Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles and functional restrictions with respect to general concept inclusion axioms and role hierarchies; early experiments indicate that this algorithm is well-suited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSpace. We investigate the limits of decidability for this family of DLs, showing that relaxing the constraints placed on the kinds of roles used in number restrictions leads to the undecidability of all inference problems. Finally, we describe a number of optimisation techniques that are crucial in obtaining implementations of the decision procedures, which, despite the hight worst-case complexity of the problem, exhibit good performance with real-life problems. 1

Abstract. FaCT is a Description Logic classifier which has been implemented as a test-bed for a highly optimised tableaux satisfiability (subsumption) testing algorithm. The correspondence between modal and description logics also allows FaCT to be used as a theorem prover for the propositional modal logics K, KT, K4 and S4. Empirical tests have demonstrated the effectiveness of the optimised implementation and, in particular, of the dependency directed backtracking optimisation. 1

Effective optimisation techniques can make a dramatic difference in the performance of knowledge representation systems based on expressive description logics. With currently-available desktop computers, systems that incorporate these techniques can effectively reason in description logics with intractable inference. Because of the correspondence between description logics and propositional modal logic, difficult problems in propositional modal logic can be effectively solved using the same techniques.

DLP (Description Logic Prover) is an experimental description logic knowledge representation system. DLP currently implements a superset of propositional dynamic logic as well as K (m) and KT (m) . Although DLP is an experimental system, it nonetheless provides a fast satisfiability checker for the above propositional modal logics as well as a fast reasonder for knowledge bases.

Abstract. While CNF propositional satisfiability (SAT) is a sub-class of the more general constraint satisfaction problem (CSP), conventional wisdom has it that some well-known CSP look-back techniques-- including backjumping and learning-- are of little use for SAT. We enhance the Tableau SAT algorithm of Crawford and Auton with look-back techniques and evaluate its performance on problems specifically designed to challenge it. The Random 3-SAT problem space has commonly been used to benchmark SAT algorithms because consistently difficult instances can be found near a region known as the phase transition. We modify Random 3-SAT in two ways which make instances even harder. First, we evaluate problems with structural regularities and find that CSP look-back techniques offer little advantage. Second, we evaluate problems in which a hard unsatisfiable instance of medium size is embedded in a larger instance, and we find the look-back enhancements to be indispensable. Without them, most instances are “exceptionally hard ”-orders of magnitude harder than typical Random 3-SAT instances with the same surface characteristics.

The performance of backtracking algorithms for solving finite-domain constraint satisfaction problems can be improved substantially by look-back and look-ahead methods. Look-back techniques extract information by analyzing failing search paths that are terminated by dead-ends. Look-ahead techniques use constraint propagation algorithms to avoid such dead-ends altogether. This survey describes a number of look-back variants including backjumping and constraint recording which recognize and avoid some unnecessary explorations of the search space. The last portion of the paper gives an overview of look-ahead methods such as forward checking and dynamic variable ordering, and discusses their combination with backjumping.

Over the past twenty veyears many backtracking algorithms have been developed for constraint satisfaction problems. This survey describes the basic backtrack search within the search space framework and then presents a number of improvements developed in the past two decades, including look-back methods such asbackjumping, constraint recording, backmarking, and look-ahead methods such as forward checking and dynamic variable ordering. 1

Using an improved backtrack algorithm with sophisticated pruning techniques, we revise previous observations correlating a high frequency of hard to solve Hamiltonian cycle instances with the G n;m phase transition between Hamiltonicity and non-Hamiltonicity. Instead all tested graphs of 100 to 1500 vertices are easily solved. When we artificially restrict the degree sequence with a bounded maximum degree, although there is some increase in difficulty, the frequency of hard graphs is still low. When we consider more regular graphs based on a generalization of knight's tours, we observe frequent instances of really hard graphs, but on these the average degree is bounded by a constant. We design a set of graphs with a feature our algorithm is unable to detect and so are very hard for our algorithm, but in these we can vary the average degree from O(1) to O(n). We have so far found no class of graphs correlated with the G n;m phase transition which asymptotically produces a high frequenc...

This chapter will discuss the implementation of the reasoning services which form the core of Description Logic based Knowledge Representation Systems. To be useful in realistic applications, such systems need both expressive logics and fast reasoners. As expressive logics inevitably have high worst-case complexities, this can only be achieved by employing highly optimised implementations of suitable reasoning algorithms. Systems based on such implementations have demonstrated that they can perform well with problems that occur in realistic applications, including problems where unoptimised reasoning is hopelessly intractable.

Effective systems for expressive description logics require a heavily-optimised subsumption checker incorporating a range of optimisation techniques. Because of the correspondence between description logics and propositional modal logic most of these techniques carry over into propositional modal logic satisfiability checking. Some of the techniques are extremely effective on various test suites for propositional modal satisfiability and others are less effective. Further, the effectiveness of a technique depends on the test performed. Description logic systems spend much of their time computing subsumption relationships between descriptions. If the system is based on an expressive description logic then the