Tableau-based Algorithm Research Articles

Efficient TBox Reasoning with Value Restrictions using the wer Reasoner

AbstractThe inexpressive Description Logic (DL) ${\cal F}{{\cal L}_0}$ , which has conjunction and value restriction as its only concept constructors, had fallen into disrepute when it turned out that reasoning in ${\cal F}{{\cal L}_0}$ w.r.t. general TBoxes is ExpTime-complete, that is, as hard as in the considerably more expressive logic ${\cal A}{\cal L}{\cal C}$ . In this paper, we rehabilitate ${\cal F}{{\cal L}_0}$ by presenting a dedicated subsumption algorithm for ${\cal F}{{\cal L}_0}$ , which is much simpler than the tableau-based algorithms employed by highly optimized DL reasoners. Our experiments show that the performance of our novel algorithm, as prototypically implemented in our ${\cal F}{{\cal L}_0}$ wer reasoner, compares very well with that of the highly optimized reasoners. ${\cal F}{{\cal L}_0}$ wer can also deal with ontologies written in the extension ${\cal F}{{\cal L}_ \bot }$ of ${\cal F}{{\cal L}_0}$ with the top and the bottom concept by employing a polynomial-time reduction, shown in this paper, which eliminates top and bottom. We also investigate the complexity of reasoning in DLs related to the Horn-fragments of ${\cal F}{{\cal L}_0}$ and ${\cal F}{{\cal L}_ \bot }$ .

Satisfiability Checking of Clinical Practice Guidelines Using an Analyzer

: A clinical practice guideline consists of the best practices required for managing a particular disease. Designing a consistent guideline is difficult and error-prone; hence, checking the consistency of guidelines is crucial. Due to the complexity of guidelines, a formal language is an appropriate choice for modeling and analyzing a guideline. IMPNL has been introduced as a metric interval-based temporal logic to model such guidelines. Moreover, a sound and complete tableau-based algorithm has been designed for checking the satisfiability of an IMPNL formula. In this paper, we introduced a clinical practice guideline analyzer suitable for modeling and checking the consistency of a guideline. The analyzer can also determine points, in which inconsistencies occur, and help designers to quickly and easily fix a guideline. Moreover, physicians can use the output of the analyzer (the calendar model) to check whether a patient is coherently treated with a specific guideline.

A Note on a Description Logic of Concept and Role Typicality for Defeasible Reasoning Over Ontologies

In this work, we propose a meaningful extension of description logics for non-monotonic reasoning. We introduce $$\mathcal {ALCH}^{\bullet }$$ , a logic allowing for the representation of and reasoning about both typical class-membership and typical instances of a relation. We propose a preferential semantics for $$\mathcal {ALCH}^{\bullet }$$ in terms of partially-ordered DL interpretations which intuitively captures the notions of typicality we are interested in. We define a tableau-based algorithm for checking $$\mathcal {ALCH}^{\bullet }$$ knowledge-base consistency that always terminates and we show that it is sound and complete w.r.t. our preferential semantics. The general framework we here propose can serve as the foundation for further exploration of non-monotonic reasoning in description logics and similarly structured logics.

Minimally inconsistent reasoning in Semantic Web.

Reasoning with inconsistencies is an important issue for Semantic Web as imperfect information is unavoidable in real applications. For this, different paraconsistent approaches, due to their capacity to draw as nontrivial conclusions by tolerating inconsistencies, have been proposed to reason with inconsistent description logic knowledge bases. However, existing paraconsistent approaches are often criticized for being too skeptical. To this end, this paper presents a non-monotonic paraconsistent version of description logic reasoning, called minimally inconsistent reasoning, where inconsistencies tolerated in the reasoning are minimized so that more reasonable conclusions can be inferred. Some desirable properties are studied, which shows that the new semantics inherits advantages of both non-monotonic reasoning and paraconsistent reasoning. A complete and sound tableau-based algorithm, called multi-valued tableaux, is developed to capture the minimally inconsistent reasoning. In fact, the tableaux algorithm is designed, as a framework for multi-valued DL, to allow for different underlying paraconsistent semantics, with the mere difference in the clash conditions. Finally, the complexity of minimally inconsistent description logic reasoning is shown on the same level as the (classical) description logic reasoning.

Conversion Algorithm of Linear-Time Temporal Logic to Buchi Automata

The size of Buchi Automata(BA) is a key factor during converting Linear-Time Temporal Logic(LTL) formulae to BA in model checking. Most algorithms forgenerating BA from LTL formulas involved intermediate automata, degeneralization algorithm and simplification of the formulas, but size of BA and time of converting can be reduced further. In this paper, we present an improved Tableau-based algorithm, which converts LTL formulae to Transition-based Buchi Automata(TBA) more efficiently. The algorithm is geared towards being used in model checking in on-the-fly fashion. By attaching the satisfyinformation of U-formula on states and transitions, we can decide whether the sequences of the BA are acceptableby using only one acceptance condition set, not multiple ones. Binary Decision Diagrams(BDD) is used to describeBA and simplify formulae. A better data structure, syntax Directed Acyclic Graph(DAG), is adopted in the algorithm. The size of product BA and computational complexity can be reduced significantly by using on-the-fly degeneralization. The algorithm can expand the state nodes while detecting the validity of nodes, removing the invalid nodes and combining equivalent states and transitions. Compared with other recent conversion tools, the algorithm proposed in this paper runs faster, with the smaller size of BA.

