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The variable hierarchy of the µcalculus is strict
 STACS 2005, PROCEEDINGS OF THE 22ND SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE. VOLUME 3404 OF LNCS., SPRINGERVERLAG
, 2005
"... Most of the logics commonly used in verification, such as LTL, CTL, CTL∗, and PDL can be embedded into the twovariable fragment of the µcalculus. It is also known that properties occurring at arbitrarily high levels of the alternation hierarchy can be formalised using only two variables. This rais ..."
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Cited by 18 (4 self)
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Most of the logics commonly used in verification, such as LTL, CTL, CTL∗, and PDL can be embedded into the twovariable fragment of the µcalculus. It is also known that properties occurring at arbitrarily high levels of the alternation hierarchy can be formalised using only two variables. This raises the question whether the number of fixedpoint variables in µformulae can be bounded in general. We answer this question negatively, and prove that the variablehierarchy of the µcalculus is semantically strict. For any k, we provide examples of formulae with k variables that are not equivalent to any formula with fewer variables. In particular, this implies that Parikh’s Game Logic is less expressive than the µcalculus, thus resolving an open issue raised by Parikh in 1983.
Guarded negation
"... Abstract. We consider restrictions of firstorder logic and of fixpoint logic in which all occurrences of negation are required to be guarded by an atomic predicate. In terms of expressive power, the logics in question, called GNFO and GNFP, extend the guarded fragment of firstorder logic and guard ..."
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Abstract. We consider restrictions of firstorder logic and of fixpoint logic in which all occurrences of negation are required to be guarded by an atomic predicate. In terms of expressive power, the logics in question, called GNFO and GNFP, extend the guarded fragment of firstorder logic and guarded least fixpoint logic, respectively. They also extend the recently introduced unary negation fragments of firstorder logic and of least fixpoint logic. We show that the satisfiability problem for GNFO and for GNFP is 2ExpTimecomplete, both on arbitrary structures and on finite structures. We also study the complexity of the associated model checking problems. Finally, we show that GNFO and GNFP are not only computationally well behaved, but also model theoretically: we show that GNFO and GNFP have the treelike model property and that GNFO has the finite model property, and we characterize the expressive power of GNFO in terms of invariance for an appropriate notion of bisimulation. 1
Unary negation
, 2011
"... We study fragments of firstorder logic and of least fixed point logic that allow only unary negation: negation of formulas with at most one free variable. These logics generalize many interesting known formalisms, including modal logic and the µcalculus, as well as conjunctive queries and monadic ..."
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Cited by 12 (4 self)
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We study fragments of firstorder logic and of least fixed point logic that allow only unary negation: negation of formulas with at most one free variable. These logics generalize many interesting known formalisms, including modal logic and the µcalculus, as well as conjunctive queries and monadic Datalog. We show that satisfiability and finite satisfiability are decidable for both fragments, and we pinpoint the complexity of satisfiability, finite satisfiability, and model checking. We also show that the unary negation fragment of firstorder logic is modeltheoretically very well behaved. In particular, it enjoys Craig interpolation and the Beth property.
Grounding for model expansion in kguarded formulas with inductive definitions
 In IJCAI
, 2007
"... Mitchell and Ternovska [2005] proposed a constraint programming framework based on classical logic extended with inductive definitions. They formulate a search problem as the problem of model expansion (MX), which is the problem of expanding a given structure with new relations so that it satisfies ..."
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Cited by 10 (5 self)
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Mitchell and Ternovska [2005] proposed a constraint programming framework based on classical logic extended with inductive definitions. They formulate a search problem as the problem of model expansion (MX), which is the problem of expanding a given structure with new relations so that it satisfies a given formula. Their longterm goal is to produce practical tools to solve combinatorial search problems, especially those in NP. In this framework, a problem is encoded in a logic, an instance of the problem is represented by a finite structure, and a solver generates solutions to the problem. This approach relies on propositionalisation of highlevel specifications, and on the efficiency of modern SAT solvers. Here, we propose an efficient algorithm which combines grounding with partial evaluation. Since the MX framework is based on classical logic, we are able to take advantage of known results for the socalled guarded fragments. In the case of kguarded formulas with inductive definitions under a natural restriction, the algorithm performs much better than naive grounding by relying on connections between kguarded formulas and tree decompositions. 1
FixedPoint Logics and Solitaire Games
 THEORY COMPUT SYSTEMS
, 2004
"... The modelchecking games associated with fixedpoint logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time. We study SolitaireLFP, a fragment of least fixedpoint logic, whose evaluation games are nested soltaire games. ..."
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Cited by 5 (2 self)
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The modelchecking games associated with fixedpoint logics are parity games, and it is currently not known whether the strategy problem for parity games can be solved in polynomial time. We study SolitaireLFP, a fragment of least fixedpoint logic, whose evaluation games are nested soltaire games. This means that on each strongly connected component of the game, only one player can make nontrivial moves. Winning sets of nested solitaire games can be computed efficiently. The model
On the Variable Hierarchy of the Modal µCalculus
"... We investigate the structure of the modal µcalculus L with respect to the question of how many different fixed point variables are necessary to define a given property. Most of the logics commonly used in verification, such as CTL, LTL, CTL , PDL, etc. can in fact be embedded into the twovariable ..."
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Cited by 2 (0 self)
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We investigate the structure of the modal µcalculus L with respect to the question of how many different fixed point variables are necessary to define a given property. Most of the logics commonly used in verification, such as CTL, LTL, CTL , PDL, etc. can in fact be embedded into the twovariable fragment of the µcalculus. It is also known that the twovariable fragment can express properties that occur at arbitrarily high levels of the alternation hierarchy. However, it is an open problem whether the variable hierarchy is strict.
AGuarded negation
"... We consider restrictions of firstorder logic and of fixpoint logic in which all occurrences of negation are required to be guarded by an atomic predicate. In terms of expressive power, the logics in question, called GNFO and GNFP, extend the guarded fragment of firstorder logic and the guarded lea ..."
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We consider restrictions of firstorder logic and of fixpoint logic in which all occurrences of negation are required to be guarded by an atomic predicate. In terms of expressive power, the logics in question, called GNFO and GNFP, extend the guarded fragment of firstorder logic and the guarded least fixpoint logic, respectively. They also extend the recently introduced unary negation fragments of firstorder logic and of least fixpoint logic. We show that the satisfiability problem for GNFO and for GNFP is 2ExpTimecomplete, both on arbitrary structures and on finite structures. We also study the complexity of the associated model checking problems. Finally, we show that GNFO and GNFP are not only computationally well behaved, but also model theoretically: we show that GNFO and GNFP have the treelike model property and that GNFO has the finite model property, and we characterize the expressive power of GNFO in terms of invariance for an appropriate notion of bisimulation. Our complexity upper bounds for GNFO and GNFP hold true even for their “cliqueguarded ” extensions CGNFO and CGNFP, in which clique guards are allowed in the place of guards.