The
Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs. Also known as
Curry–Howard isomorphism,
proofs-as-programs correspondence and
formulae-as-types correspondence, it refers to the generalization of a syntactic
analogy between systems of formal logic and computational calculi that was first discovered by the American
mathematician Haskell Curry and
logician William Alvin Howard.
At the very beginning, the Curry–Howard correspondence is
In other words, the Curry–Howard correspondence is the simple observation that two at-the-time-seemingly-unrelated families of formalisms, the proof systems on one side and the models of computation on the other side, were, on the two examples considered by Curry and Howard, in fact structurally the same kind of objects.
If one now abstracts on the peculiarities of this or that formalism, the immediate generalization is the following claim a proof is a program, the formula it proves is a type for the program. Most informally, this can be seen as an analogy which states that the return type of a function (i.e., the type of values returned by a function) is analogous to a logical theorem, subject to hypotheses corresponding to the types of the argument values passed to the function; and that the program to compute that function is analogous to a proof of that theorem. This sets a form of logic programming on a rigorous foundation proofs can be represented as programs, and especially as lambda terms, or proofs can be run.
