Back and forth between guarded and modal logics
ACM Transactions on Computational Logic (TOCL)
Automata logics, and infinite games
The monadic theory of finite representations of infinite words
Information Processing Letters
Highly acyclic groups, hypergraph covers, and the guarded fragment
Journal of the ACM (JACM)
On the bisimulation invariant fragment of monadic Σ1 in the finite
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
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Characterisation theorems for modal and guarded fragments of first-order logic are explored over finite transition systems. We show that the classical characterisations in terms of semantic invariance under the appropriate forms of bisimulation equivalence can be recovered at the level of finite model theory. The new, more constructive proofs naturallyextend to alternative proofs of the classical variants. The finite model theory version of van Benthem's characterisation of basic modal logic is due to E. Rosen. That proof is simplified and the result slightly strengthened in terms of quantitative bounds. The main theme, however, is a uniform treatment that extends to incorporate universal and inverse modalities and guarded quantification over transition systems. Technically, the present treatment exploits first-order locality in the context of a new finitary construction of locally acyclic bisimilar covers. These serve as graded finite analogues of tree unravellings, giving local control over first-order logic in finite bisimilar companion structures.