Tableau Algorithm Research Articles

An Efficient Tableau Prover using Global Caching for the Description Logic ALC

We report on our implementation of a tableau prover for the description logic ALC, which is based on the EXPTIME tableau algorithm using global caching for ALC that was developed jointly by us and Gore [9]. The prover, called TGC for Tableaux with Global Caching, checks satisfiability of a set of concepts w.r.t. a set of global assumptions by constructing an and-or graph, using tableau rules for expanding nodes. We have implemented for TGC a special set of optimizations which co-operates very well with global caching and various search strategies. The test results on the test set T98-sat of DL'98 Systems Comparison indicate that TGC is an efficient prover for ALC. This suggests that global caching together with the set of optimizations used for TGC is worth implementing and experimenting also for other modal/description logics.

Model driven engineering of a tableau algorithm for description logics

This paper presents a method for implementing tableau algorithm for description logics (DLs). The architectures of the present DL reasoners such as RACER or FaCT were developed using programming languages as Java or LISP. The implementations are not based on original definition of the abstract syntax, but they require transformation of abstract syntax into concrete syntax implementation languages use. In order to address these issues, we propose the use of model-driven engineering principles for the development of a DL reasoner where a definition of a DL abstract syntax is provided by means of metamodels. The presented approach is based on the use of a MOF-based model repository and QVT-like transformations, which transform models compliant to the DL metamodel taken from the OMG's Ontology Definition Metamodel specification into models compliant to the Tableau metamodel defined in this paper. .

Axiom Pinpointing in General Tableaux

Axiom pinpointing has been introduced in description logics (DLs) to help the user to understand the reasons why consequences hold and to remove unwanted consequences by computing minimal (maximal) subsets of the knowledge base that have (do not have) the consequence in question. Most of the pinpointing algorithms described in the DLliterature are obtained as extensions of the standard tableau-based reasoning algorithms for computing consequences from DL knowledge bases. Although these extensions are based on similar ideas, they are all introduced for a particular tableau-based algorithm for a particular DL. The purpose of this article is to develop a general approach for extending a tableau-based algorithm to a pinpointing algorithm. This approach is based on a general definition of ‘tableau algorithms,’ which captures many of the known tableau-based algorithms employed in DLs, but also other kinds of reasoning procedures.

Description Logic extension based on extenics theory

Traditional Description Logic (DL) does not fit to handle the problems with incomplete information, tracit knowledge or even contradiction premise. Therefore it is not sufficient to be the logical foundation of the Semantic Web. For this reason, the matter-element and divergence rules of Extenics were introduced to extend the traditional DL. Firstly, the semantic explanation of matter-element was given. Then the matter-element and divergence rules were used to extend the Tableau algorithm to be new Tableau-E algorithm and Tableau-E'algorithm, thus realizing the extension and consistency checking of Abox (assertion of individuals), and making up the deficiency of traditional DL.

Reasoning within extended fuzzy description logic

An extended fuzzy description logic is proposed to increase expressive power for complex fuzzy information. We introduce cut sets of the fuzzy concepts and fuzzy roles as atomic concepts and atomic roles, and inherit concept constructors of the description logics to support a new logic system for fuzzy knowledge representation. We define the syntax, semantics and knowledge base of the extended fuzzy description logic, and focus on sat-domain and consistency as main reasoning tasks. We also present sound and complete tableau algorithms for these reasoning tasks, and prove the complexity of them is PSPACE-complete. Compared with the other fuzzy description logics, the extended fuzzy description logic can express more wide fuzzy information.

Inconsistency-tolerant description logic. Part II: A tableau algorithm for [formula omitted

In the first part of this paper, we motivated and defined three systems of constructive and inconsistency-tolerant description logic. The variety of arising systems is conditioned by the variety of approaches to defining modalities in the constructive setting. We also presented sound and complete tableau calculi for the logics under consideration. Whereas these calculi were not meant to give rise to tableau algorithms, in the present second part of the paper, after providing some motivation and recalling the main definitions, we adapt methods developed by R. Dyckhoff and by I. Horrocks and U. Sattler in order to define a tableau algorithm for our basic four-valued constructive description logic CALC C . Notice that among the three logics defined in the first part of the paper, CALC C is the only logic which lends itself to applications, because for the other logics it is unknown whether they are elementarily decidable. The presented algorithm for CALC C is the first example of an elementary decision procedure for a constructive description logic.

Optimizing Terminological Reasoning for Expressive Description Logics

Tableau algorithms are currently the most widely used and empirically the fastest algorithms for reasoning in expressive description logics, including the important description logics $\mathcal{SHIQ}$ and $\mathcal{SHOIQ}$ . Achieving a high level of performance on terminological reasoning in expressive description logics when using tableau-based algorithms requires the incorporation of a wide variety of optimizations. The description logic system FaCT++ implements a wide variety of such optimizations, some present in other reasoners and some novel or refined in FaCT++.

