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

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

Semantically-guided goal-sensitive reasoning: decision procedures and the Koala prover

The main topic of this article are SGGS decision procedures for fragments of first-order logic without equality. SGGS (Semantically-Guided Goal-Sensitive reasoning) is an attractive basis for decision procedures, because it generalizes to first-order logic the Conflict-Driven Clause Learning (CDCL) procedure for propositional satisfiability. As SGGS is both refutationally complete and model-complete in the limit, SGGS decision procedures are model-constructing. We investigate the termination of SGGS with both positive and negative results: for example, SGGS decides Datalog and the stratified fragment (including Effectively PRopositional logic) that are relevant to many applications. Then we discover several new decidable fragments, by showing that SGGS decides them. These fragments have the small model property, as the cardinality of their SGGS-generated models can be upper bounded, and for most of them termination tools can be applied to test a set of clauses for membership. We also present the first implementation of SGGS - the Koala theorem prover - and we report on experiments with Koala

Rewrite-based equational theorem proving with selection and simplification

We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses. Refutation completeness is established via the use of well-founded orderings on clauses for defining a Herbrand model for a consistent set of clauses. We also formulate an abstract notion of redundancy and show that the deletion of redundant clauses during the theorem proving process preserves refutation completeness. It is often possible to compute the closure of nontrivial sets of clauses under application of non-redundant inferences. The refutation of goals for such complete sets of clauses is simpler than for arbitrary sets of clauses, in particular one can restrict attention to proofs that have support from the goals without compromising refutation completeness. Additional syntactic properties allow to restrict the search space even further, as we demonstrate for so-called quasi-Horn clauses. The results in this paper contain as special cases or generalize many known results about Knuth-Bendix-like completion procedures (for equations, Horn clauses, and Horn clauses over built-in Booleans), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses

Automated deduction with built-in theories: completeness results and constraint solving techniques

Postprint (published version

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