Automated Synthesis of Tableau Calculi

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.Comment: 32 page

Automated reasoning about metric and topology

In this paper we compare two approaches to automated reasoning about metric and topology in the framework of the logic MT introduced in [10]. MT -formulas are built from set variablesp 1,p 2,... (for arbitrary subsets of a metric space) using the Booleans ∧, ∨, →, and ¬, the distance operators∃ 0 , and the topological interior and closure operators I and C. Intended models for this logic are of the form I=(Δ,d,pI1,pI2,…) where (Δ,d) is a metric space and pIi⊆Δ . The extension φI⊆Δ of an MT -formula ϕ in I is defined inductively in the usual way, with I and C being interpreted as the interior and closure operators induced by the metric, and (∃<aφ)I={x∈Δ∣∃y∈φI d(x,y)<a} . In other words, (Iφ)I is the interior of φI , (∃<aφ)I is the open a-neighbourhood of φI , and (∃≤aφ)I is the closed one. A formula ϕ is satisfiable if there is a model I such that φI≠∅ ; ϕ is valid if ¬ϕ is not satisfiable