Type arithmetic
From HaskellWiki
(Category:Mathematics) 
(Changed Monad.Reader link refs.) 

(8 intermediate revisions by 4 users not shown)  
Line 5:  Line 5:  
A simple example of typelevel computation are operations on [[Peano numbers]]: 
A simple example of typelevel computation are operations on [[Peano numbers]]: 

+  <haskell> 

data Zero 
data Zero 

Line 12:  Line 13:  
instance Add Zero b b 
instance Add Zero b b 

instance (Add a b ab) => Add (Succ a) b (Succ ab) 
instance (Add a b ab) => Add (Succ a) b (Succ ab) 

+  </haskell> 

Many other representations of numbers are possible, including binary and 
Many other representations of numbers are possible, including binary and 

−  balanced base three. Typelevel computation may also include type 
+  balanced base tree. Typelevel computation may also include type 
representations of boolean values, lists, trees and so on. It is closely 
representations of boolean values, lists, trees and so on. It is closely 

connected to theorem proving, via 
connected to theorem proving, via 

[http://en.wikipedia.org/wiki/CurryHoward the CurryHoward isomorphism]. 
[http://en.wikipedia.org/wiki/CurryHoward the CurryHoward isomorphism]. 

−  A [http://okmij.org/ftp/Haskell/numberparameterizedtypes.html decimal representation] was put forward by [http://okmij.org/ftp/ Oleg Kiselyov] in [http://www.haskell.org/tmrwiki/NumberParamTypes "NumberParamterized Types"] in the [http://www.haskell.org/tmrwiki/IssueFive fifth issue] of [http://www.haskell.org/tmrwiki/FrontPage The Monad Reader]. 
+  A [http://okmij.org/ftp/Haskell/numberparameterizedtypes.html decimal representation] was put forward by [http://okmij.org/ftp/ Oleg Kiselyov] in [http://www.haskell.org/haskellwiki/The_Monad.Reader/Issue5/Number_Param_Types "NumberParamterized Types"] in the [http://www.haskell.org/haskellwiki/The_Monad.Reader/Issue5 fifth issue] of [http://themonadreader.wordpress.com/ The Monad Reader]. 
+  There is an implementation in the {{HackagePackageid=typelevel}} package, but unfortunately the arithmetic is really slow, because in fact it simulates Peano arithmetic with decimal numbers. 

== Library support == 
== Library support == 

Line 33:  Line 35:  
the order of 10^15 (at least). It also contains a test suite to help 
the order of 10^15 (at least). It also contains a test suite to help 

validate the somewhat unintuitive algorithms. 
validate the somewhat unintuitive algorithms. 

+  
+  More libraries: 

+  
+  * {{HackagePackageid=typelevel}} Natural numbers in decimal representation using functional dependencies and Template Haskell. However arithmetic is performed in a unary way and thus it is quite slow. 

+  * {{HackagePackageid=typeleveltf}} Similar to the typelevel package (also in speed) but uses type families instead of functional dependencies and uses the same module names as the typelevel package. Thus module name clashes are warranted if you have to use both packages. 

+  * {{HackagePackageid=typelevelnaturalnumber}} and related packages. A collection of packages where the simplest one is even Haskell2010. 

+  * {{HackagePackageid=tfp}} Decimal representation, Type families, Template Haskell. 

+  * {{HackagePackageid=typical}} Binary numbers and functional dependencies. 

+  * {{HackagePackageid=typeunary}} Unary representation and type families. 

+  * {{HackagePackageid=numtype}}, {{HackagePackageid=numtypetf}} Unary representation and functional dependencies and type families, respectively. 

== More type hackery == 
== More type hackery == 

Line 68:  Line 80:  
[http://www.cs.chalmers.se/~hallgren/Papers/wm01.html Fun with Functional Dependencies]. 
[http://www.cs.chalmers.se/~hallgren/Papers/wm01.html Fun with Functional Dependencies]. 

+  <haskell> 

module Sort where 
module Sort where 

Line 176:  Line 189:  
listQSort :: QSort xs ys => xs > ys 
listQSort :: QSort xs ys => xs > ys 

listQSort = const undefined 
listQSort = const undefined 

+  </haskell> 

And we need to be able to run this somehow, in the typechecker. So fire up ghci: 
And we need to be able to run this somehow, in the typechecker. So fire up ghci: 

+  <haskell> 

> :t listQSort list1 
> :t listQSort list1 

Cons 
Cons 

(Succ Zero) 
(Succ Zero) 

(Cons (Succ One) (Cons (Succ Two) (Cons (Succ Three) Nil))) 
(Cons (Succ One) (Cons (Succ Two) (Cons (Succ Three) Nil))) 