A Tableau Algorithm for Description Logics with Concrete Domains and General TBoxes

In order to use description logics (DLs) in an application, it is crucial to identify a DL that is sufficiently expressive to represent the relevant notions of the application domain, but for which reasoning is still decidable. Two means of expressivity required by many modern applications of DLs are concrete domains and general TBoxes. The former are used for defining concepts based on concrete qualities of their instances such as the weight, age, duration, and spatial extension. The purpose of the latter is to capture background knowledge by stating that the extension of a concept is included in the extension of another concept. Unfortunately, combining concrete domains with general TBoxes often leads to DLs for which reasoning is undecidable. In this paper, we identify a general property of concrete domains that is sufficient for proving decidability of DLs with both concrete domains and general TBoxes. We exhibit some useful concrete domains, most notably a spatial one based on the RCC-8 relations that have this property. Then, we present a tableau algorithm for reasoning in DLs equipped with concrete domains and general TBoxes.

A Modal Supervaluation Description Logic for Characterization of Vague Concepts: its semantics and a tableau algorithm for it

A Modal Supervaluation Description Logic for Characterization of Vague Concepts: its semantics and a tableau algorithm for it

On Computational Complexity of the Extended Fuzzy Description Logic with Numerical Restriction

Extended fuzzy description logic EFALCN (extended fuzzy attributive concept description language with complements and unqualified number restriction) is the fuzzy extension of the description logic with numerical restriction ALCN (attributive concept description language with complements and unqualified number restriction), but it lacks of reasoning algorithms and complexity analysis for reasoning tasks. In this paper, a constraint-propagation based tableau algorithm is proposed, and it is proved that this algorithm can be executed in PSPACE (polynomial space). For there is a polynomial time reduction that can reduce ALCN reasoning tasks into EFALCN reasoning tasks and ALCN reasoning tasks are PSPACE-complete, EFALCN reasoning tasks are

An Overview of Tableau Algorithms for Description Logics

Description logics are a family of knowledge representation formalisms that are descended from semantic networks and frames via the system Kl-one. During the last decade, it has been shown that the important reasoning problems (like subsumption and satisfiability) in a great variety of description logics can be decided using tableau-like algorithms. This is not very surprising since description logics have turned out to be closely related to propositional modal logics and logics of programs (such as propositional dynamic logic), for which tableau procedures have been quite successful. Nevertheless, due to different underlying institutions and applications, most description logics differ significantly from run-of-the-mill modal and program logics. Consequently, the research on tableau algorithms in description logics led to new techniques and results, which are, however, also of interest for modal logicians. In this article, we will focus on three features that play an important role in description logics (number restrictions, terminological axioms, and role constructors), and show how they can be taken into account by tableau algorithms.

Generating oracles from your favorite temporal logic specifications

This paper describes a generic tableau algorithm, which is the basis for a general customizable method for producing oracles from temporal logic specifications. A generic argument gives semantic rules with which to build the semantic tableau for a specification. Parameterizing the tableau algorithm by semantic rules permits it to easily accommodate a variety of temporal operators and provides a clean mechanism for fine-tuning the algorithm to produce efficient oracles.The paper develops conditions to ensure that a set of rules results in a correct tableau procedure. It gives sample rules for a variety of linear-time temporal operators and shows how rules are tailored to reduce the size of an oracle.

Automatic verification of distributed logic specifications

Formal specification is vital to the development of distributed real-time systems as these systems are inherently complex and safety-critical. It is widely acknowledged that formal specification and automatic analysdis of specifications can significantly increase system reliability. Although a number of specification techniques for real-time systems have been reported in the literature, most of these formalisms do not adequately address to the constraints that the aspects of ‘distribution’ and ‘real-time’ impose on specifications. Further, an automatic verification tool is necessary to reduce human errors in the reasoning process. In this regard, this paper is an attempt towards the development of a novel executable specification language for distributed real-time systems. First, we give a precise characterization of the syntax and semantics of DL. Subsequently, we discuss the problems of model checking, automatic verification of satisfiability of DL specifications, and testing conformance of event traces with DL specifications. Effective solutions to these problems are presented as extensions to the classical first-order tableau algorithm. The use of the proposed framework is illustrated by specifying a sample problem.

Foundations of a theorem prover for functional and mathematical uses

ABSTRACT A computational logic, PLPR (Predicate Logic using Polymorphism and Recursion) is presented. Actually this logic is the object language of an automated deduction system designed as a tool for proving mathematical theorems as well as specify and verify properties of functional programs. A useful denotationl semantics and two general deduction methods for PLPR are defined. The first one is a tableau algorithm proved to be complete and also used as a guideline for building complete calculi. The second is a sound and complete natural deduction system. Moreover a fixed point induction rule is introduced for formulas called continuous. The strategies for mechanizing the proofs of the final automated system are based on the previous deduction methods. As the examples of the use of the system show, the implemented theorem prover outperforms humans to a certain extent, retaining logic and calculi generality.

The expected complexity of analytic tableaux analyses in propositional calculus ? II

The article continues a study of the complexity of an analytic tableaux algorithm for SAT. The main result is that the average number of branches in analytic tableaux of formulae of lengthn isΘ((1.09988...)n). The maximum number of branches is also studied. Both the average and worst case complexity measures are used to compare analytic tableaux and truth tables.

Tableau algorithms for factor analysis by instrumental variable methods

Simple tableau algorithms are given for the noniterative instrumental variable (FABIN 2) and two stage regression (FABIN 3) factor loading estimates of Hagglund. Corresponding generalized least squares estimates of factor covariances and unique variances are introduced. An example is given for the purpose of illustration and comparison.