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19
Reasoning about The Past with TwoWay Automata
 In 25th International Colloqium on Automata, Languages and Programming, ICALP ’98
, 1998
"... Abstract. The pcalculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and temporal logics. It is known that the satisfiability problem for the pcalculus is EXPTIMEcomplete ..."
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Abstract. The pcalculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and temporal logics. It is known that the satisfiability problem for the pcalculus is EXPTIMEcomplete. This upper bound, however, is known for a version of the logic that has only forward modalities, which express weakest preconditions, but not backward modalities, which express strongest postconditions. Our main result in this paper is an exponential time upper bound for the satisfiability problem of the pcalculus with both forward and backward modalities. To get this result we develop a theory of twoway alternating automata on infinite trees. 1
On Logics with Two Variables
 Theoretical Computer Science
, 1999
"... This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable ..."
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Cited by 46 (9 self)
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This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2 . Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points etc. Examples of such logics are computation tree logic CTL, the modal ¯calculus L¯ , or popular description logics used in artificial intelligence. Although the additional features are usually not firstorder...
Completeness of Kozen's Axiomatisation of the Propositional µCalculus
 Inform. and Comput
, 1995
"... Propositional calculus is an extension of the propositional modal logic with the least fixpoint operator. In the paper introducing the logic Kozen posed a question about completeness of the axiomatisation which is a small extension of the axiomatisation of the modal system K. It is shown that this ..."
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Cited by 33 (1 self)
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Propositional calculus is an extension of the propositional modal logic with the least fixpoint operator. In the paper introducing the logic Kozen posed a question about completeness of the axiomatisation which is a small extension of the axiomatisation of the modal system K. It is shown that this axiomatisation is complete.
Undecidability Results on TwoVariable Logics
 IN PROCEEDINGS OF 14TH SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE STACS`97, LECTURE NOTES IN COMPUTER SCIENCE NR. 1200
, 1998
"... It is a classical result of Mortimer that L², firstorder logic with two variables, is decidable for satisfiability. We show that going beyond L² by adding any one of the following leads to an undecidable logic: ffl very weak forms of recursion, viz. (i) transitive closure operations (ii) (restrict ..."
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Cited by 33 (5 self)
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It is a classical result of Mortimer that L², firstorder logic with two variables, is decidable for satisfiability. We show that going beyond L² by adding any one of the following leads to an undecidable logic: ffl very weak forms of recursion, viz. (i) transitive closure operations (ii) (restricted) monadic fixedpoint operations ffl weak access to cardinalities, through the Hartig (or equicardinality) quantifier ffl a choice construct known as Hilbert's "operator. In fact all these extensions of L² prove to be undecidable both for satisfiability, and for satisfiability in finite models. Moreover most of them are hard for \Sigma 1 1 , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than firstorder logic.
Why Are Modal Logics So Robustly Decidable?
"... Modal logics are widely used in a number of areas in computer science, in particular ..."
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Cited by 32 (1 self)
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Modal logics are widely used in a number of areas in computer science, in particular
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 26 (4 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.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 24 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Derivation of Characteristic Formulae
, 2001
"... This paper shows how modal mucalculus formulae characterizing finitestate processes up to strong or weak bisimulation can be derived directly from the wellknown greatest fixpoint characterizations of the bisimulation relations. Our derivation simplifies earlier proofs for the strong bisimulation ..."
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Cited by 23 (1 self)
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This paper shows how modal mucalculus formulae characterizing finitestate processes up to strong or weak bisimulation can be derived directly from the wellknown greatest fixpoint characterizations of the bisimulation relations. Our derivation simplifies earlier proofs for the strong bisimulation case and, by virtue of derivation, immediately generalizes to various other bisimulationlike relations, in particular weak bisimulation.
Automata for the modal é  calculus and related results
 In 20th MFCS
, 1995
"... Abstract. The propositional calculus as introduced by Kozen in [4] is considered. The notion of disjunctive formula is dened and it is shown that every formula is semantically equivalent to a disjunctive formula. For these formulas many diculties encountered in the general case may be avoided. For ..."
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Abstract. The propositional calculus as introduced by Kozen in [4] is considered. The notion of disjunctive formula is dened and it is shown that every formula is semantically equivalent to a disjunctive formula. For these formulas many diculties encountered in the general case may be avoided. For instance, satisability checking is linear for disjunctive formulas. This kind of formula gives rise to a new notion of nite automaton which characterizes the expressive power of the calculus over all transition systems. 1