Finite and Algorithmic Model Theory

280 Martin Ottotranslation that associates p j with P j x and ♦ α ψ with ∃y(R α xy ∧ ψ(y)) so that,dually, □ α ψ is associated with ∀y(R α xy → ψ(y)). This is briefly reviewed inconnection with the model checking game for modal logic in section 6.2.3.The extension of basic modal logic with modal quantification backwardalong E α (inverse modalities) is denoted ML − ; the extension by a global modality,corresponding to the introduction of modal quantification associated withthe full binary relation, is denoted ML ∀ ; the combined extension with both theseadditions is ML − ∀. For background in connection with our treatment of modallogics and much more material on the model theory of modal logics see inparticular [17].The guarded fragment GF is defined to be a syntactic fragment of FOconsisting of formulae in which all quantifications are relativised as inϕ(x) =∃y(α(x ′ ) ∧ ψ(x ′ )), orϕ(x) =∀y(α(x ′ ) → ψ(x ′ )),where α(x ′ )isanatomicτ-formula (a relational atom, or an equality: theguard atom) such that free(ψ) ⊆ var(α) (and y is a sub-tuple of x ′ such that[x ′ ] \ [y] ⊆ [x]).The quantification pattern of guarded logic extends that of modal logic.For a modal vocabulary τ, GF[τ] properly contains (the standard first-ordertranslations of) ML[τ] and even ML − ∀[τ]. One motivation for the study of theguarded fragment stems from the analogy with modal logic, and the extensionof modal quantification patterns from Kripke structures to more generalrelational structures. Guarded fragments were proposed in [2] with a view toexplaining the good algorithmic and model theoretic properties of modal logicsin a richer fragment of first-order logic and other than the 2-variable fragment[23]; see [21]. In many ways the guarded fragment has been shown to be arather well-behaved intermediary between first-order and modal logic, in termsof its model theoretic and algorithmic properties. For instance (like modal logicand unlike FO k for k 3), GF has the finite model property and is decidable:the satisfiability problem for GF[τ] is complete for deterministic exponentialtime if τ is fixed (more precisely, for any fixed bound on the width of τ), andcomplete for doubly exponential time without this constraint [21]. Similarly tothe tree model property of modal logic (which is a consequence of bisimulationinvariance and the model transformation of tree unfolding, see in particularsection 6.3.1), GF has a generalised tree model property, which similarly stemsfrom invariance under guarded bisimulation and the availability of guarded treeunfoldings. For these considerations we refer to the discussion in section 6.4.2,