Tableau Calculus Research Articles

Tableau calculus for preference-based conditional logics

We present a tableau calculus for some fundamental systems of propositional conditional logics. We consider the conditional logics that can be characterized by preferential semantics (i.e., possible world structures equipped with a family of preference relations). For these logics, we provide a uniform completeness proof of the axiomatization with respect to the semantics, and a uniform labeled tableau procedure.

Analytic tableaux calculi for KLM logics of nonmonotonic reasoning

We present tableau calculi for the logics of nonmonotonic reasoning defined by Kraus, Lehmann and Magidor (KLM). We give a tableau proof procedure for all KLM logics, namely preferential, loop-cumulative, cumulative, and rational logics. Our calculi are obtained by introducing suitable modalities to interpret conditional assertions. We provide a decision procedure for the logics considered and we study their complexity.

Semantics, calculi, and analysis for object-oriented specifications

We present a formal semantics for an object-oriented specification language. The formal semantics is presented as a conservative shallow embedding in Isabelle/hol and the language is oriented towards ocl formulae in the context of uml class diagrams. On this basis, we formally derive several equational and tableaux calculi, which form the basis of an integrated proof environment including automatic proof support and support for the analysis of this type of specifications. We show applications of our proof environment to data refinement based on an adapted standard refinement notion. Thus, we provide an integrated formal method for refinement-based object-oriented development.

Linear response time for implicate and implicant queries

Knowledge bases can be represented as propositional Formulas. A query of such a theory typically has the form, Is a clause an implicate of the theory? Answering such queries can require exponential time. In Kautz and Selman (Proceedings of the international workshop on processing declarative knowledge (PDK), Kaiserslautern, Germany, 1991), knowledge compilation was proposed as a solution to this problem: Pay the exponential penalty once by compiling the knowledge base into a target language that would guarantee fast response to queries. The reduced implicate trie (ri-trie), introduced in Murray and Rosenthal (Proceedings of the international conference TABLEAUX 2005—analytic tableaux and related methods, Koblenz, Germany. Lecture notes in artificial intelligence, vol 3702. Springer, Berlin, pp 231–244, 2005), may be used as a target language for knowledge compilation. It has the property that a query is processed in time linear in the size of the query, regardless of the size of the compiled knowledge base. In this paper, structures dual to ri-tries, the reduced implicant tries are investigated, and the dual problem—determining the implicants of a formula—is considered. The main result is that, for a given formula, the two structures can be merged into a single reduced implicate/implicant trie that can serve dual roles, representing both implicates and implicants. Furthermore, rii-tries can be computed directly, without separately computing and then merging the dual structures.

HTab: a Terminating Tableaux System for Hybrid Logic

Hybrid logic is a formalism that is closely related to both modal logic and description logic. A variety of proof mechanisms for hybrid logic exist, but the only widely available implemented proof system, HyLoRes , is based on the resolution method. An alternative to resolution is the tableaux method, already widely used for both modal and description logics. Tableaux algorithms have also been developed for a number of hybrid logics, and the goal of the present work is to implement one of them. In this article we present the implementation of a terminating tableaux algorithm for the hybrid logic H ( @ , A ) . The performance of the tableaux algorithm is compared with the performances of HyLoRes , HyLoTab (a system based on a different tableaux algorithm) and RacerPro . HTab is written in the functional language Haskell, using the Glasgow Haskell Compiler (GHC). The code is released under the GNU GPL and can be downloaded from http://hylo.loria.fr/intohylo/htab.php .

The polynomial bounds of proof complexity in Frege systems

The concept of a determinative set of variables for a propositional formula was introduced by one of the authors, which made it possible to distinguish the set of hard-determinable formulas. The proof complexity of a formula of this sort has exponential lower bounds in some proof systems of classical propositional calculus (cut-free sequent system, resolution system, analytic tableaux, cutting planes, and bounded Frege systems). In this paper we prove that the property of hard-determinability is insufficient for obtaining a superpolynomial lower bound of proof lines (sizes) in Frege systems: an example of a sequence of hard-determinable formulas is given whose proof complexities are polynomially bounded in every Frege system.

