Abstract Canonical Inference

An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewrite-system reduction are connected to proof orderings. Fairness of deductive mechanisms is defined in terms of proof orderings, distinguishing between (ordinary) "fairness," which yields completeness, and "uniform fairness," which yields saturation.Comment: 28 pages, no figures, to appear in ACM Trans. on Computational Logi

Abstract Canonical Inference Systems

Colloque sur invitation. internationale.International audienceWe provide a general proof theoretical setting under which the so-called ``completion processes'' (as used for equational reasoning) can be modeled, understood, studied, proved and generalized. This framework---based on a well-founded ordering on proofs---allows us to derive saturation processes and redundancy criteria abstractly

On Constructive Connectives and Systems

Canonical inference rules and canonical systems are defined in the framework of non-strict single-conclusion sequent systems, in which the succeedents of sequents can be empty. Important properties of this framework are investigated, and a general non-deterministic Kripke-style semantics is provided. This general semantics is then used to provide a constructive (and very natural), sufficient and necessary coherence criterion for the validity of the strong cut-elimination theorem in such a system. These results suggest new syntactic and semantic characterizations of basic constructive connectives

New results on rewrite-based satisfiability procedures

Program analysis and verification require decision procedures to reason on theories of data structures. Many problems can be reduced to the satisfiability of sets of ground literals in theory T. If a sound and complete inference system for first-order logic is guaranteed to terminate on T-satisfiability problems, any theorem-proving strategy with that system and a fair search plan is a T-satisfiability procedure. We prove termination of a rewrite-based first-order engine on the theories of records, integer offsets, integer offsets modulo and lists. We give a modularity theorem stating sufficient conditions for termination on a combinations of theories, given termination on each. The above theories, as well as others, satisfy these conditions. We introduce several sets of benchmarks on these theories and their combinations, including both parametric synthetic benchmarks to test scalability, and real-world problems to test performances on huge sets of literals. We compare the rewrite-based theorem prover E with the validity checkers CVC and CVC Lite. Contrary to the folklore that a general-purpose prover cannot compete with reasoners with built-in theories, the experiments are overall favorable to the theorem prover, showing that not only the rewriting approach is elegant and conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page

Efficient Encodings of First-Order Horn Formulas in Equational Logic

We present several translations from first-order Horn formulas to equational logic. The goal of these translations is to allow equational theorem provers to efficiently reason about non-equational problems. Using these translations we were able to solve 37 problems of rating 1.0 (i.e. which had not previously been automatically solved) from the TPTP

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

Theory Combination: Beyond Equality Sharing

International audienceSatisfiability is the problem of deciding whether a formula has a model. Although it is not even semidecidable in first-order logic, it is decidable in some first-order theories or fragments thereof (e.g., the quantifier-free fragment). Satisfiability modulo a theory is the problem of determining whether a quantifier-free formula admits a model that is a model of a given theory. If the formula mixes theories, the considered theory is their union, and combination of theories is the problem of combining decision procedures for the individual theories to get one for their union. A standard solution is the equality-sharing method by Nelson and Oppen, which requires the theories to be disjoint and stably infinite. This paper surveys selected approaches to the problem of reasoning in the union of disjoint theories, that aim at going beyond equality sharing, including: asymmetric extensions of equality sharing, where some theories are unrestricted, while others must satisfy stronger requirements than stable infiniteness; superposition-based decision procedures; and current work on conflict-driven satisfiability (CDSAT)

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

Abstract canonical inference

An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewrite-system reduction are connected to proof orderings. Fairness of deductive mechanisms is defined in terms of proof orderings, distinguishing between (ordinary) “fairness, ” which yields completeness, and “uniform fairness, ” which yields saturation