+  </haskell> 

== A Really Advanced Example : TypeLevel Lambda Calculus == 
== A Really Advanced Example : TypeLevel Lambda Calculus == 

Line 192:  Line 208:  
calculus (with only one variable): 
calculus (with only one variable): 

+  <haskell> 

{# OPTIONS fglasgowexts #} 
{# OPTIONS fglasgowexts #} 

data X 
data X 

Line 209:  Line 226:  
instance Eval (Lam t) (Lam t) 
instance Eval (Lam t) (Lam t) 

instance (Eval s s', Apply s' t u) => Eval (App s t) u 
instance (Eval s s', Apply s' t u) => Eval (App s t) u 

+  </haskell> 

Now, lets evaluate some lambda expressions: 
Now, lets evaluate some lambda expressions: 

+  <haskell> 

> :t undefined :: Eval (App (Lam X) X) u => u 
> :t undefined :: Eval (App (Lam X) X) u => u 

undefined :: Eval (App (Lam X) X) u => u :: X 
undefined :: Eval (App (Lam X) X) u => u :: X 

+  </haskell> 

Ok good, and: 
Ok good, and: 

+  <haskell> 

> :t undefined :: Eval (App (Lam (App X X)) (Lam (App X X)) ) u => u 
> :t undefined :: Eval (App (Lam (App X X)) (Lam (App X X)) ) u => u 

^CInterrupted. 
^CInterrupted. 

+  </haskell> 

diverges ;) 
diverges ;) 

−  +  == Turingcompleteness == 

−  == Turingcomplete example == 

It's possible to embed the Turingcomplete [[Type_SKSK combinator calculus]] at the type level. 
It's possible to embed the Turingcomplete [[Type_SKSK combinator calculus]] at the type level. 

Line 236:  Line 258:  
[[Category:Idioms]] 
[[Category:Idioms]] 

[[Category:Mathematics]] 
[[Category:Mathematics]] 

