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Completions of µalgebras
 In Proceedings of the Twentieth Annual IEEE Symposium on Logic in Computer Science (LICS 2005
, 2005
"... A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning ..."
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A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of µalgebras contains a µalgebra that has no embedding into a complete µalgebra. We focus then on modal µalgebras, i.e. algebraic models of the propositional modal µcalculus. We prove that free modal µalgebras satisfy a condition – reminiscent of Whitman’s condition for free lattices – which allows us to prove that (i) modal operators are adjoints on free modal µalgebras, (ii) least prefixed points of Σ1operations satisfy the constructive relation µx.f = W n≥0 f n (⊥). These properties imply the following statement: the MacNeilleDedekind completion of a free modal µalgebra is a complete modal µalgebra and moreover the canonical embedding preserves all the operations in the class Comp(Σ1, Π1) of the fixed point alternation hierarchy.
A Solver for Modal Fixpoint Logics
"... We presentMLSolver, a tool for solving the satisfiability and validity problems for modal fixpoint logics. The underlying technique is based on characterisations of satisfiability through infinite (cyclic) tableaux in which branches have an inner thread structure mirroring the regeneration of least ..."
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We presentMLSolver, a tool for solving the satisfiability and validity problems for modal fixpoint logics. The underlying technique is based on characterisations of satisfiability through infinite (cyclic) tableaux in which branches have an inner thread structure mirroring the regeneration of least and greatest fixpoint constructs in the infinite. Wellfoundedness for unfoldings of least fixpoints is checked using deterministic parity automata. This reduces the satisfiability and validity problems to the problem of solving a parity game. MLSolver then uses a parity game solver in order to decide satisfiability and derives example models from the winning strategies in the parity game. Currently supported logics are the modal and lineartime
Game Over: The Foci Approach to LTL Satisfiability and Model Checking
, 2004
"... Focus games have been shown to yield gametheoretical characterisations for the satisfiability and the model checking problem for various temporal logics. One of the players is given a tool  the focus  that enables him to show the regeneration of temporal operators characterised as least or grea ..."
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Focus games have been shown to yield gametheoretical characterisations for the satisfiability and the model checking problem for various temporal logics. One of the players is given a tool  the focus  that enables him to show the regeneration of temporal operators characterised as least or greatest fixpoints. His strategy usually is build upon a priority list of formulas and, thus, is not positional. This paper defines foci games for satisfiability of LTL formulas. Strategies in these games are trivially positional since they parallelise all of the focus player's choices, thus resulting in a 1player game in e#ect. The games are shown to be correct and to yield smaller (counter)models than the focus games. Finally, foci games for model checking LTL are defined as well.
Model checking for action abstraction ⋆
"... Abstract. We endow action sets of transition systems with a partial order that expresses the degree of specialization of actions, and with an intuitive but flexible consistency predicate that constrains the extension of such orders with more specialized actions. We develop a satisfaction relation fo ..."
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Abstract. We endow action sets of transition systems with a partial order that expresses the degree of specialization of actions, and with an intuitive but flexible consistency predicate that constrains the extension of such orders with more specialized actions. We develop a satisfaction relation for such models and the µcalculus. We then prove that this satisfaction relation is sound for Thomsen’s extended bisimulation as our refinement notion for models, even for consistent extensions of ordered action sets. We then demonstrate how this satisfaction relation can be reduced, fairly efficiently, to classical µcalculus model checking. These results provide formal support for change management of models and their validation (e.g. in modelcentric software development), and enable verification of concrete systems with respect to properties specified for abstract actions. 1
Model Checking Nash Equilibria in MAD Distributed Systems
"... Abstract—We present a symbolic model checking algorithm for verification of Nash equilibria in finite state mechanisms modeling Multiple Administrative Domains (MAD) distributed systems. Given a finite state mechanism, a proposed protocol for each agent and an indifference threshold for rewards, our ..."
