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On the Decision Problem for TwoVariable FirstOrder Logic
, 1997
"... We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity ..."
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Cited by 48 (1 self)
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We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finitemodel property, which means that if an FO²sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIMEcomplete.
BisimulationInvariant Ptime and HigherDimensional µCalculus
 THEORETICAL COMPUTER SCIENCE
, 1998
"... Consider the class of all those properties of worlds in finite Kripke structures (or of states in finite transition systems), that are ffl recognizable in polynomial time, and ffl closed under bisimulation equivalence. It is shown that the class of these bisimulationinvariant Ptime queries has a ..."
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Cited by 16 (1 self)
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Consider the class of all those properties of worlds in finite Kripke structures (or of states in finite transition systems), that are ffl recognizable in polynomial time, and ffl closed under bisimulation equivalence. It is shown that the class of these bisimulationinvariant Ptime queries has a natural logical characterization. It is captured by the straightforward extension of propositional µcalculus to arbitrary finite dimension. Bisimulationinvariant Ptime, or the modal fragment of Ptime, thus proves to be one of the very rare cases in which a logical characterization is known in a setting of unordered structures. It is also shown that higherdimensional µcalculus is undecidable for satisfiability in finite structures, and even \Sigma 1 1 hard over general structures.
Loosely Guarded Fragment of FirstOrder Logic Has the Finite Model Property
, 2000
"... We show that the loosely guarded and packed fragments of firstorder logic have the finite model property. We use a construction of Herwig. We point out some consequences in temporal predicate logic and algebraic logic. AMS classification: Primary 03B20; Secondary 03B45, 03C07, 03C13, 03C30, 03G1 ..."
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Cited by 15 (3 self)
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We show that the loosely guarded and packed fragments of firstorder logic have the finite model property. We use a construction of Herwig. We point out some consequences in temporal predicate logic and algebraic logic. AMS classification: Primary 03B20; Secondary 03B45, 03C07, 03C13, 03C30, 03G15 Keywords: finite structures, modal logic, modal fragment, packed fragment 1 Introduction Perhaps because beginning students of modal logic are often told that modal logic is more expressive than firstorder logic and indeed has some secondorder expressive power, or perhaps because they are hoping for something new, it can come as a surprise to them that every modal formula has a `standard translation' into firstorder logic. For example, (p !q) is translated to 9y(R(x;y) ^ (P(y) ! 8z(R(y;z) ! Q(z)))): (1) The translation mimics the Kripke semantics for modal logic. Not every firstorder formula (with one free variable in the appropriate signature) is the translation of a modal formu...
Bisimulation Invariance and Finite Models
 IN LOGIC COLLOQUIUM ’02
, 2006
"... We study bisimulation invariance over finite structures. This investigation leads to a new, quite elementary proof of the van BenthemRosen characterisation of basic modal logic as the bisimulation invariant fragment of firstorder logic. The ramification of this characterisation for the finer no ..."
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Cited by 6 (0 self)
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We study bisimulation invariance over finite structures. This investigation leads to a new, quite elementary proof of the van BenthemRosen characterisation of basic modal logic as the bisimulation invariant fragment of firstorder logic. The ramification of this characterisation for the finer notion of global twoway bisimulation equivalence is based on bisimulation respecting constructions of models that recover in finite models some of the desirable properties of the usually in finite bisimilar unravellings.
Guarded Fixed Point Logics and the Monadic Theory of Countable Trees
, 2001
"... Different variants of guarded logics (a powerful generalization of modal logics) are surveyed and an elementary proof for the decidability of guarded fixed point logics is presented. In a joint paper with Igor Walukiewicz, we proved that the satisfiability problems for guarded fixed point logics ar ..."
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Cited by 3 (0 self)
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Different variants of guarded logics (a powerful generalization of modal logics) are surveyed and an elementary proof for the decidability of guarded fixed point logics is presented. In a joint paper with Igor Walukiewicz, we proved that the satisfiability problems for guarded fixed point logics are decidable and complete for deterministic double exponential time (E. Grädel and I. Walukiewicz, Guarded fixed point logic, Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 4554). That proof relies on alternating automata on trees and on a forgetful determinacy theorem for games on graphs with unbounded branching. The exposition given here emphasizes the tree model property of guarded logics: every satisfiable sentence has a model of bounded tree width. Based on the tree model property, we show that the satisfiability problem for guarded fixed point formulae can be reduced to the monadic theory of countable trees (SωS), or to the µcalculus with backwards modalities.
Exploring the iterated update universe
, 2006
"... Abstract. We investigate the asymptotic properties of the logical system for information update developped by Baltag, Moss and Solecki [2]. We build on the idea of looking at update logics as dynamical systems. We show that every epistemic formula either always holds or is always refuted from certai ..."
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Cited by 2 (0 self)
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Abstract. We investigate the asymptotic properties of the logical system for information update developped by Baltag, Moss and Solecki [2]. We build on the idea of looking at update logics as dynamical systems. We show that every epistemic formula either always holds or is always refuted from certain moment on, in the course of update with factual epistemic events, i.e. events with only propositional prerequisite formulas, or signals. We characterize in terms of a pebble game the class of frames such that iterated update with factual epistemic events built over them gives rise only to finite sets of reachable states. The characterization is nontrivial, and so the ’Finite Evolution Conjecture ’ (see van Benthem [4]) is refuted. Finally, after giving some basic insights into the dissipative nature of update with general, nonfactual epistemic events, we show the distinctive stabilizing nature of epistemically ordered multiS5 events events in which agents can be ordered in terms of how much they know. 1.
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(This is a sample cover image for this issue. The actual cover is not yet available at this time.) This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: