Itauto: An Extensible Intuitionistic SAT Solver

We present the design and implementation of itauto, a Coq reflexive tactic for intuitionistic propositional logic. The tactic inherits features found in modern SAT solvers: definitional conjunctive normal form; lazy unit propagation and conflict driven backjumping. Formulae are hash-consed using native integers thus enabling a fast equality test and a pervasive use of Patricia Trees. We also propose a hybrid proof by reflection scheme whereby the extracted solver calls user-defined tactics on the leaves of the propositional proof search thus enabling theory reasoning and the generation of conflict clauses. The solver has decent efficiency and is more scalable than existing tactics on synthetic benchmarks and preliminary experiments are encouraging for existing developments

Refutation Systems : An Overview and Some Applications to Philosophical Logics

Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics

Representing scope in intuitionistic deductions

AbstractIntuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever — rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how λ-terms are to be extracted from its deductions

Proceedings of the Workshop on Linear Logic and Logic Programming

Declarative programming languages often fail to effectively address many aspects of control and resource management. Linear logic provides a framework for increasing the strength of declarative programming languages to embrace these aspects. Linear logic has been used to provide new analyses of Prolog\u27s operational semantics, including left-to-right/depth-first search and negation-as-failure. It has also been used to design new logic programming languages for handling concurrency and for viewing program clauses as (possibly) limited resources. Such logic programming languages have proved useful in areas such as databases, object-oriented programming, theorem proving, and natural language parsing. This workshop is intended to bring together researchers involved in all aspects of relating linear logic and logic programming. The proceedings includes two high-level overviews of linear logic, and six contributed papers. Workshop organizers: Jean-Yves Girard (CNRS and University of Paris VII), Dale Miller (chair, University of Pennsylvania, Philadelphia), and Remo Pareschi, (ECRC, Munich)

A Semantic Framework for Proof Evidence

International audienceTheorem provers produce evidence of proof in many different formats, such as proof scripts, natural deductions, resolution refutations, Herbrand expansions, and equational rewritings. In implemented provers, numerous variants of such formats are actually used: consider, for example, such variants of or restrictions to resolution refu-tations as binary resolution, hyper-resolution, ordered-resolution, paramodulation, etc. We propose the foundational proof certificates (FPC) framework for defining the semantics of a broad range of proof evidence. This framework allows both producers of proof certificates and the checkers of those certificates to have a clear formal definition of the semantics of a wide variety of proof evidence. Employing the FPC framework will allow one to separate a proof from its provenance and to allow anyone to construct their own proof checker for a given style of proof evidence. The foundation on which FPC relies is that of proof theory, particularly recent work into focused proof systems: such proof systems provide protocols by which a checker extracts information from the certificate (mediated by the so called clerks and experts) as well as performs various deterministic and non-deterministic computations. While we shall limit ourselves to first-order logic in this paper, we shall not limit ourselves in many other ways. The FPC framework is described for both classical and intuitionistic logics and for proof structures as diverse as resolution refutations, natural deduction, Frege proofs, and equality proofs

Representing Scope in Intuitionistic Deductions

Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever—rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how λ-terms are to be extracted from its deductions

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established