Terminating Tableau Calculi for Hybrid Logics Extending K

This article builds on work by Bolander and Blackburn [Thomas Bolander and Patrick Blackburn. Termination for hybrid tableaus. Journal of Logic and Computation , 17(3):517–554, 2007] on terminating tableau systems for the minimal hybrid logic K . We provide (for the basic uni-modal hybrid language) terminating tableau systems for a number of non-transitive hybrid logics extending K , such as the logic of irreflexive frames, antisymmetric frames, and so on; these systems don't employ loop-checks. We also provide (for hybrid tense logic enriched with the universal modality) a terminating tableau calculus for the logic of transitive frames; this system makes use of loop-checks.

Termination of rewriting under strategies

A termination proof method for rewriting under strategies, based on an explicit induction on the termination property, is presented and instantiated for the innermost, outermost, and local strategies. Rewriting trees are simulated by proof trees generated with an abstraction mechanism, narrowing and constraints representing sets of ground terms. Abstraction introduces variables to represent normal forms without computing them and to control the narrowing mechanism, well known to easily diverge. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. It is established that termination under strategy is equivalent to the construction of finite proof trees schematizing terminating rewriting trees. Sufficient effective conditions to ensure finiteness are studied and the method is illustrated on several examples for each specific strategy.

Systematic Methods for the Implementation of Nonlinear Wave-Digital Structures

Wave-digital (WD) structures containing adaptors with memory (characterized by port reflection filters) and nonlinear elements are suitable for the modeling of a wide range of nonlinear circuits and physical structures. In this paper, we propose two methods for automating the construction of algorithms that efficiently implement such structures, starting from their symbolic description. The former is based on the solution of state-space equations, while the latter is based on direct structural inspection. The state-space approach starts from the blockwise construction of a tableau matrix for the direct implementation of a generic WD structure and, for this reason, is here referred to as the wave tableau (WT) method. It has very general applicability as it works for a generic WD structure. The second technique (binary connection tree) implements a WD structure through a direct inspection (scanning) of the treelike topological representation of the reference model. Although valid for a slightly less general range of cases, this approach turns out to be much more efficient and flexible than that of the WT method. Such methods are particularly interesting for an interactive and immediate prototyping of physical models for the synthesis of sounds as they bring nonlinear WD structures with dynamic adaptors to a level of practical usability for a wide range of users while enabling the modeling of a wide variety of time-varying nonlinear physical models in an automatic fashion. The proposed solutions have been extensively tested on applications for the automatic modeling of acoustic interactions of musical interest.

PSPACE bounds for rank-1 modal logics

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE . This leads to a unified derivation of tight PSPACE -bounds for a number of logics, including K , KD , coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant proof-theoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

Tableaux and Resource Graphs for Separation Logic

Separation logic (SL) is often presented as an assertion language for reasoning about mutable data structures. As recent results about verification in SL have mainly been achieved from a model-checking point of view, our aim in this article is to study SL from a complementary proof-theoretic perspective in order to provide results about proof search in SL. We begin our study with a fragment of SL, denoted SLP, where first-order quantifiers, variables and equality are removed. We first define specific structures, called resource graphs, that capture SLP models by considering heaps as resources via a labelling process. We then provide a tableau calculus that allows us to build such resource graphs from which either proofs, or countermodels can be generated. We finally prove soundess, completeness and termination of our tableau calculus before discussing extensions to various fragments of SL (including full SL) and the related decidability issues.

A tableau calculus for Propositional Intuitionistic Logic with a refined treatment of nested implications

Since 1993, when Hudelmaier developed an O(n log n)-space decision procedure for propositional Intuitionistic Logic, a lot of work has been done to improve the efficiency of the related proof-search algorithms. In this paper a tableau calculus using the signs T, F and Fc with a new set of rules to treat signed formulas of the kind T((A → B) → C) is provided. The main feature of the calculus is the reduction of both the non-determinism in proof-search and the width of proofs with respect to Hudelmaier's one. These improvements have a significant influence on the performances of the implementation.

