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Curry-Howard-Lambek correspondence

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Haskell theoretical foundations

General:
Mathematics - Category theory
Research - Curry/Howard/Lambek

Lambda calculus:
Alpha conversion - Beta reduction
Eta conversion - Lambda abstraction

Other:
Recursion - Combinatory logic
Chaitin's construction - Turing machine
Relational algebra

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


1 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).

2 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).

3 Connectives

Of course, atomic propositions contribute little towards knowledge, and the Haskell type system incorporates the logical connectives \and and \or, though heavily disguised. Haskell handles \or conjuction in the manner described by Intuitionistic Logic. When a program has type A \or B, 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 \and conjuction is handled via an isomorphism in Closed Cartesian Categories in general (Haskell types belong to this category): \mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y). That is, instead of a function from X \times Y to Z, we can have a function that takes an argument of type X and returns another function of type Y \rightarrow Z, that is, a function that takes Y to give (finally) a result of type Z: this technique is (known as currying) logically means A \and B \rightarrow C \iff A \rightarrow (B \rightarrow C).

(insert quasi-funny example here)

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

4 Theorems for free!

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

(.) :: (a -> b) -> (b -> c) -> (a -> c)
(.) f g x = f (g x)
The type is, actually,
forall a b c. (a -> b) -> (b -> c) -> (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!)

5 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

6 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 advertized 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 = filter (<= x) xs
                           higher = filter (> 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.

7 Multi-parameter type classes

Haskell makes frequent use of multiparameter type classes. Type classes use a logic language (Prolog-like), and for multiparamter type classes they define a relation between types.

7.1 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.

8 Indexed types

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