Beyond Pure Axioms: Node Creating Rules in Hybrid Tableaux

We present a method of extending the tableau calculus for the basic hybrid language which automatically yields completeness results for many frame classes that cannot be defined by means of pure axioms (for example, Church-Rosser frames). The extended calculus makes use of node-creating rules. These rules trade on the idea of using nominals to perform skolemization on formulas of the strong hybrid language. Alternatively, viewing them from a Hilbert-style perspective, such rules can be viewed as a systematic generalization of Gabbay’s irreflexivity rule. Our completeness result covers all frame classes definable by pure nominal-free universal existential sentences of the strong hybrid language. This properly includes all frame classes definable by universal existential first-order sentences. 1 Basic Hybrid Logic Basic hybrid logic is the result of extending modal logic with nominals and the @-operator. Suppose we are given a set σ of modalities, and two (countably infinite) disjoint sets PROP (whose elements are typically written p, q, and r, possibly subscripted, and called proposition letters) and NOM (whose elements are typically written i, j, k, and l, possibly subscripted, and called nominals). Then the basic hybrid language over σ, PROP and NOM is defined as follows: φ ::= p | i | ¬φ | φ ∧ ψ | 4(φ1, . . . φn) | @iφ. Here p is a proposition letter, i is a nominal, and 4 is an n-ary modality (an element of σ). Thus, except for the clauses for i and @iφ, this is the standard definition of a modal language with arbitrary arity modalities (see, for example, Definition 1.12 in [6]). We follow the usual convention of writing 3φ rather than 4(φ) when working with unary modalities. What do the clauses for i and @iφ give us? Nominals are special proposition letters that are true at precisely one node in any model: they ‘name’, or ‘label’, the unique node they are true at. The @ operator allows us to assert that a formula is true at a named node: @iφ says that φ is true at the node named by the nominal i. In short, by hybridizing the modal language we make it referential: it can now talk about individual nodes in Kripke models. Let’s be precise. A model for the basic hybrid language over σ, PROP, and NOM, is a Kripke model M = (W, (R)4∈σ, V ), such that the valuation V assigns singleton subsets of W to nominals; such valuations are sometimes called hybrid valuations. Apart from this restriction, everything is standard: W is a non-empty set of nodes, and for all 4 in σ, if 4 is an n-ary modality, then R is an n+ 1-ary relation. Following standard terminology we call the pair (W, (R)4∈σ) the frame underlying the model. Given such a hybrid model M, we interpret our language as follows: M, w |= a iff w ∈ V (a), where a ∈ PROP ∪ NOM M, w |= ¬φ iff M, w 6|= φ M, w |= φ ∧ ψ iff M, w |= φ and M, w |= ψ M, w |= 4(φ1, . . . , φn) iff there are v1, . . . , vn ∈ W such that (w, v1, . . . , vn) ∈ R 4 and M, v1 |= φ1 . . . M, vn |= φn M, w |= @iφ iff M, v |= φ, where V (i) = {v} Readers unfamiliar with arbitrary arity modalities should note that in the unary case the clause for modalities simplifies down to the more familiar: M, w |= 3φ iff there is a v ∈ W such that (w, v) ∈ R and M, v |= φ. If M, w |= φ then we say that φ is satisfied in M at w. For any frame F, if φ is satisfied in every model (F, V ) at every w in F no matter which (hybrid) valuation V we choose, then we say that φ is valid on F. A formula is valid if it is valid on every frame. A formula is valid on a class of frames F if it is valid on every frame in F. As promised, the hybridized language is referential. That nominals name is hardwired into the definition of valuations, and the clause for @iφ says “evaluate φ at the node that i names”. Notice that @ij says that the nominals i and j name the same node, that @i3j means that the node named i has the node named j as an R-successor, and that @i 4 (j1, . . . , jn) means that the n+ 1 nodes named i, j1,. . . ,jn, stand in the R 4 relation. It’s worth mentioning that the language we have just defined is a very