A Tableau Based Decision Procedure for an Expressive Fragment of Hybrid Logic with Binders, Converse and Global Modalities

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 FHL (full hybrid logic with the binder, including the global and converse modalities), where no occurrence of a universal operator is in the scope of a binder. By means of a satisfiability preserving translation of formulae, the calculus can be turned into a satisfiability decision procedure for the fragment FHL \ 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 universal operator.The calculus is based on tableaux 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.…”

“…where b occurs in the branch where b is a fresh nominal Table 1 are the same as those presented in [9], but for the fact that the modal rules (2, 3 and 3 − ) are here reformulated to address the multi-modal case. The equality rule (=) does not add any node to the branch, but modifies the labels of its nodes.…”

“…Treating nominal equalities by means of substitution, like in [6,7,9,11], is essential to ensure such a property. By the effect of substitution, however, distinct node labels may become equal, though the corresponding nodes are still distinct elements of the branch.…”

“…In particular, [9] presents a tableau calculus constituting a satisfiability decision procedure for HL(@, ↓, E, 3 − ) \ 2↓2. Such a procedure is here extended to multi-modal hybrid logic HL m (@, ↓, E, 3 − , Trans, ) \ 2↓2: a tableau calculus is presented, which terminates and is sound and complete for formulae in the fragment HL m (@, ↓, E, 3 − , Trans, ) \ ↓2, i.e.…”

“…The presentation that follows is somewhat simplified, and the reader is referred to [9] for the more formal approach. Let us assume that when the A rule is applied, beyond the premiss shown in Table 1, the branch contains a node called the minor premiss of the rule application (which will be defined further on, in Definition 5).…”