On completeness of program synthesis systems

Abstract

This paper addresses a question of completeness of systems extracting programs from proofs. We consequently consider specifications of total functions, partial functions, predicates and partial predicates. Our aim is to investigate the problem of existence of a logical system which allows to extract all total recursive function (all partial recursive functions, all decidable predicates) from proofs in this logical system. Another natural question is whether such systems exist among generally known systems. We prove that it is impossible to extract programs for all total recursive functions from the proofs of specifications of the form ∀¯x∃yOut(x, y) and it is impossible to extract algorithms deciding all decidable predicates from the proofs of the specifications of the form ∀¯x(P(¯x)V⌝P(¯x)). Concerning specifications of the partial functions and the partial predicates we prove that in the system HA+MP consisting from the intuitionistic arithmetic with the Markov's principle it is possible to extract all partial recursive functions from the specifications of the form ∀¯x(In(¯x)⊃∃yOut(¯x, y)) and algorithms deciding all decidable predicates from the specifications of the form ∀¯x(In(¯x)⊃(P(¯x)V⌝P(¯x))).