Invited Lectures.- Tableau Algorithms for Description Logics.- Modality and Databases.- Local Symmetries in Propositional Logic.- Comparison.- Design and Results of TANCS-2000 Non-classical (Modal) Systems Comparison.- Consistency Testing: The RACE Experience.- Benchmark Analysis with FaCT.- MSPASS: Modal Reasoning by Translation and First-Order Resolution.- TANCS-2000 Results for DLP.- Evaluating *SAT on TANCS 2000 Benchmarks.- Research Papers.- A Labelled Tableau Calculus for Nonmonotonic (Cumulative) Consequence Relations.- A Tableau System for Godel-Dummett Logic Based on a Hypersequent Calculus.- An Analytic Calculus for Quantified Propositional Godel Logic.- A Tableau Method for Inconsistency-Adaptive Logics.- A Tableau Calculus for Integrating First-Order and Elementary Set Theory Reasoning.- Hypertableau and Path-Hypertableau Calculi for some Families of Intermediate Logics.- Variants of First-Order Modal Logics.- Complexity of Simple Dependent Bimodal Logics.- Properties of Embeddings from Int to S4.- Term-Modal Logics.- A Subset-Matching Size-Bounded Cache for Satisfiability in Modal Logics.- Dual Intuitionistic Logic Revisited.- Model Sets in a Nonconstructive Logic of Partial Terms with Definite Descriptions.- Search Space Compression in Connection Tableau Calculi Using Disjunctive Constraints.- Matrix-Based Inductive Theorem Proving.- Monotonic Preorders for Free Variable Tableaux.- The Mosaic Method for Temporal Logics.- Sequent-Like Tableau Systems with the Analytic Superformula Property for the Modal Logics KB, KDB, K5, KD5.- A Tableau Calculus for Equilibrium Entailment.- Towards Tableau-Based Decision Procedures for Non-Well-Founded Fragments of Set Theory.- Tableau Calculus for Only Knowing and Knowing At Most.- A Tableau-Like Representation Framework for Efficient Proof Reconstruction.- The Semantic Tableaux Version of the Second Incompleteness Theorem Extends Almost to Robinson's Arithmetic Q.- System Descriptions.- Redundancy-Free Lemmatization in the Automated Model-Elimination Theorem Prover AI-SETHEO.- E-SETHEO: An Automated3 Theorem Prover.

Automated Reasoning with Analytic Tableaux and Related Methods: International Conference, TABLEAUX 2000 St Andrews, Scotland, UK, July 3-7, 2000 Proceedings

Equilibrium logic is a general purpose nonmonotonic reasoning formalism closely aligned with answer set programming (ASP). In particular it provides a logical foundation for ASP as well as an extension of the basic syntax of answer set programs. We present an overview of equilibrium logic and its main properties and uses.

Equilibrium logic

We investigate a peculiar intuitionistic modal logic, called Propositional Lax Logic (PLL), which has promising applications to the formal verification of computer hardware. The logic has emerged from an attempt to express correctness up to behavioural constraints?a central notion in hardware verification?as a logical modality. As a modal logic it is special since it features a single modal operator ? that has a flavour both of possibility and of necessity. In the paper we provide the motivation for PLL and present several technical results. We investigate some of its proof-theoretic properties, presenting a cut-elimination theorem for a standard Gentzen-style sequent presentation of the logic. We go on to define a new class of fallible two-frame Kripke models for PLL. These models are unusual since they feature worlds with inconsistent information; furthermore, the only frame condition imposed is that the ?-frame be a subrelation of the ?-frame. We give a natural translation of these models into Goldblatt's J-space models of PLL. Our completeness theorem for these models yields a Godel-style embedding of PLL into a classical bimodal theory of type (S4,S4) and underpins a simple proof of the finite model property. We proceed to prove soundness and completeness of several theories for specialized classes of models. We conclude with a brief exploration of two concrete and rather natural types of model from hardware verification for which the modality ? models correctness up to timing constraints. We obtain decidability of ?-free fragment of the logic of the first type of model, which coincides with the stable form of Maksimova's intermediate logicL?.

