Completeness of Synthesis under Realizability Assumptions using Superposition
For automated program synthesis, this work provides a theoretical completeness guarantee for a specific calculus, addressing a known limitation in existing approaches.
The paper refines a superposition-based calculus for program synthesis to ensure completeness: if a computable recursion-free program exists, the calculus will find it. The refined calculus is proven sound and complete for this class.
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.