Mark Kaminski

Terminating Tableau Systems for Hybrid Logic with Difference and Converse

This paper contributes to the principled construction of tableau-based decision procedures for hybrid logic with global, difference, and converse modalities. We also consider reflexive and transitive relations. For converse-free formulas we present a terminating control that does not rely on the usual chain-based blocking scheme. Our tableau systems are based on a new model existence theorem.

Spartacus: A Tableau Prover for Hybrid Logic

Spartacus is a tableau prover for hybrid multimodal logic with global modalities and reflexive and transitive relations. Spartacus is the first system to use pattern-based blocking for termination. To achieve a competitive performance, Spartacus implements a number of optimization techniques, including a new technique that we call lazy branching. We evaluate the practical impact of pattern-based blocking and lazy branching for the basic modal logic K and observe high effectiveness of both techniques.

Terminating tableaux for hybrid logic with eventualities

We present the first terminating tableau system for hybrid logic with eventualities. The system is designed as a basis for gracefully degrading reasoners. Eventualities are formulas of the form

Terminating Tableaux for Graded Hybrid Logic with Global Modalities and Role Hierarchies

\Diamond^*s

Correctness and worst-case optimality of Pratt-style decision procedures for modal and hybrid logics

that hold for a state if it can reach in n≥0 steps a state satisfying the formula s. The system is prefix-free and employs a novel clausal form that abstracts away from propositional reasoning. It comes with an elegant correctness proof. We discuss some optimizations for decision procedures.

Terminating Tableaux for \(\mathcal{SOQ}\) with Number Restrictions on Transitive Roles

We present a terminating tableau calculus for graded hybrid logic with global modalities, reflexivity, transitivity and role hierarchies. Termination of the system is achieved through pattern-based blocking. Previous approaches to related logics all rely on chain-based blocking. Besides being conceptually simple and suitable for efficient implementation, the pattern-based approach gives us a NExpTime complexity bound for the decision procedure.

Clausal graph tableaux for hybrid logic with eventualities and difference

We extend Pratts worst-case optimal decision procedure for PDL to a richer logic with nominals, difference modalities, and inverse actions. We prove correctness and worst-case optimality. Our correctness proof is based on syntactic models called demos. The main theorem states that a formula is satisfiable if and only if it is contained in a demo. From this theorem the correctness of the decision procedure is easily obtained. Our development is modular and we extend it stepwise from modal logic with eventualities to the full logic.

Terminating Tableaux for Hybrid Logic with the Difference Modality and Converse

We show that the description logic \(\mathcal{SOQ}\) with number restrictions on transitive roles is decidable by a terminating tableau calculus. The language decided by the calculus includes the universal role, which allows us to internalize TBox axioms. Termination of the system is achieved through pattern-based blocking.

Clausal Tableaux for Hybrid PDL

We introduce the method of clausal graph tableaux at the example of hybrid logic with difference and star modalities. Clausal graph tableaux are prefix-free and terminate by construction. They provide an abstract method of establishing the small model property of modal logics. In contrast to the filtration method, clausal graph tableaux result in goal-directed decision procedures. Until now no goal-directed decision procedure for the logic considered in this paper was known. There is the promise that clausal graph tableaux lead to a new class of effective decision procedures.

Hybrid Tableaux for the Difference Modality

A camera which permits automatic and manual focusing operation includes an objective lens structure capable of assuming a condition ready for automatic focusing and a condition ready for manual focusing. The objective lens structure has a focusing lens, a manual focusing operating member capable of moving the focusing lens by extraneous operation when the lens structure is in the condition ready for the manual focusing, and signal means for indicating which of the two conditions the lens structure assumes. The camera further includes an automatically focusing device for measuring the distance to an object to be photographed and generating an output for moving the focusing lens in accordance with the measured distance, and means responsive to the signal means to render the automatically focusing device inoperative when the signal means is indicating the condition ready for manual focusing.