103 research outputs found
Superposition with Equivalence Reasoning andDelayed Clause Normal Form Transformation
This report describes a superposition calculus where quantifiers are eliminated lazily. Superposition and simplification inferences may employ equivalences that have arbitrary formulas at their smaller side. A closely related calculus is implemented in the Saturate system and has shown useful on many examples, in particular in set theory. The report presents a completeness proof and reports on practical experience obtained with the Saturate system
Superposition as a logical glue
The typical mathematical language systematically exploits notational and
logical abuses whose resolution requires not just the knowledge of domain
specific notation and conventions, but not trivial skills in the given
mathematical discipline. A large part of this background knowledge is expressed
in form of equalities and isomorphisms, allowing mathematicians to freely move
between different incarnations of the same entity without even mentioning the
transformation. Providing ITP-systems with similar capabilities seems to be a
major way to improve their intelligence, and to ease the communication between
the user and the machine. The present paper discusses our experience of
integration of a superposition calculus within the Matita interactive prover,
providing in particular a very flexible, "smart" application tactic, and a
simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311
Shostak Light
We represent the essential ingredients of Shostak's procedure at a high level of abstraction, and as a refinement of the Nelson-Oppen procedure. We analyze completeness issues of the method based on a general notion of theories. We also formalize a notion of #-models and show that on the basis of Shostak's procedure we cannot distinguish a theory from its approximation represented by the class of its #-models
Authors ’ Addresses
The most efficient techniques that have been developed to date for equality handling in first-order theorem proving are based on superposition calculi. Superposition is a refinement of paramodulation in that various ordering constraints are imposed on inferences. For practical purposes, a key aspect of superposition is its compatibility with powerful simplification techniques. In this paper we solve a long-standing open problem by showing that strict superposition—that is, superposition without equality factoring—is refutationally complete. The difficulty of the problem arises from the fact that the strict calculus, in contrast to the standard calculus with equality factoring, is not compatible with arbitrary removal of tautologies, so that the usual techniques for proving the (refutational) completeness of paramodulation calculi are not directly applicable. We deal with the problem by introducing a suitable notion of direct rewrite proof and modifying proof techniques based on candidate models and counterexamples in that we define these concepts, no
Shostak Light
Abstract. We represent the essential ingredients of Shostak’s procedure at a high level of abstraction, and as a refinement of the Nelson-Oppen procedure. We analyze completeness issues of the method based on a general notion of theories. We also formalize a notion of σ-models and show that on the basis of Shostak’s procedure we cannot distinguish a theory from its approximation represented by the class of its σ-models.
Strict Basic Superposition and Chaining
The most efficient techniques that have been developed to date for equality handling in first-order theorem proving are based on superposition calculi. Superposition is a refinement of paramodulation in that various ordering constraints are imposed on inferences. For practical purposes, a key aspect of superposition is its compatibility with powerful simplification techniques. In this paper we solve a long-standing open problem by showing that strict superposition---that is, superposition without equality factoring---is refutationally complete. The difficulty of the problem arises from the fact that the strict calculus, in contrast to the standard calculus with equality factoring, is not compatible with arbitrary removal of tautologies, so that the usual techniques for proving the (refutational) completeness of paramodulation calculi are not directly applicable. We deal with the problem by introducing a suitable notion of direct rewrite proof and modifying proof techniques based on can..
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