Scavenger 0.1: A Theorem Prover Based on Conflict Resolution

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving new clauses by (a first-order generalization of) conflict-driven clause learning.Comment: Published at CADE 201

On the Expressivity and Applicability of Model Representation Formalisms

A number of first-order calculi employ an explicit model representation formalism for automated reasoning and for detecting satisfiability. Many of these formalisms can represent infinite Herbrand models. The first-order fragment of monadic, shallow, linear, Horn (MSLH) clauses, is such a formalism used in the approximation refinement calculus. Our first result is a finite model property for MSLH clause sets. Therefore, MSLH clause sets cannot represent models of clause sets with inherently infinite models. Through a translation to tree automata, we further show that this limitation also applies to the linear fragments of implicit generalizations, which is the formalism used in the model-evolution calculus, to atoms with disequality constraints, the formalisms used in the non-redundant clause learning calculus (NRCL), and to atoms with membership constraints, a formalism used for example in decision procedures for algebraic data types. Although these formalisms cannot represent models of clause sets with inherently infinite models, through an additional approximation step they can. This is our second main result. For clause sets including the definition of an equivalence relation with the help of an additional, novel approximation, called reflexive relation splitting, the approximation refinement calculus can automatically show satisfiability through the MSLH clause set formalism.Comment: 15 page

SCL: Clause Learning from Simple Models

International audienceSeveral decision procedures for the Bernays-Schoenfinkel (BS) fragment of first-order logic rely on explicit model assumptions. In particular, the procedures differ in their respective model representation formalisms. We introduce a new decision procedure SCL deciding the BS fragment. SCL stands for clause learning from simple models. Simple models are solely built on ground literals. Nevertheless, we show that SCL can learn exactly the clauses other procedures learn with respect to more complex model representation formalisms. Therefore, the overhead of complex model representation formalisms is not always needed. SCL is sound and complete for full first-order logic without equality

On the Expressivity and Applicability of Model Representation Formalisms

International audienceA number of first-order calculi employ an explicit model representation formalism in support of non-redundant inferences and for detecting satisfiability. Many of these formalisms can represent infinite Herbrand models. The first-order fragment of monadic, shallow, linear, Horn (MSLH) clauses, is such a formalism used in the approximation refinement calculus (AR). Our first result is a finite model property for MSLH clause sets. Therefore, MSLH clause sets cannot represent models of clause sets with inherently infinite models. Through a translation to tree automata, we further show that this limitation also applies to the linear fragments of implicit generalizations, which is the formalism used in the model-evolution calculus (ME), to atoms with disequality constraints, the formalisms used in the non-redundant clause learning calculus (NRCL), and to atoms with membership constraints, a formalism used for example in decision procedures for algebraic data types. Although these formalisms cannot represent models of clause sets with inherently infinite models, through an additional approximation step they can. This is our second main result. For clause sets including the definition of an equivalence relation with the help of an additional, novel approximation, called reflexive relation splitting, the approximation refinement calculus can automatically show satisfiability through the MSLH clause set formalism

On the Combination of the Bernays–Schönfinkel–Ramsey Fragment with Simple Linear Integer Arithmetic

International audienceIn general, first-order predicate logic extended with linear integer arithmetic is undecidable. We show that the Bernays-Schönfinkel-Ramsey fragment (∃ * ∀ *-sentences) extended with a restricted form of linear integer arithmetic is decidable via finite ground instantiation. The identified ground instances can be employed to restrict the search space of existing automated reasoning procedures considerably, e.g., when reasoning about quantified properties of array data structures formalized in Bradley, Manna, and Sipma's array property fragment. Typically, decision procedures for the array property fragment are based on an exhaustive instantiation of universally quantified array indices with all the ground index terms that occur in the formula at hand. Our results reveal that one can get along with significantly fewer instances

SGGS decision procedures

SGGS (Semantically-Guided Goal-Sensitive reasoning) is a conflict-driven first-order theorem-proving method which is refutationally complete and model complete in the limit. These features make it attractive as a basis for decision procedures. In this paper we show that SGGS decides the stratified fragment which generalizes EPR, the PVD fragment, and a new fragment that we dub restrained. The new class has the small model property, as the size of SGGS-generated models can be upper-bounded, and is also decided by hyperresolution and ordered resolution. We report on experiments with a termination tool implementing a restrainedness test, and with an SGGS prototype named Koala