on the positional determinacy of edge labeled games it is well known that games with the parity winning condition admit positional determinacy the winner has always a positional memoryless strategy this property continues to hold if edges rather than vertices are labeled we show that in this latter case the converse is also true that is a winning condition over arbitrary set of colors admits positional determinacy in all games if and only if it can be reduced to a parity condition with some finite number of priorities

an automata theoretic decision procedure for the propositional mu calculus the propositional mu calculus is a propositional logic of programs which incorporates a least fixpoint operator and subsumes the propositional dynamic logic of fischer and ladner the infinite looping construct of streett and the game logic of parikh we give an elementary time decision procedure using a reduction to the emptiness problem for automata on infinite trees a small model theorem is obtained as a corollary

graded modalities in strategy logic abstract strategy logic sl is a logical formalism for strategic reasoning in multi agent systems its main feature is that it has variables for strategies that are associated to specific agents using a binding operator in this paper we introduce graded strategy logic graded sl an extension of sl by graded quantifiers over tuples of strategy variables i e there exist at least g different tuples x 1 x n of strategies where g is a cardinal from the set n 0 1 2 0 we prove that the model checking problem of graded sl is decidable we then turn to the complexity of fragments of graded sl when the g s are restricted to finite cardinals written graded n sl the complexity of model checking is no harder than for sl i e it is non elementary in the quantifier block rank we illustrate our formalism by showing how to count the number of different strategy profiles that are nash equilibria ne by analysing the structure of the specific formulas involved we conclude that the important problem of checking for the existence of a unique ne can be solved in 2 exptime which is not harder than merely checking for the existence of such an equilibrium

dp lower bounds for equivalence checking and model checking of one counter automata we present a general method for proving dp hardness of problems related to formal verification of one counter automata for this we show a reduction of the sat unsat problem to the truth problem for a fragment of presburger arithmetic the fragment contains only special formulas with one free variable and is particularly apt for transforming to simulation like equivalences on one counter automata in this way we show that the membership problem for any relation subsuming bisimilarity and subsumed by simulation preorder is dp hard even for one counter nets where the counter cannot be tested for zero we also show dp hardness for deciding simulation between one counter automata and finite state systems in both directions and for the model checking problem with one counter nets and the branching time temporal logic ef

compositional verification and 3 valued abstractions join forces abstract two of the most promising approaches to fighting the state explosion problem are abstraction and compositional verification in this work we join their forces to obtain a novel fully automatic compositional technique that can determine the truth value of the full calculus with respect to a given system given a system m m 1 m 2 we view each component m i as an abstraction m i of the global system the abstract component m i is defined using a 3 valued semantics so that whenever a calculus formula has a definite value true or false on m i the same value holds also for m thus can be checked on either m 1 or m 2 or both and if any of them returns a definite result then this result holds also for m if both checks result in an indefinite value the composition of the components needs to be considered however instead of constructing the composition of m 1 and m 2 our approach identifies and composes only the parts of the components in which their composition is necessary in order to conclude the truth value of indeed relevant it ignores the parts which can be handled separately the resulting model is often can potentially be significantly smaller than the full system furthermore we explain how our compositional approach can be combined with abstraction of the components in order to further reduce the size of the checked components the result is an incremental compositional abstraction refinement framework which resembles automatic assume guarantee reasoning

modular strategies for recursive game graphs many problems in formal verification and program analysis can be formalized as computing winning strategies for two player games on graphs in this paper we focus on solving games in recursive game graphs which can model the control flow in sequential programs with recursive procedure calls while such games can be viewed as the pushdown games studied in the literature the natural notion of winning in our framework requires the strategies to be modular with only local memory that is resolution of choices within a module does not depend on the context in which the module is invoked but only on the history within the current invocation of the module while reachability in global pushdown games is known to be exptime complete we show reachability in modular games to be np complete we present a fixed point computation algorithm for solving modular games such that in the worst case the number of iterations is exponential in the total number of returned values from the modules if the strategy within a module does not depend on the global history but can remember the history of the past invocations of this module that is if memory is local but persistent we show that reachability becomes undecidable

