# Curry-Howard-Lambek correspondence

The **Curry-Howard-Lambek correspondance** is a three way isomorphism between types (in programming languages), propositions (in logic) and objects of a Cartesian closed category. Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs in logic (and *vice versa*).

## Contents

## Life, the Universe and Everything

As is well established by now,

```
theAnswer :: Integer
theAnswer = 42
```

The logical interpretation of the program is that the type `Integer`

is inhabited (by the value `42`

), so the existence of this program *proves* the proposition `Integer`

(a type without any value is the "bottom" type, a proposition with no proof).

## Inference

A (non-trivial) Haskell function maps a value (of type `a`

, say) to another value (of type `b`

), therefore, *given* a value of type `a`

(a proof of `a`

), it *constructs* a value of type `b`

(so the proof is *transformed* into a proof of `b`

)! So `b`

is inhabited if `a`

is, and a proof of `a -> b`

is established (hence the notation, in case you were wondering).

```
representation :: Bool -> Integer
representation False = 0
representation True = 1
```

says, for example, if `Boolean`

is inhabited, so is `Integer`

(well, the point here is demonstration, not discovery).

## Connectives

Of course, atomic propositions contribute little towards knowledge, and the Haskell type system incorporates the logical connectives and , though heavily disguised. Haskell handles conjuction in the manner described by Intuitionistic Logic. When a program has type , the value returned itself indicates which one. The algebraic data types in Haskell has a tag on each alternative, the constructor, to indicate the injections:

```
data Message a = OK a | Warning a | Error a
p2pShare :: Integer -> Message String
p2pShare n | n == 0 = Warning "Share! Freeloading hurts your peers."
| n < 0 = Error "You cannot possibly share a negative number of files!"
| n > 0 = OK ("You are sharing " ++ show n ++ " files.")
```

So any one of `OK String`

, `Warning String`

or `Error String`

proves the proposition `Message String`

, leaving out any two constructors would not invalidate the program. At the same time, a proof of `Message String`

can be pattern matched against the constructors to see which one it proves.
On the other hand, to prove `String`

is inhabited from the proposition `Message String`

, it has to be proven that you can prove `String`

from any of the alternatives...

```
show :: Message String -> String
show (OK s) = s
show (Warning s) = "Warning: " ++ s
show (Error s) = "ERROR! " ++ s
```

The conjuction is handled via an isomorphism in Closed Cartesian Categories in general (Haskell types belong to this category): . That is, instead of a function from to , we can have a function that takes an argument of type and returns another function of type , that is, a function that takes to give (finally) a result of type : this technique is (known as currying) logically means .

*(insert quasi-funny example here)*

So in Haskell, currying takes care of the connective. Logically, a proof of is a pair of proofs of the propositions. In Haskell, to have the final value, values of both and have to be supplied (in turn) to the (curried) function.

## Theorems for free!

Things get interesting when polymorphism comes in. The composition operator in Haskell proves a very simple theorem.

```
(.) :: (b -> c) -> (a -> b) -> (a -> c)
(.) f g x = f (g x)
```

The type is, actually, `forall a b c. (b -> c) -> (a -> b) -> (a -> c)`

, to be a bit verbose, which says, logically speaking, for all propositions `a, b`

and `c`

, if from `a`

, `b`

can be proven, and if from `b`

, `c`

can be proven, then from `a`

, `c`

can be proven (the program says how to go about proving: just compose the given proofs!)

## Negation

Of course, there's not much you can do with just truth. `forall b. a -> b`

says that given `a`

, we can infer anything. Therefore we will take `forall b. a -> b`

as meaning `not a`

. Given this, we can prove several more of the axioms of logic.

```
type Not x = (forall a. x -> a)
doubleNegation :: x -> Not (Not x)
doubleNegation k pr = pr k
contraPositive :: (a -> b) -> (Not b -> Not a)
contraPositive fun denyb showa = denyb (fun showa)
deMorganI :: (Not a, Not b) -> Not (Either a b)
deMorganI (na, _) (Left a) = na a
deMorganI (_, nb) (Right b) = nb b
deMorganII :: Either (Not a) (Not b) -> Not (a,b)
deMorganII (Left na) (a, _) = na a
deMorganII (Right nb) (_, b) = nb b
```

## Type classes

A type class in Haskell is a proposition *about* a type.

```
class Eq a where
(==) :: a -> a -> Bool
(/=) :: a -> a -> Bool
```

means, logically, there is a type `a`

for which the type `a -> a -> Bool`

is inhabited, or, from `a`

it can be proved that `a -> a -> Bool`

(the class promises two different proofs for this, having names `==`

and `/=`

).
This proposition is of existential nature (not to be confused with existential type). A proof for this proposition (that there is a type that conforms to the specification) is (obviously) a set of proofs of the advertised proposition (an implementation), by an `instance`

declaration:

```
instance Eq Bool where
True == True = True
False == False = True
_ == _ = False
(/=) a b = not (a == b)
```

A not-so-efficient sort implementation would be:

```
sort [] = []
sort (x : xs) = sort lower ++ [x] ++ sort higher
where (lower,higher) = partition (< x) xs
```

Haskell infers its type to be `forall a. (Ord a) => [a] -> [a]`

. It means, if a type `a`

satisfies the proposition about propositions `Ord`

(that is, has an ordering defined, as is necessary for comparison), then `sort`

is a proof of `[a] -> [a]`

. For this to work, somewhere, it should be proved (that is, the comparison functions defined) that `Ord a`

is true.

## Multi-parameter type classes

Haskell makes frequent use of multiparameter type classes. Type classes constitute a Prolog-like logic language, and multiparameter type classes define a relation between types.

### Functional dependencies

These type level functions are set-theoretic. That is, `class TypeClass a b | a -> b`

defines a relation between types `a`

and `b`

, and requires that there would not be different instances of `TypeClass a b`

and `TypeClass a c`

for different `b`

and `c`

, so that, essentially, `b`

can be inferred as soon as `a`

is known. This is precisely functions as relations as prescribed by set theory.

## Indexed types

*(please someone complete this, should be quite interesting, I have no idea what it should look like logically)*