double (Bool, (Bool, Bool)) One
100+rainning :: Matrix Double (G, (S, R)) One
73−rainning = row [0,1] . kp2 . kp2
101+rainning = row [0,1] . sndM . sndM
74
102
75-- Alcuin Puzzle
103-- Alcuin Puzzle
76
104
77data Being = Farmer | Fox | Goose | Beans
105data Being = Farmer | Fox | Goose | Beans
78− deriving (Bounded, Enum, Eq, Show, Generic)
106+ deriving (Bounded, Enum, Eq, Show, G.Generic)
79
107
80data Bank = LeftB | RightB
108data Bank = LeftB | RightB
81− deriving (Bounded, Enum, Eq, Show, Generic)
109+ deriving (Bounded, Enum, Eq, Show, G.Generic)
82
110
83eats :: Being -> Being -> Bool
111eats :: Being -> Being -> Bool
84eats Fox Goose = True
112eats Fox Goose = True
@@ -93,21 +121,21 @@ cross LeftB = RightB
93cross RightB = LeftB
121cross RightB = LeftB
94
122
95crossR :: R.Relation Bank Bank
123crossR :: R.Relation Bank Bank
96−crossR = R.fromF' cross
124+crossR = R.fromF cross
97
125
98-- | Initial state, everyone in the left bank
126-- | Initial state, everyone in the left bank
99locationLeft :: Being -> Bank
127locationLeft :: Being -> Bank
100locationLeft _ = LeftB
128locationLeft _ = LeftB
101
129
102locationLeftR :: R.Relation Being Bank
130locationLeftR :: R.Relation Being Bank
103−locationLeftR = R.fromF' locationLeft
131+locationLeftR = R.fromF locationLeft
104
132
105-- | Initial state, everyone in the right bank
133-- | Initial state, everyone in the right bank
106locationRight :: Being -> Bank
134locationRight :: Being -> Bank
107locationRight _ = RightB
135locationRight _ = RightB
108
136
109locationRightR :: R.Relation Being Bank
137locationRightR :: R.Relation Being Bank
110−locationRightR = R.fromF' locationRight
138+locationRightR = R.fromF locationRight
111
139
112-- Properties
140-- Properties
113
141
@@ -124,7 +152,7 @@ inv :: R.Relation Being Bank -> Bool
124inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)
152inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)
125 where
153 where
126 farmer :: R.Relation Being Being
154 farmer :: R.Relation Being Being
127− farmer = R.fromF' (const Farmer)
155+ farmer = R.fromF (const Farmer)
128
156
129-- Arbitrary state
157-- Arbitrary state
130bankState :: Being -> Bank -> Bool
158bankState :: Being -> Bank -> Bool
@@ -138,19 +166,19 @@ bankStateR :: R.Relation Being Bank
138bankStateR = R.toRel bankState
166bankStateR = R.toRel bankState
139
167
140-- Main
168-- Main
141−main :: IO ()
169+exec :: IO ()
142−main = do
170+exec = do
143 putStrLn "Monty Hall Problem solution:"
171 putStrLn "Monty Hall Problem solution:"
144 prettyPrint (secondChoice . firstChoice)
172 prettyPrint (secondChoice . firstChoice)
145 putStrLn "\n Sum of dices probability:"
173 putStrLn "\n Sum of dices probability:"
146− prettyPrint (sumSSM `comp` khatri die die)
174+ prettyPrint (sumSSM . kr die die)
147 putStrLn "\n Conditional dice throw:"
175 putStrLn "\n Conditional dice throw:"
148 prettyPrint conditionalThrows
176 prettyPrint conditionalThrows
149 putStrLn "\n Checking that the last result is indeed a distribution: "
177 putStrLn "\n Checking that the last result is indeed a distribution: "
150− prettyPrint (bang . sumSSM . khatri die die)
178+ prettyPrint (bang . sumSSM . kr die die)
151 putStrLn "\n Probability of grass being wet:"
179 putStrLn "\n Probability of grass being wet:"
152− prettyPrint (grass_wet . state)
180+ prettyPrint (grass_wet (state grass sprinkler rain))
153 putStrLn "\n Probability of rain:"
181 putStrLn "\n Probability of rain:"
154− prettyPrint (rainning . state)
182+ prettyPrint (rainning . state grass sprinkler rain)
155 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
183 putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
156 print (inv bankStateR)
184 print (inv bankStateR)
test/Examples/Spreadsheets.hsnew file+47
@@ -0,0 +1,47 @@
1+{-# LANGUAGE DeriveGeneric #-}
2+
3+module Examples.Spreadsheets
4+
5+import LAoP.Utils
6+import LAoP.Matrix.Type
7+import GHC.Generics
8+import Prelude hiding (id, (.))
9+
10+data Student = Student1 | Student2 | Student3 | Student4
11+ deriving (Eq, Show, Enum, Bounded, Generic)
12+
13+data Question = Question1 | Question2 | Question3 | Question4
14+ deriving (Eq, Show, Enum, Bounded, Generic)
15+
16+data Results = Exam | Test | Final
17+ deriving (Eq, Show, Enum, Bounded, Generic)
18+
19+test :: Matrix Float One Results
20+test = point Test
21+
22+exam :: Matrix Float One Results
23+exam = point Exam
24+
25+final :: Matrix Float One Results
26+final = point Final
27+
28+m :: Matrix Float Question Student
29+m = fromLists [[95, 90, 100, 40], [20, 90, 90, 0], [30, 20, 95, 0], [50, 80, 100, 30]]
30+
31+w :: Matrix Float Question One
32+w = fromLists [[0.2, 0.3, 0.2, 0.3]]
33+
34+xls :: Matrix Float Question One
35+ -> Matrix Float Question Student
36+ -> Matrix Float Student One
37+ -> Matrix Float (Either Question Results) (Either One Student)
38+xls w m t = join (fork w m) (fork zeros r)
39+ where
40+ rExam = m . tr w
41+ rTest = tr t
42+ rFinal = rTest `maxPW` rExam
43+ r = (rExam . tr exam) + (rTest . tr test) + (rFinal . tr final)
44+
45+-- | Overloaded, point-wise 'max' function
46+maxPW :: Ord e => Matrix e a b -> Matrix e a b -> Matrix e a b
47+maxPW = zipWithM max
test/Spec.hs+7−1
@@ -1,2 +1,8 @@
1+module Main (main) where
2+
3+import qualified Examples.Readme as RD
4+
1main :: IO ()
5main :: IO ()
2−main = putStrLn "Test suite not yet implemented"
6+main = do
7+ putStrLn ("Testing if README works" :: String)
8+ RD.exec