Beth Definability in Expressive Description Logics

The Beth definability property, a well-known property from classical logic, is investigated in the context of description logics: if a general L-TBox implicitly defines an L-concept in terms of a given signature, where L is a description logic, then does there always exist over this signature an explicit definition in L for the concept? This property has been studied before and used to optimize reasoning in description logics. In this paper a complete classification of Beth definability is provided for extensions of the basic description logic ALC with transitive roles, inverse roles, role hierarchies, and/or functionality restrictions, both on arbitrary and on finite structures. Moreover, we present a tableau-based algorithm which computes explicit definitions of at most double exponential size. This algorithm is optimal because it is also shown that the smallest explicit definition of an implicitly defined concept may be double exponentially long in the size of the input TBox. Finally, if explicit definitions are allowed to be expressed in first-order logic, then we show how to compute them in single exponential time.

Formally Verified Tableau-Based Reasoners for a Description Logic

Description Logics are a family of logics used to represent and reason about conceptual and terminological knowledge. One of the most basic description logics is ${\mathcal{ALC}}$ , used as a basis from which to obtain others. Description logics are particularly important to provide a logical basis for the web ontology languages (such as OWL) used in the Semantic Web. In order to increase the reliability of the Semantic Web, formal methods can be applied, and in particular formal verification of its reasoning services can be carried out. In this paper, we present the formal verification of a tableau-based satisfiability algorithm for the logic ${\mathcal{ALC}}$ . The verification has been completed in several stages. First, we develop an abstract formalization of satisfiability-checking of ${\mathcal{ALC}}$ -concepts. Secondly, we define and formally verify a tableau-based algorithm in which the order of rule application and branch selection can be flexibly specified, using a methodology of refinements to transfer the main properties from the ${\mathcal{ALC}}$ abstract formalization. Finally, we obtain verified and executable reasoners from the algorithm via a process of instantiation.

An On-the-fly Tableau-based Decision Procedure for PDL-satisfiability

We present a tableau-based algorithm for deciding satisfiability for propositional dynamic logic ( PDL ) which builds a finite rooted tree with ancestor loops and passes extra information from children to parents to separate good loops from bad loops during backtracking. It is easy to implement, with potential for parallelisation, because it constructs a pseudo-model “on the fly” by exploring each tableau branch independently. But its worst-case behaviour is 2EXPTIME rather than EXPTIME. A prototype implementation in the TWB ( http://twb.rsise.anu.edu.au ) is available.

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.

Automata can show PS pace results for description logics

In the area of Description Logic (DL), both tableau-based and automata-based algorithms are frequently used to show decidability and complexity results for basic inference problems such as satisfiability of concepts. Whereas tableau-based algorithms usually yield worst-case optimal algorithms in the case of PS pace-complete logics, it is often very hard to design optimal tableau-based algorithms for E xpT ime-complete DLs. In contrast, the automata-based approach is usually well-suited to prove E xpT ime upper-bounds, but its direct application will usually also yield an E xpT ime-algorithm for a PS pace-complete logic since the (tree) automaton constructed for a given concept is usually exponentially large. In the present paper, we formulate conditions under which an on-the-fly construction of such an exponentially large automaton can be used to obtain a PS pace-algorithm. We illustrate the usefulness of this approach by proving a new PS pace upper-bound for satisfiability of concepts with respect to acyclic terminologies in the DL SI , which extends the basic DL ALC with transitive and inverse roles.

Constructing Formally Verified Reasoners for the [formula omitted] Description Logic

Description Logics are a family of logics used to represent and reason about conceptual and terminological knowledge. Recently, its importance has been increased since they are used as a basis for the Ontology Web Language (OWL) used for the Semantic Web. In previous work, we have developed in PVS a generic framework for reasoning in the ALC description logic, proving its termination, soundness and completeness. In this paper we present the construction, from the generic framework, of a formally verified generic tableau-based algorithm for checking satisfiability of ALC-concepts. We do it using a methodology of refinements to transfer the properties from the framework to the algorithm. We also obtain some verified reasoners from the algorithm by a process of instantiation.

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++.

Tableau-based automata construction for dynamic linear time temporal logic*

We present a tableau-based algorithm for obtaining a Buchi automaton from a formula in Dynamic Linear Time Temporal Logic (DLTL), a logic which extends LTL by indexing the until operator with regular programs. The construction of the states of the automaton is similar to the standard construction for LTL, but a different technique must be used to verify the fulfillment of until formulas. The resulting automaton is a Buchi automaton rather than a generalized one. The construction can be done on-the-fly, while checking for the emptiness of the automaton. We also extend the construction to the Product Version of DLTL.

A Tableau Decision Algorithm for Modalized ALC with Constant Domains

The aim of this paper is to construct a tableau decision algorithm for the modal description logic K ALC with constant domains. More precisely, we present a tableau procedure that is capable of deciding, given an ALC-formula ϕ with extra modal operators (which are applied only to concepts and TBox axioms, but not to roles), whether ϕ is satisfiable in a model with constant domains and arbitrary accessibility relations. Tableau-based algorithms have been shown to be 'practical' even for logics of rather high complexity. This gives us grounds to believe that, although the satisfiability problem for K ALC is known to be NEXPTIME-complete, by providing a tableau decision algorithm we demonstrate that highly expressive description logics with modal operators have a chance to be implementable. The paper gives a solution to an open problem of Baader and Laux [5].