Completeness of Synthesis under Realizability Assumptions using Superposition

TL;DR

This paper extends superposition calculus for program synthesis, proving its soundness and completeness for recursion-free programs under realizability assumptions, enabling automatic derivation of correct programs.

Contribution

It introduces a refined superposition calculus supporting synthesis of recursion-free programs with uncomputable symbols, and proves its soundness and completeness.

Findings

Refined superposition calculus supports synthesis of recursion-free programs.

The calculus is proven sound for the synthesis task.

Completeness is established when at least one computable program exists.

Abstract

Program synthesis is the task of automatically deriving a program that has been specified by a user in advance. Combining automated theorem proving with program synthesis enables the automated construction of proven-to-be-correct programs, thereby ensuring software reliability. In this paper, we consider the superposition-based calculus extended to support synthesis of recursion-free programs allowing reasoning with uncomputable symbols. We present cases where the calculus fails and refine it to solve them. We prove that the refined calculus is sound. Finally, we also prove completeness in the following sense: if at least one computable program satisfying the given specification exists, we show that the modified calculus finds one.