Tableaux for Quantified Hybrid Logic

Abstract

Abstract. We present a (sound and complete) tableau calculus for Quantified Hybrid Logic (QHL). QHL is an extension of orthodox quantified modal logic: as well as the usual ✷ and ✸ modalities it contains names for (and variables over) states, operators @s for asserting that a formula holds at a named state, and a binder ↓ that binds a variable to the current state. The first-order component contains equality and rigid and non-rigid designators. As far as we are aware, ours is the first tableau system for QHL. Completeness is established via a variant of the standard translation to first-order logic. More concretely, a valid QHL-sentence is translated into a valid first-order sentence in the correspondence language. As it is valid, there exists a first-order tableau proof for it. This tableau proof is then converted into a QHL tableau proof for the original sentence. In this way we recycle a well-known result (completeness of first-order logic) instead of a well-known proof. The tableau calculus is highly flexible. We only present it for the constant domain semantics, but slight changes render it complete for varying, expanding or contracting domains. Moreover, completeness with respect to specific frame classes can be obtained simply by adding extra rules or axioms (this can be done for every first-order definable class of frames which is closed under and reflects generated subframes).

An efficient approach to nominal equalities in hybrid logic tableaux

“…They are essentially the same rules of other internalized calculi for HL(@) [12,9,11,31] modulo a reformulation of the calculi from the "nodes as formulae" to the "nodes as sets" style. The boolean, modal and label rules are called logical rules.…”

“…The tableau system we are now going to present, and which will be called H, "internalizes" prefixes, like in [9,11,13], (partially) in [31], one of the systems in [14] and one of the calculi in [12]. Tableau nodes are in fact labelled by sets of satisfaction statements, i.e.…”

“…The tableau calculus proposed for first-order HL(@, ∀, ∃) in [11] (an extension of [9], which treats only the propositional case) has quite "natural" rules for treating nominal equalities (reflexivity, a rule which "passes" formulae from a nominal to an equal one and a rule that makes worlds, equal to a state seen by a given one, also visible by it). However, such a treatment of equalities, although very intuitive and natural, leads to a non terminating calculus even for the propositional (decidable) fragment HL(@).…”

“…Several proof systems have been defined for hybrid logics (see, for instance, [4,9,11,12,14,15,26]) and some provers have also been implemented, among which [5,18,21,22,30]. Since nominals and the satisfaction operator are the main peculiarities of hybrid languages, it is important to devise efficient ways of treating them.…”

Coalgebraic Hybrid Logic

“…The sequent calculus shares one characteristic trait with the tableaux calculus studied in [6]: sequents are composed of @-prefixed formulas. As a consequence, the sequent calculus presented here can be understood as (the dual of) a labelled tableau [34] by reading an @-prefixed formula @ i φ as a labelled sequent i : φ.…”