on temporal logics with data variable quantifications although data values are available in almost every computer system reasoning about them is a challenging task due to the huge data size or even infinite data domains temporal logics are the well known specification formalisms for reactive and concurrent systems various extensions of temporal logics have been proposed to reason about data values mostly in the last decade among them one natural idea is to extend temporal logics with variable quantifications ranging over an infinite data domain grumberg kupferman and sheinvald initiated the research on this topic recently and obtained several interesting results however there is still a lack of systematic investigation on the theoretical aspects of the variable extensions of temporal logics our goal in this paper is to fill this gap around this goal we make the following choices 1 we consider the variable extensions of two widely used temporal logics linear temporal logic ltl and computation tree logic ctl and allow arbitrary nestings of variable quantifications with boolean and temporal operators the resulting logics are called respectively variable ltl in brief vltl and variable ctl in brief vctl 2 we investigate the decidability and complexity of both the satisfiability and model checking problems over both finite and infinite words trees in particular we obtain the following results existential variable quantifiers or one single universal quantifier in the beginning already entail the undecidability of the satisfiability problems of both vltl and vctl over both finite and infinite words trees if only existential path quantifiers are used in vctl then the satisfiability problem over finite trees is decidable no matter which variable quantifiers are available for vltl formulae with one single universal variable quantifier in the beginning if the occurrences of the non parameterized atomic propositions are guarded by the positive occurrences of the quantified variables then its satisfiability problem becomes decidable over both finite and infinite words based on these results of the satisfiability problem we deduce the un decidability results of the model checking problem

formally reasoning about quality in recent years there has been a growing need and interest in formally reasoning about the quality of software and hardware systems as opposed to traditional verification in which one considers the question of whether a system satisfies a given specification or not reasoning about quality addresses the question of how well the system satisfies the specification we distinguish between two approaches to specifying quality the first propositional quality extends the specification formalism with propositional quality operators which prioritize and weight different satisfaction possibilities the second temporal quality refines the eventually operators of the specification formalism with discounting operators whose semantics takes into an account the delay incurred in their satisfaction in this article we introduce two quantitative extensions of linear temporal logic ltl one by propositional quality operators and one by discounting operators in both logics the satisfaction value of a specification is a number in 0 1 which describes the quality of the satisfaction we demonstrate the usefulness of both extensions and study the decidability and complexity of the decision and search problems for them as well as for extensions of ltl that combine both types of operators

fuzzy alternating automata over distributive lattices abstract nondeterminism gives computation models the power of existential choice as a generalization of nondeterminism alternation gives computation models the power of existential choice and universal choice simultaneously in this paper we extend fuzzy nondeterministic automata to a model called fuzzy alternating automata over distributive lattices compared with the previous work a weight labels a leaf node of the run tree rather than be involved in the edge between states when executing a transition one advantage of our setting is that it is easy to complement a given fuzzy alternating automaton it suffices to take the dual operation on the transition function and negate final costs on states moreover we show that fuzzy alternating automata have the same expressive power as fuzzy nondeterministic automata and the former ones are exponentially more succinct than the latter ones in addition we illustrate that such exponential blow up is unavoidable

games for the m calculus abstract given a formula of the propositional calculus we construct a tableau of the formula and define an infinite game of two players of which one wants to show that the formula is satisfiable and the other seeks the opposite the strategy for the first player can be further transformed into a model of the formula while the strategy for the second forms what we call a refutation of the formula using martin s determinacy theorem we prove that any formula has either a model or a refutation this completeness result is a starting point for the completeness theorem for the calculus to be presented elsewhere however we argue that refutations have some advantages of their own they are generated by a natural system of sound logical rules and can be presented as regular trees of the size exponential in the size of a refuted formula this last aspect completes the small model theorem for the calculus established by emerson and jutla 1988 thus on a more practical side refutations can be used as small objects testifying incorrectness of a program specification expressed by a formula we illustrate this point by an example

complexity of equivalence problems for concurrent systems of finite agents abstract a concurrent system of synchronous communicating agents is assembled from simpler sequential agents by parallel composition and hiding for example hide a1 alin p1 p2 pn describes the system of communicating agentsp1 pnin which the communication eventsa1 alare hidden consider descriptions of two systemspandqof synchronously communicating finite state agents assume that one wants to check whetherp qfor one of the commonly used equivalence we show that this question is pspace hard for all equivalences that lie between strong bisimulation and trace equivalences for some equivalences exponential lower and upper bounds are proven we also show that this problem is np hard and co np hard even for a class of very simple finite agents

