# laop - Linear Algebra of Programming library

The LAoP discipline generalises relations and functions treating them as
Boolean matrices and in turn consider these as arrows.

**LAoP** is a library for algebraic (inductive) construction and manipulation of matrices
in Haskell. See my Msc Thesis for the
motivation behind the library, the underlying theory, and implementation details.

This module offers many of the combinators mentioned in the work of
Macedo (2012) and Oliveira (2012).

See the package in hackage here

A Functional Pearl has been written and can be regarded as the reference document for this library.

## Features

This library offers 4 main matrix programming modules:

- One in which matrices are typed with type level natural numbers;
- One in which matrices are typed with arbitrary generic data types;
- One in which matrices are regarded as relations, i.e. boolean matrices;

There's also an experimental module that uses matrices to represent probability
distributions.

The most interesting feature is that matrices are represented as an inductive data type.
Given this formulation, matrix algorithms can be expressed in a much more elegant and
calculational style than the traditional vector of vectors representation. This data type
guarantees that a matrix will always have valid dimensions. It can also express block matrix
computations naturally, which leads to total, efficient and statically typed manipulation
and transformation functions.

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.

Given this, this new matrix formulation compared to other libraries:

- Is more compositional and polymorphic and does not have partial matrix manipulation functions (hence less chances for usage errors);
- 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.

## Known issues

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).

## Notes

This is still a work in progress, any feedback is welcome!

