Towards a Model Theory of Ordered Logics: Expressivity and Interpolation

Authors Bartosz Bednarczyk , Reijo Jaakkola



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Author Details

Bartosz Bednarczyk
  • Computational Logic Group, Technische Universität Dresden, Germany
  • Institute of Computer Science, University of Wrocław, Poland
Reijo Jaakkola
  • Tampere University, Finland

Acknowledgements

We would like to thank Antti Kuusisto and Jean Christoph Jung for fruitful discussions on related topics, as well as Tim Lyon and Emanuel Kieroński for language corrections. We would also like to thank the anonymous reviewers for their very helpful comments.

Cite AsGet BibTex

Bartosz Bednarczyk and Reijo Jaakkola. Towards a Model Theory of Ordered Logics: Expressivity and Interpolation. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.15

Abstract

We consider the family of guarded and unguarded ordered logics, that constitute a recently rediscovered family of decidable fragments of first-order logic (FO), in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. While the complexities of their satisfiability problems are now well-established, their model theory, however, is poorly understood. Our paper aims to provide some insight into it. We start by providing suitable notions of bisimulation for ordered logics. We next employ bisimulations to compare the relative expressive power of ordered logics, and to characterise our logics as bisimulation-invariant fragments of FO à la van Benthem. Afterwards, we study the Craig Interpolation Property (CIP). We refute yet another claim from the infamous work by Purdy, by showing that the fluted and forward fragments do not enjoy CIP. We complement this result by showing that the ordered fragment and the guarded ordered logics enjoy CIP. These positive results rely on novel and quite intricate model constructions, which take full advantage of the "forwardness" of our logics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
Keywords
  • ordered fragments
  • fluted fragment
  • guarded fragment
  • model theory
  • Craig Interpolation Property
  • expressive power
  • model checking

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