+  [[Category:Typelevel programming]] 
Revision as of 16:26, 19 July 2012
Type arithmetic (or typelevel computation) are calculations on the typelevel, often implemented in Haskell using functional dependencies to represent functions.
A simple example of typelevel computation are operations on Peano numbers:
data Zero data Succ a class Add a b ab  a b > ab, a ab > b instance Add Zero b b instance (Add a b ab) => Add (Succ a) b (Succ ab)
Many other representations of numbers are possible, including binary and balanced base tree. Typelevel computation may also include type representations of boolean values, lists, trees and so on. It is closely connected to theorem proving, via the CurryHoward isomorphism.
A decimal representation was put forward by Oleg Kiselyov in "NumberParamterized Types" in the fifth issue of The Monad Reader. There is an implementation in the typelevel package, but unfortunately the arithmetic is really slow, because in fact it simulates Peano arithmetic with decimal numbers.
Contents 
1 Library support
Robert Dockins has gone as far as to write a library for type level arithmetic, supporting the following operations on type level naturals: addition, subtraction, multiplication, division, remainder, GCD, and also contains the following predicates: test for zero, test for equality and < > <= >=
This library uses a binary representation and can handle numbers at the order of 10^15 (at least). It also contains a test suite to help validate the somewhat unintuitive algorithms.
More libraries:
 typelevel Natural numbers in decimal representation using functional dependencies and Template Haskell. However arithmetic is performed in a unary way and thus it is quite slow.
 typeleveltf Similar to the typelevel package (also in speed) but uses type families instead of functional dependencies and uses the same module names as the typelevel package. Thus module name clashes are warranted if you have to use both packages.
 typelevelnaturalnumber and related packages. A collection of packages where the simplest one is even Haskell2010.
 tfp Decimal representation, Type families, Template Haskell.
 typical Binary numbers and functional dependencies.
 typeunary Unary representation and type families.
 numtype, numtypetf Unary representation and functional dependencies and type families, respectively.
2 More type hackery
Not to be outdone, Oleg Kiselyov has written on invertible, terminating, 3place addition, multiplication, exponentiation relations on typelevel Peano numerals, where any two operands determine the third. He also shows the invertible factorial relation. Thus providing all common arithmetic operations on Peano numerals, including nbase discrete logarithm, nth root, and the inverse of factorial. The inverting method can work with any representation of (typelevel) numerals, binary or decimal.
Oleg says, "The implementation of RSA on the type level is left for future work".
3 Djinn
Somewhat related is Lennart Augustsson's tool Djinn, a theorem prover/coding wizard, that generates Haskell code from a given Haskell type declaration.
Djinn interprets a Haskell type as a logic formula using the CurryHoward isomorphism and then a decision procedure for Intuitionistic Propositional Calculus.
4 An Advanced Example : TypeLevel Quicksort
An advanced example: quicksort on the type level.
Here is a complete example of advanced type level computation, kindly provided by Roman Leshchinskiy. For further information consult Thomas Hallgren's 2001 paper Fun with Functional Dependencies.
module Sort where  natural numbers data Zero data Succ a  booleans data True data False  lists data Nil data Cons a b  shortcuts type One = Succ Zero type Two = Succ One type Three = Succ Two type Four = Succ Three  example list list1 :: Cons Three (Cons Two (Cons Four (Cons One Nil))) list1 = undefined  utilities numPred :: Succ a > a numPred = const undefined class Number a where numValue :: a > Int instance Number Zero where numValue = const 0 instance Number x => Number (Succ x) where numValue x = numValue (numPred x) + 1 numlHead :: Cons a b > a numlHead = const undefined numlTail :: Cons a b > b numlTail = const undefined class NumList l where listValue :: l > [Int] instance NumList Nil where listValue = const [] instance (Number x, NumList xs) => NumList (Cons x xs) where listValue l = numValue (numlHead l) : listValue (numlTail l)  comparisons data Less data Equal data Greater class Cmp x y c  x y > c instance Cmp Zero Zero Equal instance Cmp Zero (Succ x) Less instance Cmp (Succ x) Zero Greater instance Cmp x y c => Cmp (Succ x) (Succ y) c  put a value into one of three lists according to a pivot element class Pick c x ls eqs gs ls' eqs' gs'  c x ls eqs gs > ls' eqs' gs' instance Pick Less x ls eqs gs (Cons x ls) eqs gs instance Pick Equal x ls eqs gs ls (Cons x eqs) gs instance Pick Greater x ls eqs gs ls eqs (Cons x gs)  split a list into three parts according to a pivot element class Split n xs ls eqs gs  n xs > ls eqs gs instance Split n Nil Nil Nil Nil instance (Split n xs ls' eqs' gs', Cmp x n c, Pick c x ls' eqs' gs' ls eqs gs) => Split n (Cons x xs) ls eqs gs listSplit :: Split n xs ls eqs gs => (n, xs) > (ls, eqs, gs) listSplit = const (undefined, undefined, undefined)  zs = xs ++ ys class App xs ys zs  xs ys > zs instance App Nil ys ys instance App xs ys zs => App (Cons x xs) ys (Cons x zs)  zs = xs ++ [n] ++ ys  this is needed because   class CCons x xs xss  x xs > xss  instance CCons x xs (Cons x xs)   doesn't work class App' xs n ys zs  xs n ys > zs instance App' Nil n ys (Cons n ys) instance (App' xs n ys zs) => App' (Cons x xs) n ys (Cons x zs)  quicksort class QSort xs ys  xs > ys instance QSort Nil Nil instance (Split x xs ls eqs gs, QSort ls ls', QSort gs gs', App eqs gs' geqs, App' ls' x geqs ys) => QSort (Cons x xs) ys listQSort :: QSort xs ys => xs > ys listQSort = const undefined
And we need to be able to run this somehow, in the typechecker. So fire up ghci:
> :t listQSort list1 Cons (Succ Zero) (Cons (Succ One) (Cons (Succ Two) (Cons (Succ Three) Nil)))
5 A Really Advanced Example : TypeLevel Lambda Calculus
Again, thanks to Roman Leshchinskiy, we present a simple lambda calculus encoded in the type system (and with nonterminating typechecking fun!)
Below is an example which encodes a strippeddown version of the lambda calculus (with only one variable):
{# OPTIONS fglasgowexts #} data X data App t u data Lam t class Subst s t u  s t > u instance Subst X u u instance (Subst s u s', Subst t u t') => Subst (App s t) u (App s' t') instance Subst (Lam t) u (Lam t) class Apply s t u  s t > u instance (Subst s t u, Eval u u') => Apply (Lam s) t u' class Eval t u  t > u instance Eval X X instance Eval (Lam t) (Lam t) instance (Eval s s', Apply s' t u) => Eval (App s t) u
Now, lets evaluate some lambda expressions:
> :t undefined :: Eval (App (Lam X) X) u => u undefined :: Eval (App (Lam X) X) u => u :: X
Ok good, and:
> :t undefined :: Eval (App (Lam (App X X)) (Lam (App X X)) ) u => u ^CInterrupted.
diverges ;)
6 Turingcompleteness
It's possible to embed the Turingcomplete SK combinator calculus at the type level.
7 Theory
See also dependent type theory.
8 Practice
Extensible records (which are used e.g. in type safe, declarative relational algebra approaches to database management)