## Example

```
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE DataKinds #-}
module Main where
import Matrix.Type
import Utils
import Dist
import GHC.TypeLits
import Data.Coerce
import GHC.Generics
import Control.Category hiding (id)
import Prelude hiding ((.))
-- Monty Hall Problem
data Outcome = Win | Lose
deriving (Bounded, Enum, Eq, Show, Generic)
switch :: Outcome -> Outcome
switch Win = Lose
switch Lose = Win
firstChoice :: Matrix Double () Outcome
firstChoice = col [1/3, 2/3]
secondChoice :: Matrix Double Outcome Outcome
secondChoice = fromF' switch
-- Dice sum
type SS = Natural 1 6 -- Sample Space
sumSS :: SS -> SS -> Natural 2 12
sumSS = coerceNat (+)
sumSSM = fromF' (uncurry sumSS)
condition :: (Int, Int) -> Int -> Int
condition (fst, snd) thrd = if fst == snd
then fst * 3
else fst + snd + thrd
conditionSS :: (SS, SS) -> SS -> Natural 3 18
conditionSS = coerceNat2 condition
conditionalThrows = fromF' (uncurry conditionSS) . khatri (khatri die die) die
die :: Matrix Double () SS
die = col $ map (const (1/6)) [nat @1 @6 1 .. nat 6]
-- Sprinkler
rain :: Matrix Double () Bool
rain = col [0.8, 0.2]
sprinkler :: Matrix Double Bool Bool
sprinkler = fromLists [[0.6, 0.99], [0.4, 0.01]]
grass :: Matrix Double (Bool, Bool) Bool
grass = fromLists [[1, 0.2, 0.1, 0.01], [0, 0.8, 0.9, 0.99]]
state :: Matrix Double () (Bool, (Bool, Bool))
state = khatri grass identity . khatri sprinkler identity . rain
grass_wet :: Matrix Double (Bool, (Bool, Bool)) One
grass_wet = row [0,1] . kp1
rainning :: Matrix Double (Bool, (Bool, Bool)) One
rainning = row [0,1] . kp2 . kp2
-- Alcuin Puzzle
data Being = Farmer | Fox | Goose | Beans
deriving (Bounded, Enum, Eq, Show, Generic)
data Bank = LeftB | RightB
deriving (Bounded, Enum, Eq, Show, Generic)
eats :: Being -> Being -> Bool
eats Fox Goose = True
eats Goose Beans = True
eats _ _ = False
eatsR :: R.Relation Being Being
eatsR = R.toRel eats
cross :: Bank -> Bank
cross LeftB = RightB
cross RightB = LeftB
crossR :: R.Relation Bank Bank
crossR = R.fromF' cross
-- | Initial state, everyone in the left bank
locationLeft :: Being -> Bank
locationLeft _ = LeftB
locationLeftR :: R.Relation Being Bank
locationLeftR = R.fromF' locationLeft
-- | Initial state, everyone in the right bank
locationRight :: Being -> Bank
locationRight _ = RightB
locationRightR :: R.Relation Being Bank
locationRightR = R.fromF' locationRight
-- Properties
-- Being at the same bank
sameBank :: R.Relation Being Bank -> R.Relation Being Being
sameBank = R.ker
-- Risk of somebody eating somebody else
canEat :: R.Relation Being Bank -> R.Relation Being Being
canEat w = sameBank w `R.intersection` eatsR
-- "Starvation" property.
inv :: R.Relation Being Bank -> Bool
inv w = (w `R.comp` canEat w) `R.sse` (w `R.comp` farmer)
where
farmer :: R.Relation Being Being
farmer = R.fromF' (const Farmer)
-- Arbitrary state
bankState :: Being -> Bank -> Bool
bankState Farmer LeftB = True
bankState Fox LeftB = True
bankState Goose RightB = True
bankState Beans RightB = True
bankState _ _ = False
bankStateR :: R.Relation Being Bank
bankStateR = R.toRel bankState
-- Main
main :: IO ()
main = do
putStrLn "Monty Hall Problem solution:"
prettyPrint (secondChoice . firstChoice)
putStrLn "\n Sum of dices probability:"
prettyPrint (sumSSM `comp` khatri die die)
putStrLn "\n Conditional dice throw:"
prettyPrint conditionalThrows
putStrLn "\n Checking that the last result is indeed a distribution: "
prettyPrint (bang . sumSSM . khatri die die)
putStrLn "\n Probability of grass being wet:"
prettyPrint (grass_wet . state)
putStrLn "\n Probability of rain:"
prettyPrint (rainning . state)
putStrLn "\n Is the arbitrary state a valid state? (Alcuin Puzzle)"
print (inv bankStateR)
```

```
Monty Hall Problem solution:
┌ ┐
│ 0.6666666666666666 │
│ 0.3333333333333333 │
└ ┘
Sum of dices probability:
┌ ┐
│ 2.7777777777777776e-2 │
│ 5.555555555555555e-2 │
│ 8.333333333333333e-2 │
│ 0.1111111111111111 │
│ 0.1388888888888889 │
│ 0.16666666666666666 │
│ 0.1388888888888889 │
│ 0.1111111111111111 │
│ 8.333333333333333e-2 │
│ 5.555555555555555e-2 │
│ 2.7777777777777776e-2 │
└ ┘
Conditional dice throw:
┌ ┐
│ 2.7777777777777776e-2 │
│ 9.259259259259259e-3 │
│ 1.8518518518518517e-2 │
│ 6.481481481481481e-2 │
│ 5.555555555555555e-2 │
│ 8.333333333333333e-2 │
│ 0.12962962962962962 │
│ 0.1111111111111111 │
│ 0.1111111111111111 │
│ 0.12962962962962962 │
│ 8.333333333333333e-2 │
│ 5.555555555555555e-2 │
│ 6.481481481481481e-2 │
│ 1.8518518518518517e-2 │
│ 9.259259259259259e-3 │
│ 2.7777777777777776e-2 │
└ ┘
Checking that the last result is indeed a distribution:
┌ ┐
│ 1.0 │
└ ┘
Probability of grass being wet:
┌ ┐
│ 0.4483800000000001 │
└ ┘
Probability of rain:
┌ ┐
│ 0.2 │
└ ┘
Is the arbitrary state a valid state? (Alcuin Puzzle)
False
```