probabilistic temporal logics via the modal mu calculus this paper presents a mu calculus based modal logic for describing properties of reactive probabilistic labeled transition systems rpltss and develops a model checking algorithm for determining whether or not states in finite state rpltss satisfy formulas in the logic the logic is based on the distinction between probabilistic systems and nonprobabilistic observations using the modal mu calculus one may specify sets of observations and the semantics of our logic then enable statements to be made about the measures of such sets at various system states the logic may be used to encode a variety of probabilistic modal and temporal logics in addition the model checking problem for it may be reduced to the calculation of solutions to systems of non linear equations finally the logic induces an equivalence on rpltss that coincides with accepted notions of probabilistic bisimulation in the literature

alternating pushdown and stack automata the classes of languages accepted by alternating pushdown automata alternating stack automata and alternating nonerasing stack automata both with and without an auxiliary space bounded worktape are characterized in terms of complexity classes defined by time bounded deterministic turing machines it is also shown that alternating 2 way finite state machines accept only regular languages

characterization of the shape stability for nonlinear elliptic problems abstract we characterize all geometric perturbations of an open set for which the solution of a nonlinear elliptic pde of p laplacian type with dirichlet boundary condition is stable in the l norm the necessary and sufficient conditions are jointly expressed by a geometric property associated to the p convergence if the dimension n of the space satisfies n 1 p n and if the number of the connected components of the complements of the moving domains are uniformly bounded a simple characterization of the uniform convergence can be derived in a purely geometric frame in terms of the hausdorff complementary convergence several examples are presented

results on the propositional μ calculus abstract in this paper we define and study a propositional calculus l which consists essentially of propositional modal logic with a least fixpoint operator l is syntactically simpler yet strictly more expressive than propositional dynamic logic pdl for a restricted version we give an exponential time decision procedure small model property and complete deductive system theory subsuming the corresponding results for pdl

the complexity of word problems this time with interleaving abstract we consider regular expressions extended with the interleaving operator and investigate the complexity of membership and inequivalence problems for these expressions for expressions using the operators union concatenation kleene star and interleaving we show that the inequivalence problem deciding whether two given expressions do not describe the same set of words is complete for exponential space without kleene star we show that the inequivalence problem is complete for the class p 2 at the second level of the polynomial time hierarchy certain cases of the membership problem deciding whether a given word is in the language described by a given expression are shown to be np complete it is also shown that certain languages can be described exponentially more succinctly by using interleaving

simulation relations for alternating buchi automata quotienting by simulation equivalences is a well established technique for reducing the size of nondeterministic buchi automata we adapt this technique to alternating buchi automata to this end we suggest two new quotients namely minimax and semi elective quotients prove that they preserve the recognized languages and show that computing them is not more difficult than computing quotients for nondeterministic buchi automata our approach is game theoretic the proofs rely on a specifically tailored join operation for strategies in simulation games which is interesting in its own right we explain the merits of our quotienting procedures with respect to converting alternating buchi automata into nondeterministic ones

on the complexity of checking semantic equivalences between pushdown processes and finite state processes simulation preorder equivalence and bisimulation equivalence are the most commonly used equivalences in concurrency theory their standard definitions are often called strong simulation bisimulation while weak simulation bisimulation abstracts from internal t actions we study the computational complexity of checking these strong and weak semantic preorders equivalences between pushdown processes and finite state processes we present a complete picture of the computational complexity of these problems and also study fixed parameter tractability in two important input parameters x the size of the finite control of the pushdown process and y the size of the finite state process all simulation problems are generally exptime complete and only become polynomial if both parameters x and y are fixed weak bisimulation equivalence is pspace complete but becomes polynomial if and only if parameter x is fixed strong bisimulation equivalence is pspace complete but becomes polynomial if either parameter x or y is fixed

improved model checking of hierarchical systems we present a unified game based approach for branching time model checking of hierarchical systems such systems are exponentially more succinct than standard state transition graphs as repeated sub systems are described only once early work on model checking of hierarchical systems shows that one can do better than a naive algorithm that flattens the system and removes the hierarchy given a hierarchical system s and a branching time specification j for it we reduce the model checking problem does s satisfy j to the problem of solving a hierarchical game obtained by taking the product of s with an alternating tree automaton a j for j our approach leads to clean uniform and improved model checking algorithms for a variety of branching time temporal logics in particular by improving the algorithm for solving hierarchical parity games we are able to solve the model checking problem for the m calculus in pspace and time complexity that is only polynomial in the depth of the hierarchy our approach also leads to an abstraction refinement paradigm for hierarchical systems the abstraction maintains the hierarchy and is obtained by merging both states and sub systems into abstract states

