What are the most interesting equivalences arising from the Curry-Howard Isomorphism?

Question

I came upon the Curry-Howard Isomorphism relatively late in my programming life, and perhaps this contributes to my being utterly fascinated by it. It implies that for every pro

Answer 1

While it's not a simple isomorphism, this discussion of constructive LEM is a very interesting result. In particular, in the conclusion section, Oleg Kiselyov discusses how the use of monads to get double-negation elimination in a constructive logic is analogous to distinguishing computationally decidable propositions (for which LEM is valid in a constructive setting) from all propositions. The notion that monads capture computational effects is an old one, but this instance of the Curry--Howard isomorphism helps put it in perspective and helps get at what double-negation really "means".

Answer 2

You're muddying things a little bit regarding nontermination. Falsity is represented by uninhabited types, which by definition can't be non-terminating because there's nothing of that type to evaluate in the first place.

Non-termination represents contradiction--an inconsistent logic. An inconsistent logic will of course allow you to prove anything, including falsity, however.

Ignoring inconsistencies, type systems typically correspond to an intuitionistic logic, and are by necessity constructivist, which means certain pieces of classical logic can't be expressed directly, if at all. On the other hand this is useful, because if a type is a valid constructive proof, then a term of that type is a means of constructing whatever you've proven the existence of.

A major feature of the constructivist flavor is that double negation is not equivalent to non-negation. In fact, negation is rarely a primitive in a type system, so instead we can represent it as implying falsehood, e.g., not P becomes P -> Falsity. Double negation would thus be a function with type (P -> Falsity) -> Falsity, which clearly is not equivalent to something of just type P.

However, there's an interesting twist on this! In a language with parametric polymorphism, type variables range over all possible types, including uninhabited ones, so a fully polymorphic type such as ∀a. a is, in some sense, almost-false. So what if we write double almost-negation by using polymorphism? We get a type that looks like this: ∀a. (P -> a) -> a. Is that equivalent to something of type P? Indeed it is, merely apply it to the identity function.

But what's the point? Why write a type like that? Does it mean anything in programming terms? Well, you can think of it as a function that already has something of type P somewhere, and needs you to give it a function that takes P as an argument, with the whole thing being polymorphic in the final result type. In a sense, it represents a suspended computation, waiting for the rest to be provided. In this sense, these suspended computations can be composed together, passed around, invoked, whatever. This should begin to sound familiar to fans of some languages, like Scheme or Ruby--because what it means is that double-negation corresponds to continuation-passing style, and in fact the type I gave above is exactly the continuation monad in Haskell.

Answer 3

Your chart is not quite right; in many cases you have confused types with terms.

function type              implication
function                   proof of implication
function argument          proof of hypothesis
function result            proof of conclusion
function application RULE  modus ponens
recursion                  n/a [1]
structural induction       fold (foldr for lists)
mathematical induction     fold for naturals (data N = Z | S N)
identity function          proof of A -> A, for all A
non-terminating function   n/a [2]
tuple                      normal proof of conjunction
sum                        disjunction
n/a [3]                    first-order universal quantification
parametric polymorphism    second-order universal quantification
currying                   (A,B) -> C -||- A -> (B -> C), for all A,B,C
primitive type             axiom
types of typeable terms    theory
function composition       syllogism
substitution               cut rule
value                      normal proof

[1] The logic for a Turing-complete functional language is inconsistent. Recursion has no correspondence in consistent theories. In an inconsistent logic/unsound proof theory you could call it a rule which causes inconsistency/unsoundness.

[2] Again, this is a consequence of completeness. This would be a proof of an anti-theorem if the logic were consistent -- thus, it can't exist.

[3] Doesn't exist in functional languages, since they elide first-order logical features: all quantification and parametrization is done over formulae. If you had first-order features, there would be a kind other than *, * -> *, etc.; the kind of elements of the domain of discourse. For example, in Father(X,Y) :- Parent(X,Y), Male(X), X and Y range over the domain of discourse (call it Dom), and Male :: Dom -> *.

Answer 4

2-continuation           | Sheffer stoke
n-continuation language  | Existential graph
Recursion                | Mathematical Induction

One thing that is important, but have not yet being investigated is the relationship of 2-continuation (continuations that takes 2 parameters) and Sheffer stroke. In classic logic, Sheffer stroke can form a complete logic system by itself (plus some non-operator concepts). Which means the familiar and, or, not can be implemented using only the Sheffer stoke or nand.

This is an important fact of its programming type correspondence because it prompts that a single type combinator can be used to form all other types.

The type signature of a 2-continuation is (a,b) -> Void. By this implementation we can define 1-continuation (normal continuations) as (a,a) -> Void, product type as ((a,b)->Void,(a,b)->Void)->Void, sum type as ((a,a)->Void,(b,b)->Void)->Void. This gives us an impressive of its power of expressiveness.

If we dig further, we will find out that Piece's existential graph is equivalent to a language with the only data type is n-continuation, but I didn't see any existing languages is in this form. So inventing one could be interesting, I think.