Synthesis of Functional Programs with Help of First-Order Intuitionistic Logic

Curry-Howard isomorphism makes it possible to obtain functional programs from proofs in logic. We analyse the problem of program synthesis for ML programs with algebraic types and relate it to the proof search problems in appropriate logics. The problem of synthesis for closed programs is easily equivalent to the proof construction in intuitionistic propositional logic and thus fits in the class of PSPACE-complete problems. We focus further attention on the synthesis problem relative to a given external library of functions. It turns out that the problem is undecidable for unbounded instantiation in ML. However its restriction to instantiations with atomic types only results in a case equivalent to proof search in a restricted fragment of intuitionistic first-order logic, being the core of Sigma_1 level of the logic in the Mints hierarchy. This results in EXPSPACE-completeness for this special case of the ML program synthesis problem.