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

Abstract

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.