Propositional Lax Logic

Developing modern software typically involves composing functionality from existing libraries. This task is difficult because libraries may expose many methods to the developer. To help developers in such scenarios, we present a technique that synthesizes and suggests valid expressions of a given type at a given program point. As the basis of our technique we use type inhabitation for lambda calculus terms in long normal form. We introduce a succinct representation for type judgements that merges types into equivalence classes to reduce the search space, then reconstructs any desired number of solutions on demand. Furthermore, we introduce a method to rank solutions based on weights derived from a corpus of code. We implemented the algorithm and deployed it as a plugin for the Eclipse IDE for Scala. We show that the techniques we incorporated greatly increase the effectiveness of the approach. Our evaluation benchmarks are code examples from programming practice; we make them available for future comparisons.

/pdf/complete-completion-using-types-and-weights-2wq24qn1al.pdf

Complete completion using types and weights

First‐order logic

Duplication-free tableau calculi and related cut-free sequent calculi for the interpolable propositional intermediate logics

In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent ? we can extract a Kripke counter-model for ?. Finally, we provide a procedure that given a sequent ? returns either a proof of ? in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.

/pdf/contraction-free-linear-depth-sequent-calculi-for-uz5vodk6xi.pdf

Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

We present FCUBE, a theorem prover for Intuitionistic propositional logic based on a tableau calculus. The main novelty of FCUBE is that it implements several optimization techniques that allow to prune the search space acting on different aspects of proof-search. We tested the efficiency of our techniques by comparing FCUBE with other theorem provers. We found that our prover outperforms the other provers on several interesting families of formulas.

/pdf/fcube-an-efficient-prover-for-intuitionistic-propositional-tikw3xux0f.pdf

FCUBE: an efficient prover for intuitionistic propositional logic

A proof is given that the set of maximal intermediate propositional logics with the disjunction property and the set of maximal intermediate predicate logics with the disjunction property and the explicit definability property have the power of continuum. To prove our results, we introduce various notions which might be interesting by themselves. In particular, we illustrate a method to generate wide sets of pairwise “constructively incompatible constructive logics”. We use a notion of “semiconstructive” logic and define wide sets of “constructive” logics by representing the “constructive” logics as “limits” of decreasing sequences of “semiconstructive” logics. Also, we introduce some generalizations of the usual filtration techniques for propositional logics. For instance, “fitrations over rank formulas” are used to show that any two different logics belonging to a suitable uncountable set of “constructive” logics are “constructively incompatible”.

Counting the Maximal Intermediate Constructive Logics

It is well known that G3i, the sequent calculus for intuitionistic propositional logic where weakening and contraction are absorbed into the rules, is not terminating. Indeed, due to the contraction in the rule for left implication, the naive goal-oriented proof-search strategy, consisting in applying the rules of the calculus bottom up until possible, can generate branches of infinite length. The usual solution to this problem is to support the proof-search procedure with a loop checking mechanism that prevents the generation of infinite branches by storing and analyzing some information regarding the branch under development. In this article, we propose a new technique based on evaluation functions. An evaluation function is a lightweight computational mechanism that, analyzing only the current goal of the proof search, allows one to drive the application of rules to guarantee termination and to avoid useless backtracking. We describe an evaluation-driven proof-search procedure that given a sequent σ returns either a G3i-derivation of σ or a countermodel for σ. We prove that such a procedure is terminating and correct, and that the depth of the G3i-trees generated during proof search is quadratic in the size of σ. Finally, we discuss the overhead time introduced by evaluation functions in the proof-search procedure.

Mauro Ferrari

Papers

Duplication-free tableau calculi and related cut-free sequent calculi for the interpolable propositional intermediate logics

Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

FCUBE: an efficient prover for intuitionistic propositional logic

Counting the Maximal Intermediate Constructive Logics

An Evaluation-Driven Decision Procedure for G3i