Curry-Howard-Lambek correspondence
Haskell theoretical foundations
General: Lambda calculus: Other: |
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.
(.) :: (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!)
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 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.
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.
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)