Automated Synthesis of Tableau Calculi

Abstract

Abstract. This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules that can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a well-defined first-order semantics then adding a general blocking mechanism provides a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for a description logic with transitive roles and propositional intuitionistic logic.

Tableau Development for a Bi-intuitionistic Tense Logic

“…This follows by the results of the tableau synthesis framework and atomic rule refinement [ST11,TS13], because we can show the semantics of biskt defined in Section 3 is well-defined in the sense of [ST11].…”

“…We first expressed the semantics in first-order logic and then converted the formulae to inference rules following the tableau synthesis method described in [ST11]. We do not describe the conversion here because it is completely analogous to the conversion for intuitionistic logic considered as a case study in [ST11, Section 9].…”

“…We just note for the rule refinement step in the synthesis process atomic rule refinement as introduced in [TS13] is sufficient. Atomic rule refinement is a specialization of a general rule refinement technique described in [ST11] with the distinct advantage that it is automatic because separate proofs do no need to be given.…”

“…Semantic tableau deduction calculi in particular are easy to develop. In this paper we follow the methodology of tableau calculus synthesis and refinement as introduced in [ST11,TS13] to develop a tableau calculus for the logic. We give soundness, completeness and termination results as consequences of results of tableau synthesis and that biskt has the effective finite model property.…”

Unification in modal and description logics

“…This has been achieved in [41] (improving a previous attempt from [40]), where proof systems for admissibility are presented: in such proof systems, the basic objects are sequent rules, not just ordinary sequents. Another approach is developed in [18]: here methods for synthetizing tableau calculi from [65] are applied to a first-order specification of the class of models used in [61] in order to test rule admissibility.…”