Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration
Roberto Sebastiani
TL;DR
This work tackles the ambiguity in defining when a partial assignment satisfies a formula, revealing two notions—verification and entailment—that behave differently outside CNF. It proves that entailment is a more robust, universally applicable notion, especially for non-CNF and existential formulas, while verification remains a cheap, necessary condition. The authors argue for adopting entailment as the standard notion of partial-satisfaction and using verification as an efficient proxy, with significant practical benefits for enumeration procedures (e.g., AllSAT/AllSMT, projected enumeration, and model counting) where early detection of satisfaction can prune large search subtrees. The study also clarifies CNF-related scenarios, discusses CNF-ization limitations, and outlines avenues for entailment-based enhancements in formula compilation and search strategies, including potential empirical evaluations and integration into existing tooling.
Abstract
Many procedures for SAT-related problems, in particular for those requiring the complete enumeration of satisfying truth assignments, rely their efficiency and effectiveness on the detection of (possibly small) partial assignments satisfying an input formula. Surprisingly, there seems to be no unique universally-agreed definition of formula satisfaction by a partial assignment in the literature. In this paper we analyze in deep the issue of satisfaction by partial assignments, raising a flag about some ambiguities and subtleties of this concept, and investigating their practical consequences. We identify two alternative notions that are implicitly used in the literature, namely verification and entailment, which coincide if applied to CNF formulas but differ and present complementary properties if applied to non-CNF or to existentially-quantified formulas. We show that, although the former is easier to check and as such is implicitly used by most current search procedures, the latter has better theoretical properties, and can improve the efficiency and effectiveness of enumeration procedures.