Clausal Tableaux for Multimodal Logics of Belief

We develop clausal tableau calculi for six multimodal logics variously designed for reasoning about multi-degree belief, reasoning about distributed systems of belief and for reasoning about epistemic states of agents in multi-agent systems. Our tableau calculi are sound, complete, cut-free and have the analytic superformula property, thereby giving decision procedures for all of these logics. We also use our calculi to obtain complexity results for five of these logics. The complexity of the remaining logic was known.

Visualizing Proof Search for Theorem Prover Development

We describe an interactive visualization tool for large natural deduction proof searches. The tool permits the display of a search as it progresses, as well as the proof tree itself. We discuss the feature set and architecture of the tool, including aspects of extensibility and the interface for interaction with other user-provided analysis and visualization code.

Abduction via C-tableaux and δ-resolution

The formalization of abductive reasoning has received increasing attention from logicians. However, few work is found beyond abduction in propositional logic, given that in a first order formalism, the undecidability problem naturally appears, and therefore an abductive problem cannot even be appropriately formulated. Still, many applications in artificial intelligence allow finite domains to work with, and this gives an opportunity to apply abduction in first order logic with restricted domains. In this paper, we present an approach to abductive reasoning in C-structures, first order structures with a finite domain in which each of its elements has a corresponding constant representing its interpretation. By using semantic tableaux with bounded depth, that is, C-tableaux (a modification of N-tableaux, introduced by the authors in previous papers) and δ-resolution calculus, we build an effective procedure for the searching of minimal abductive solutions within the proposed semantics.

ALC + T: a Preferential Extension of Description Logics

We extend the Description Logic ALC with a operator T that allows us to reason about the prototypical properties and inheritance with exceptions. The resulting logic is called ALC + T. The typicality operator is intended to select the most normal or most instances of a concept. In our framework, knowledge bases may then contain, in addition to ordinary ABoxes and TBoxes, subsumption relations of the form T(C) is subsumed by P, expressing that typical C-members have the property P. The semantics of a typicality operator is defined by a set of postulates that are strongly related to Kraus-Lehmann-Magidor axioms of preferential logic P. We first show that T enjoys a simple semantics provided by ordinary structures equipped with a preference relation. This allows us to obtain a modal interpretation of the typicality operator. We show that the satisfiability of anALC+Tknowledge base is decidable and it is precisely EXPTIME. We then present a tableau calculus for deciding satisfiability of ALC + T knowledge bases. Our calculus gives a (suboptimal) nondeterministic-exponential time decision procedure for ALC + T. We finally discuss how to extend ALC + T in order to infer defeasible properties of (explicit or implicit) individuals. We propose two alternatives: (i) a nonmonotonic completion of a knowledge base; (ii) a model semantics for ALC + T whose intuition is that minimal models are those that maximise typical instances of concepts.

Some Computational Results on MPI Parallel Implementation of Dense Simplex Method

There are two major variants of the Simplex Algorithm: the revised method and the standard, or tableau method. Today, all serious implementations are based on the revised method because it is more efficient for sparse linear programming problems. Moreover, there are a number of applications that lead to dense linear problems so our aim in this paper is to present some computational results on parallel implementation of dense Simplex Method. Our implementation is implemented on a SMP cluster using C programming language and the Message Passing Interface MPI. Preliminary computational results on randomly generated dense linear programs support our results.

Tableaux for Public Announcement Logic

Public announcement logic extends multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. In this article, we propose a labelled tableau calculus for this logic, and show that it decides satisfiability of formulas in deterministic polynomial space. Since this problem is known to be PSPACE-complete, it follows that our proof method is optimal.

Supervaluationism and Logical Consequence: A Third Way

It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness. This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion.

Analytic Methods for the Logic of Proofs

The logic of proofs (lp) was proposed as Godel's missed link between Intuitionistic and S4-proofs, but so far the tableau-based methods proposed for lp have not explored this closeness with S4 and contain rules whose analycity is not immediately evident. We study possible formulations of analytic tableau proof methods for lp that preserve the subformula property. Two sound and complete tableau decision methods of increasing degree of analycity are proposed, KELP and preKELP. The latter is particularly inspired on S4-proofs. The crucial role of proof constants in the structure of lp-proofs methods is analysed. In particular, a method for the abduction of proof constant specifications in strongly analytic preKELP proofs is presented; abduction heuristics and the complexity of the method are discussed.