Subsumption Demodulation in First-Order Theorem Proving

Motivated by applications of first-order theorem proving to software analysis, we introduce a new inference rule, called subsumption demodulation, to improve support for reasoning with conditional equalities in superposition-based theorem proving. We show that subsumption demodulation is a simplification rule that does not require radical changes to the underlying superposition calculus. We implemented subsumption demodulation in the theorem prover Vampire, by extending Vampire with a new clause index and adapting its multi-literal matching component. Our experiments, using the TPTP and SMT-LIB repositories, show that subsumption demodulation in Vampire can solve many new problems that could so far not be solved by state-of-the-art reasoners

Subsumption Demodulation in First-Order Theorem Proving

Motivated by applications of first-order theorem proving to software analysis, we introduce a new inference rule, called subsumption demodulation, to improve support for reasoning with conditional equalities in superposition-based theorem proving. We show that subsumption demodulation is a simplification rule that does not require radical changes to the underlying superposition calculus. We implemented subsumption demodulation in the theorem prover Vampire, by extending Vampire with a new clause index and adapting its multi-literal matching component. Our experiments, using the TPTP and SMT-LIB repositories, show that subsumption demodulation in Vampire can solve many new problems that could so far not be solved by state-of-the-art reasoners

{SCL(EQ)}: {SCL} for First-Order Logic with Equality

International audienceAbstract We propose a new calculus SCL(EQ) for first-order logic with equality that only learns non-redundant clauses. Following the idea of CDCL (Conflict Driven Clause Learning) and SCL (Clause Learning from Simple Models) a ground literal model assumption is used to guide inferences that are then guaranteed to be non-redundant. Redundancy is defined with respect to a dynamically changing ordering derived from the ground literal model assumption. We prove SCL(EQ) sound and complete and provide examples where our calculus improves on superposition

SCL(EQ): SCL for First-Order Logic with Equality

We propose a new calculus SCL(EQ) for first-order logic with equality thatonly learns non-redundant clauses. Following the idea of CDCL (Conflict DrivenClause Learning) and SCL (Clause Learning from Simple Models) a ground literalmodel assumption is used to guide inferences that are then guaranteed to benon-redundant. Redundancy is defined with respect to a dynamically changingordering derived from the ground literal model assumption. We prove SCL(EQ)sound and complete and provide examples where our calculus improves onsuperposition.<br

Set of support, demodulation, paramodulation: a historical perspective

This article is a tribute to the scientific legacy of automated reasoning pioneer and JAR founder Lawrence T. (Larry) Wos. Larry's main technical contributions were the set-of-support strategy for resolution theorem proving, and the demodulation and paramodulation inference rules for building equality into resolution. Starting from the original definitions of these concepts in Larry's papers, this survey traces their evolution, unearthing the often forgotten trails that connect Larry's original definitions to those that became standard in the field

Efficient reasoning procedures for complex first-order theories

The complexity of a set of first-order formulas results from the size of the set and the complexity of the problem described by its formulas. Decision Procedures for Ontologies This thesis presents new superposition based decision procedures for large sets of formulas. The sets of formulas may contain expressive constructs like transitivity and equality. The procedures decide the consistency of knowledge bases, called ontologies, that consist of several million formulas and answer complex queries with respect to these ontologies. They are the first superposition based reasoning procedures for ontologies that are at the same time efficient, sound, and complete. The procedures are evaluated using the well-known ontologies YAGO, SUMO and CYC. The results of the experiments, which are presented in this thesis, show that these procedures decide the consistency of all three above-mentioned ontologies and usually answer queries within a few seconds. Reductions for General Automated Theorem Proving Sophisticated reductions are important in order to obtain efficient reasoning procedures for complex, particularly undecidable problems because they restrict the search space of theorem proving procedures. In this thesis, I have developed a new powerful reduction rule. This rule enables superposition based reasoning procedures to find proofs in sets of complex formulas. In addition, it increases the number of problems for which superposition is a decision procedure.Die Komplexität einer Formelmenge für einen automatischen Theorembeweiser in Prädikatenlogik 1. Stufe ergibt sich aus der Anzahl der zu betrachtenden Formeln und aus der Komplexität des durch die Formeln beschriebenen Problems. Entscheidungsprozeduren für Ontologien Diese Arbeit entwickelt effiziente auf Superposition basierende Beweisprozeduren für sehr große entscheidbare Formelmengen, die ausdrucksstarke Konstrukte, wie Transitivität und Gleichheit, enthalten. Die Prozeduren ermöglichen es Wissenssammlungen, sogenannte Ontologien, die aus mehreren Millionen Formeln bestehen, auf Konsistenz hin zu überprüfen und Antworten auf komplizierte Anfragen zu berechnen. Diese Prozeduren sind die ersten auf Superposition basierten Beweisprozeduren für große, ausdrucksstarke Ontologien, die sowohl korrekt und vollständig, als auch effizient sind. Die entwickelten Prozeduren werden anhand der weit bekannten Ontologien YAGO, SUMO und CYC evaluiert. Die Experimente zeigen, dass diese Prozeduren die Konsistenz aller untersuchten Ontologien entscheiden und Anfragen in wenigen Sekunden beantworten. Reduktionen für allgemeines Theorembeweisen Um effiziente Prozeduren für das Beweisen in sehr schwierigen und insbesondere in unentscheidbaren Formelmengen zu erhalten, sind starke Reduktionsregeln, die den Beweisraum einschränken, von essentieller Bedeutung. Diese Arbeit entwickelt eine neue mächtige Reduktionsregel, die es Superposition ermöglicht Beweise in sehr schwierigen Formelmengen zu finden und erweitert die Menge von Problemen, für die Superposition eine Entscheidungsprozedur ist

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

Contextual Rewriting in SPASS

First-order theorem proving with equality is undecidable, in general. However, it is semi-decidable in the sense that it is refutationally complete. The basis for a (semi)-decision procedure for first-order clauses with equality is a calculus composed of inference and reduction rules. The inference rules of the calculus generate new clauses whereas the reduction rules delete clauses or transform them into simpler ones. If, in particular, strong reduction rules are available, decidability of certain subclasses of first-order logic can be shown. Hence, sophisticated reductions are essential for progress in automated theorem proving. In this thesis we consider the superposition calculus and in particular the sophisticated reduction rule Contextual Rewriting. However, it is in general undecidable whether contextual rewriting can be applied. Therefore, to make the rule applicable in practice, it has to be further refined. In this work we develop an instance of contextual rewriting which effectively performs contextual rewriting and we implement this in the theorem prover Spass