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Abstract—We present a symbolic model checking algorithm for verification of Nash equilibria in finite state mechanisms modeling Multiple Administrative Domains (MAD) distributed systems. Given a finite state mechanism, a proposed protocol for each agent and an indifference threshold for rewards, our model checker returns PASS if the proposed protocol is a Nash equilibrium (up to the given indifference threshold) for the given mechanism, FAIL otherwise. We implemented our model checking algorithm inside the NuSMV model checker and present experimental results showing its effectiveness for moderate size mechanisms. I.
Local Strategy Improvement for Parity Game Solving
"... The problem of solving a parity game is at the core of many problems in model checking, satisfiability checking and program synthesis. Some of the best algorithms for solving parity game are strategy improvement algorithms. These are global in nature since they require the entire parity game to be p ..."
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The problem of solving a parity game is at the core of many problems in model checking, satisfiability checking and program synthesis. Some of the best algorithms for solving parity game are strategy improvement algorithms. These are global in nature since they require the entire parity game to be present at the beginning. This is a distinct disadvantage because in many applications one only needs to know which winning region a particular node belongs to, and a witnessing winning strategy may cover only a fractional part of the entire game graph. We present a local strategy improvement algorithm which explores the game graph onthefly whilst performing the improvement steps. We also compare it empirically with existing global strategy improvement algorithms and the currently only other local algorithm for solving parity games. It turns out that local strategy improvement can outperform these others by several orders of magnitude. 1
Abstract GDV 2004 Preliminary Version Game Over: The Foci Approach to LTL Satisfiability and Model Checking
"... Focus games have been shown to yield gametheoretical characterisations for the satisfiability and the model checking problem for various temporal logics. One of the players is given a tool – the focus – that enables him to show the regeneration of temporal operators characterised as least or greate ..."
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Focus games have been shown to yield gametheoretical characterisations for the satisfiability and the model checking problem for various temporal logics. One of the players is given a tool – the focus – that enables him to show the regeneration of temporal operators characterised as least or greatest fixpoints. His strategy usually is build upon a priority list of formulas and, thus, is not positional. This paper defines foci games for satisfiability of LTL formulas. Strategies in these games are trivially positional since they parallelise all of the focus player’s choices, thus resulting in a 1player game in effect. The games are shown to be correct and to yield smaller (counter)models than the focus games. Finally, foci games for model checking LTL are defined as well.
Satisfiability Games for CTL*
"... This paper defines games for the full branching time logic CTL # . They provide a direct method to check satisfiability of a formula since they work on its subformulas only. Thus, this method avoids determinisation of automata or translations into other logics. Instead, it employs a simple tool ..."
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This paper defines games for the full branching time logic CTL # . They provide a direct method to check satisfiability of a formula since they work on its subformulas only. Thus, this method avoids determinisation of automata or translations into other logics. Instead, it employs a simple tool called focus which the players use to highlight one particular formula in a set of subformulas. Focus games have been proved to be a useful method for working with temporal logics. For instance, model checking focus games for CTL# and LTL, as well as satisfiability checking focus games for LTL and CTL have been shown to work. Indeed, the games of this paper are an extension of the latter ones. They can be used as a basis of a decision procedure for CTL # that matches the known lower bound of double exponential time.
Model Checking Games for a Fair BranchingTime Temporal Epistemic Logic ⋆
"... Abstract. Model checking games are instances of Hintikka’s game semantics for logic used for purposes of debugging systems verification models. Previous work in the area has developed these games for branching time logic. The paper develops an extension to a logic that adds epistemic operators, and ..."
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Abstract. Model checking games are instances of Hintikka’s game semantics for logic used for purposes of debugging systems verification models. Previous work in the area has developed these games for branching time logic. The paper develops an extension to a logic that adds epistemic operators, and interprets the branching time operators with respect to fairness constraints. The implementation of the extended games in the epistemic model checker MCK is described. 1