A Tableaux Based Decision Procedure for a Broad Class of Hybrid Formulae with Binders

Abstract

In this paper we provide the first (as far as we know) direct calculus deciding satisfiability of formulae in negation normal form in the fragment of hybrid logic with the satisfaction operator and the binder, where no occurrence of the 2 operator is in the scope of a binder. A preprocessing step, rewriting formulae into equisatisfiable ones, turns the calculus into a satisfiability decision procedure for the fragment HL(@, ↓) \2↓2, i.e. formulae in negation normal form where no occurrence of the binder is both in the scope of and contains in its scope a 2 operator.The calculus is based on tableaux, where nominal equalities are treated by means of substitution, and termination is achieved by means of a form of anywhere blocking with indirect blocking. Direct blocking is a relation between nodes in a tableau branch, holding whenever the respective labels (formulae) are equal up to (a proper form of) nominal renaming. Indirect blocking is based on a partial order on the nodes of a tableau branch, which arranges them into a tree-like structure.

A Proof Procedure for Hybrid Logic with Binders, Transitivity and Relation Hierarchies

“…This work is a continuation of previous works, where terminating tableau calculi for decidable fragments of Hybrid Logic with the binder have been defined [8,9]. In particular, [9] presents a tableau calculus constituting a satisfiability decision procedure for HL(@, ↓, E, 3 − ) \ 2↓2.…”

“…The above reported arguments showing decidability of fragments of hybrid logic with binders are all of semantical nature. The first proof procedures constituting satisfiability decision procedures for such fragments are defined in [7,8]. In particular, [8] presents a tableau based satisfiability decision procedure for HL(@, ↓, E, ✸ − ) \ ✷↓✷, and such a procedure is extended to multi-modal hybrid logic HL m (@, ↓, E, ✸ − , Trans, ⊑ ) \ ✷↓✷ in [9].…”

A Prover Dealing with Nominals, Binders, Transitivity and Relation Hierarchies

“…A sound and complete satisfiability decision procedure for HL \ 2↓2 is defined in [16] (extending [12][13][14], where proof procedures for sublogics of HL\2↓2 are introduced), thus showing that this rich fragment of hybrid logic, subsuming the description logic SHOIQ, is decidable. This work describes the Sibyl prover, an implementation of the above mentioned proof procedure.…”

GUBS, a Behaviour-Based Language for Design in Synthetic Biology

“…An unobservable program can be assimilated to a programming error. The detection of such errors can be carried out at compile time using tableaux method [10] that automatically determines whether a formula is satisfiable in a model. Indeed Gubs uses a fragment of H(A, @) named HL(@) logic which is decidable.…”