Automating Unrealizability Logic: Hoare-style Proof Synthesis for Infinite Sets of Programs

Unrealizability logic (UL) was proposed by Kim et al. as the first Hoare-style proof system for proving properties that hold for an infinite set of programs (defined by a regular tree grammar). The goal of our work is to automate reasoning and proof generation for UL. A key ingredient in UL is the notion of nonterminal summaries-inductive facts that characterize recursive nonterminals in the grammar that defines the set of programs. They are analogous to procedure summaries in Hoare logic. The goal of automating UL led us to reformulate the inference rules-in particular, introducing a unified rule for nonterminal summaries, called the rule of adaptation, which draws inspiration from how procedure summaries are handled in Hoare logic. In the same way that verification conditions can be used to synthesize loop invariants for Hoare logic proofs, our reformulation of UL reduces the problem of synthesizing a nonterminal summary to a Syntax-Guided Synthesis problem. We implement Wuldo, the first checker and synthesizer for UL. Wuldo can express proofs beyond the reach of existing tools, including proofs that establish how infinitely many programs behave on infinitely many inputs, and in some cases Wuldo can even synthesize the needed nonterminal summaries.