Expressive completeness through logically tractable models

Abstract

How can we prove that some fragment of a given logic has the power to define precisely all structural properties that satisfy some characteristic semantic preservation condition? This issue is a fundamental one for classical model theory and applications in non-classical settings alike. While methods differ greatly, and while the classical methods can usually not be matched for instance in the setting of finite model theory, this note surveys some interesting commonality revolving around the use and availability of tractable representatives in the relevant model classes. The construction of models in which simple invariants like partial types based on some weak fragment control all the relevant structural properties, may be seen at the heart of such questions. We highlight some constructions involving degrees of acyclicity and saturation that can be achieved in finite model constructions, and discuss their uses towards expressive completeness w.r.t. bisimulation based equivalences in hypergraphs and relational structures. The emphasis is on the combinatorial challenges in such more constructive approaches that work in non-classical settings and especially in finite model theory. One new result concerns expressive completeness w.r.t. guarded negation bisimulation, a back-and-forth equivalence involving local homomorphisms.

Some Model Theory of Guarded Negation

“…Analogous results on arbitrary structures have been established for both GFO [2] and GNFO [8]. In the context of finite model theory, Otto [42,40] provided Van Benthem-style characterizations of GFO and of the "k-bounded fragment of GNFO" indexed by a number k. Central to these results are the notions of guarded bisimulation and guarded negation bisimulation that play similar roles in the model theory of GFO, respectively, GNFO as does bisimulation in the model theory of modal logic. For a comprehensive survey the interested reader should turn to [29].…”

“…In [8] guarded-negation bisimulations (GN-bisimulations) were introduced, and it was shown that GNFO expresses the first-order logic properties that are invariant under GN-bisimulations. A related characterization over finite structures for the k-variable fragment of GNFO is given in [40]. Here we will work over all structures, giving a characterization theorem for a simpler kind of simulation relation, which we call a strong GN-bisimulation.…”

Bisimulation and Coverings for Graphs and Hypergraphs

“…Technically, the generalisation of good model-theoretic features, and especially of the finite model property, involves reductions to Corollary 3.2 above. An analogue of Theorem 3.1 has recently been outlined in [22].…”

“…branching behaviour so as to produce finite structures for which guarded bisimulation equivalence to sufficiently high finite depth entails first-order equivalence up to some given quantifier rank. See [23,22] for details and the general context. This yields the following.…”

Guarded Negation

“…GNFO is the ≈ GN -invariant fragment of FO, and for all k ≥ 1, GNFO k is the ≈ k GN -invariant fragment of FO on arbitrary structures. The finite variant of Theorem 6.3, showing that GNFO k captures the ≈ k GN -invariant fragment of FO on finite structures has recently been established inOtto [2012].…”