Abstract
We survey different notions of bisimulation equivalence that provide flexible and powerful concepts for understanding the expressive power as well as the model-theoretic and algorithmic properties of modal logics and of more and more powerful variants of guarded logics. An appropriate notion of bisimulation for a logic allows us to study the expressive power of that logic in terms of semantic invariance and logical indistinguishability. As bisimilar nodes or tuples in two structures cannot be distinguished by formulae of the logic, bisimulations may be used to control the complexity of the models under consideration. In this manner, bisimulation-respecting model constructions and transformations lead to results about model-theoretic properties of modal and guarded logics, such as the tree model property of modal logics and the fact that satisfiable guarded formulae have models of bounded tree width. A highlight of the bisimulation-based analysis are the characterisation theorems: inside a classical level of logical expressiveness such as first-order or monadic second-order definability, these provide a tight match between bisimulation invariance and logical definability. Typically such characterisation theorems state that a modal or guarded logic is not only invariant under bisimulation but, conversely, also expressively complete for the class of all bisimulation invariant properties at that level. Finally, the bisimulation-based analysis of modal and guarded logics also leads to important insights concerning their algorithmic properties. Since satisfiable formulae always admit simple models, for instance tree-like ones, and since modal and guarded logics can be embedded or interpreted in monadic second-order logic on trees, powerful automata theoretic methods become available for checking satisfiability and for evaluating formulae.
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Notes
- 1.
If \(\mathbf{x}\mathbf{y}\) consists of a single variable symbol \(z\), \(\alpha \) can be the equality \(z=z\).
- 2.
One should except the initial position from the guardedness requirement in order to match the liberal treatment of (outermost) free variables in \(\mathrm {GF}\).
- 3.
We attach the empty set as a root to \(I({\mathfrak {A}})\) and join it to every guarded set to obtain a natural tree unfolding for our purposes, rather than a forest.
- 4.
It may be worth to point out that, unlike the finite bisimilar coverings obtained for graph-like structures in [18], the bisimilar coverings of relational structures or of hypergraphs will necessarily be branched coverings, and do not provide unique liftings.
- 5.
Caveat: \(\pi (h({\mathfrak {C}})) \subseteq {\mathfrak {A}}\) need not itself be acyclic.
- 6.
References
Andréka H, van Benthem J, Németi I (1998) Modal languages and bounded fragments of predicate logic. J Philoso Logic 27:217–274
BĂ¡rĂ¡ny V, Gottlob G, Otto M (2014) Querying the guarded fragment. Logical Methods Comput Sci (to appear)
BĂ¡rĂ¡ny V, ten Cate B, Segoufin L (2011) Guarded negation. In: Proceedings of ICALP, pp 356–367
Beeri C, Fagin R, Maier D, Yannakakis M (1983) On the desirability of acyclic database schemes. J ACM 30:497–513
van Benthem J (1983) Modal logic and classical logic. Bibliopolis, Napoli
van Benthem J (2005) Guards, bounds, and generalized semantics. J Logic Lang Inform 14(3):263–279
van Benthem J (2007) A new modal Lindström theorem. Log Univers 1:125–138
van Benthem J, ten Cate B, Väänänen J (2007) Lindström theorems for fragments of first-order logic. In: Proceedings of 22nd IEEE symposium on logic in computer science, LICS 2007, pp 280–292
Berge C (1973) Graphs and hypergraphs. North-Holland, Amsterdam
ten Cate B, Segoufin L (2011) Unary negation. In: Proceedings of STACS, pp 344–355
Dawar A, Otto M (2009) Modal characterisation theorems over special classes of frames. Ann Pure Appl Logic 161:1–42
Grädel E (1999) On the restraining power of guards. J Symbolic Logic 64:1719–1742
Grädel E (2002) Guarded fixed point logics and the monadic theory of countable trees. Theoret Comput Sci 288:129–152
Grädel E, Hirsch C, Otto M (2002) Back and forth between guarded and modal logics. ACM Trans Comput Logics 3:418–463
Grädel E, Walukiewicz I (1999) Guarded fixed point logic. In: Proceedings of 14th IEEE symposium on logic in computer science, LICS 1999, pp 45–54
Herwig B (1995) Extending partial isomorphisms on finite structures. Combinatorica 15:365–371
Herwig B, Lascar D (2000) Extending partial isomorphisms and the profinite topology on free groups. Trans AMS 352:1985–2021
Otto M (2004) Modal and guarded characterisation theorems over finite transition systems. Ann Pure Appl Logic 164(12):1418–1453
Otto M (2006) Bisimulation invariance and finite models. In: Colloquium logicum 2002. Lecture notes in logic, pp 276–298, ASL
Otto M (2011) Model theoretic methods for fragments of FO and special classes of (finite) structures. In: Esparza J, Michaux C, Steinhorn C (eds) Finite and algorithmic model theory, volume 379 of LMS lecture notes, pp 271–341. CUP
Otto M (2012a) Highly acyclic groups, hypergraph covers and the guarded fragment. J ACM 59:1
Otto M (2012b) On groupoids and hypergraphs. arXiv:1211.5656 (preprint)
Otto M (2013a) Expressive completeness through logically tractable models. Ann Pure Appl Logic 164(12):1418–1453
Otto M (2013b) Groupoids, hypergraphs and symmetries in finite models. In: Proceedings of 28th IEEE symposium on logic in computer science, LICS 2013
Otto M (2014) Finite groupoids, finite coverings and symmetries in finite structures. arXiv:1404.4599 (preprint)
Otto M, Piro R (2008) A Lindström characterisation of the guarded fragment and of modal logic with a global modality. In: Areces C, Goldblatt R (eds) Advances in modal logic 7, pp 273–288
Rabin M (1969) Decidability of second-order theories and automata on infinite trees. Trans AMS 141:1–35
Rosen E (1997) Modal logic over finite structures. J Logic Lang Inform 6:427–439
Rossman B (2008) Homomorphism preservation theorems. J ACM 55:1–53
Vardi M (1998) Reasoning about the past with two-way automata. In: Automata, languages and programming ICALP 98. Lecture notes in computer science, vol 1443. Springer, pp 628–641
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Grädel, E., Otto, M. (2014). The Freedoms of (Guarded) Bisimulation. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_1
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