Question

Could you please explain me what is the basic connection between the fundamentals of logical programming and the phenomenon of syntactic similarity between type systems and conventional logic?

Answer 1

The Curry-Howard correspondence is not about logic programming, but functional programming. The fundamental mechanic of Prolog is justified in proof theory by John Robinson’s resolution technique, which shows how it is possible to check whether logical formulae expressed as Horn clauses are satisfiable, that is, whether you can find terms to substitue for their logic variables that make them true.

Thus logic programming is about specifying programs as logical formulae, and the calculation of the program is some form of proof inference, in Prolog reolution, as I have said. By contrast the Curry-Howard correspondence shows how proofs in a special formulasition of logic, called natural deduction, correspond to programs in the lambda calculus, with the type of the program corresponding to the formula that the proof proves; computation in the lambda calculus corresponds to an important phenomenon in proof theory called normalisation, which transforms proofs into new, more direct proofs. So logic programming and functional programming correspond to different levels in these logics: logic programs match formulae of a logic, whilst functional programs match proofs of formulae.

There’s another difference: the logics used are generally different. Logic programming generally uses simpler logics — as I said, Prolog is founded on Horn clauses, which are a highly restricted class of formulae where implications may not be nested, and there are no disjunctions, although Prolog recovers the full strength of classical logic using the cut rule. By contrast, functional programming languages such as Haskell make heavy use of programs whose types have nested implications, and are decorated by all kinds of forms of polymorphism. They are also based on intuitionistic logic, a class of logics that forbids use of the principle of the excluded middle, which Robinson’s computational mechanism is based on.

Some other points:

1. It is possible to base logic programming on more sophisticated logics than Horn clauses; for example, Lambda-prolog is based on intuitionistic logic, with a different computation mechanism than resolution.
2. Dale Miller has called the proof-theoretic paradigm behind logic programming the proof search as programming metaphor, to contrast with the proofs as programs metaphor that is another term used for the Curry-Howard correspondence.