And-or tableaux for fixpoint logics with converse: LTL, CTL, PDL and CPDL

Over the last forty years, computer scientists have invented or borrowed numerous logics for reasoning about digital systems. Here, I would like to concentrate on three of them: Linear Time Temporal Logic (LTL), branching time Computation Tree temporal Logic (CTL), and Propositional Dynamic Logic (PDL), with and without converse. More specifically, I would like to present results and techniques on how to solve the satisfiability problem in these logics, with global assumptions, using the tableau method. The issues that arise are the typical tensions between computational complexity, practicality and scalability. This is joint work with Linh Anh Nguyen, Pietro Abate, Linda Postniece, Florian Widmann and Jimmy Thomson

Hypertableau Reasoning for Description Logics

We present a novel reasoning calculus for the description logic SHOIQ^+---a knowledge representation formalism with applications in areas such as the Semantic Web. Unnecessary nondeterminism and the construction of large models are two primary sources of inefficiency in the tableau-based reasoning calculi used in state-of-the-art reasoners. In order to reduce nondeterminism, we base our calculus on hypertableau and hyperresolution calculi, which we extend with a blocking condition to ensure termination. In order to reduce the size of the constructed models, we introduce anywhere pairwise blocking. We also present an improved nominal introduction rule that ensures termination in the presence of nominals, inverse roles, and number restrictions---a combination of DL constructs that has proven notoriously difficult to handle. Our implementation shows significant performance improvements over state-of-the-art reasoners on several well-known ontologies

Reasoning Algebraically with Description Logics

Semantic Web applications based on the Web Ontology Language (OWL) often require the use of numbers in class descriptions for expressing cardinality restrictions on properties or even classes. Some of these cardinalities are specified explicitly, but quite a few are entailed and need to be discovered by reasoning procedures. Due to the Description Logic (DL) foundation of OWL, those reasoning services are offered by DL reasoners. Existing DL reasoners employ reasoning procedures that are arithmetically uninformed and substitute arithmetic reasoning by "don't know" non-determinism in order to cover all possible cases. This lack of information about arithmetic problems dramatically degrades the performance of DL reasoners in many cases, especially with ontologies relying on the use of Nominals and Qualied Cardinality Restrictions. The contribution of this thesis is twofold: on the theoretical level, it presents algebra�ic reasoning with DL (ReAl DL) using a sound, complete, and terminating reasoning procedure for the DL SHOQ. ReAl DL combines tableau reasoning procedures with algebraic methods, namely Integer Programming, to ensure arithmetically better informed reasoning. SHOQ extends the standard DL ALC with transitive roles, role hierarchies, qualified cardinality restrictions (QCRs), and nominals, and forms an expressive subset of OWL. Although the proposed algebraic tableau is double exponential in the worst case, it deals with cardinalities with an additional level of information and properties that make the calculus amenable and well suited for optimizations. In order for ReAl DL to have a practical merit, suited optimizations are proposed towards achieving an efficient reasoning approach that addresses the sources of complexity related to nominals and QCRs. On the practical level, a running prototype reasoner (HARD) is implemented based on the proposed calculus and optimizations. HARD is used to evaluate the practical merit of ReAl DL, as well as the effectiveness of the proposed optimizations. Experimental results based on real world and synthetic ontologies show that ReAl DL outperforms existing reasoning approaches in handling the interactions between nominals and QCRs. ReAl DL also comes with some interesting features such as the ability to handle ontologies with cyclic descriptions without adopting special blocking strategies. ReAl DL can form a basis to provide more efficient reasoning support for ontologies using nominals or QCRs

Tableau-based decision procedure for the multi-agent epistemic logic with all coalitional operators for common and distributed knowledge

We develop a conceptually clear, intuitive, and feasible decision procedure for testing satisfiability in the full multi-agent epistemic logic CMAEL(CD) with operators for common and distributed knowledge for all coalitions of agents mentioned in the language. To that end, we introduce Hintikka structures for CMAEL(CD) and prove that satisfiability in such structures is equivalent to satisfiability in standard models. Using that result, we design an incremental tableau-building procedure that eventually constructs a satisfying Hintikka structure for every satisfiable input set of formulae of CMAEL(CD) and closes for every unsatisfiable input set of formulae.Comment: Substantially extended and corrected version of arXiv:0902.2125. To appear in: Logic Journal of the IGPL, special issue on Formal Aspects of Multi-Agent System

A hybrid ABox calculus using algebraic reasoning for the Description Logic SHIQ

We present a hybrid tableau calculus for the description logic (DL) SHIQ. The presented algorithm decides SHIQ ABox consistency and uses an algebraic approach for more informed reasoning about qualified number restrictions (QNRs). Benefiting from integer linear programming and several optimization techniques to deal with the interaction of QNRs and inverse roles, our approach provides a more deterministic and informed calculus. In addition, a prototype reasoner based on the hybrid calculus has been implemented that decides concept satisfiability for ALCHIQ. We provide a set of benchmarks that demonstrate the effectiveness of our hybrid reasoner in comparison to other DL reasoners