the complexity of mean payoff pushdown games two player games on graphs are central in many problems in formal verification and program analysis such as synthesis and verification of open systems in this work we consider solving recursive game graphs or pushdown game graphs that model the control flow of sequential programs with recursion while pushdown games have been studied before with qualitative objectives such as reachability and regular objectives in this work we study for the first time such games with the most well studied quantitative objective the mean payoff objective in pushdown games two types of strategies are relevant 1 global strategies which depend on the entire global history and 2 modular strategies which have only local memory and thus do not depend on the context of invocation but rather only on the history of the current invocation of the module our main results are as follows 1 one player pushdown games with mean payoff objectives under global strategies are decidable in polynomial time 2 two player pushdown games with mean payoff objectives under global strategies are undecidable 3 one player pushdown games with mean payoff objectives under modular strategies are np hard 4 two player pushdown games with mean payoff objectives under modular strategies can be solved in np i e both one player and two player pushdown games with mean payoff objectives under modular strategies are np complete we also establish the optimal strategy complexity by showing that global strategies for mean payoff objectives require infinite memory even in one player pushdown games and memoryless modular strategies are sufficient in two player pushdown games finally we also show that all the problems have the same complexity if the stack boundedness condition is added where along with the mean payoff objective the player must also ensure that the stack height is bounded

model checking of pushdown systems for projection temporal logic abstract in this paper we study the model checking problem of pushdown systems for projection temporal logic ptl an interval based temporal logic that is as expressive as the full regular languages for this we provide an algorithm to decide whether a pushdown system determined by a pushdown automaton satisfies a desired logic property by using relevant automata notations and operations the algorithm terminates in exponential time for normal ptl properties i e properties specified as normal ptl formulas to show the lower bound of complexity of the model checking problem we also construct a polynomial time reduction from the acceptance problem of a linearly bounded alternating turing machine as a result the model checking problem of pushdown systems for normal ptl properties is exptime complete

the verification and control of interacting similar discrete event systems this paper explore issues related to the control and verification of similar module systems in the discrete event systems framework similar module systems are distributed systems comprised of subsystem modules that exhibit isomorphic local behavior coordinated on global event occurrences when given a global model of these systems it is shown how to decompose the global model into the component subsystems in polynomial time it is also shown how to perform various verification tasks for these interacting systems while mitigating common state explosion difficulties by taking advantage of the special similar module system structure control properties of the similar module systems are also discussed it is assumed that the local modules are supervised by exactly one local controller and the controllers enforce the same local control policy necessary and sufficient conditions for achieving local and global control specifications in this setting are identified

simulation preorder over simple process algebras we consider the problem of simulation preorder equivalence between infinite state process and finite state ones first we describe a general method how to utilize the decidability of bisimulation problems to solve certain instances of the corresponding simulation problems for certain process classes the method allows us to design effective reductions of simulation problems to their bisimulation counterparts and some new decidability border for the problem of simulation preorder equivalence are decidable in exptime between pushdown and finite state ones on the other hand simulation preorder is undecidable between pa and finite state process in both directions these results also hold for those pa and finite state process which are deterministic and normed and trace equivalence is also shown to be undecidable for pa finally we prove that simulation preorder in both directions and simulation equivalence are interact able between all classes of infinite state systems in the hierarchy of process rewrite systems and finite state ones this result is obtained by showing that the problem whether a bpa or bpp process simulation a finite state one is pspace hard and the other direction is co np hard consequently simulation equivalence between bpa or bpp and finite state processes is also co np hard

ccs expressions finite state processes and three problems of equivalence abstract we examine the computational complexity of testing finite state processes for equivalence in milner s calculus of communicating systems ccs the equivalence problems in ccs are presented as refinements of the familiar problem of testing whether two nondeterministic finite automata nfa are equivalent i e accept the same language three notions of equivalence proposed for ccs are investigated namely observational equivalence strong observational equivalence and failure equivalence we show that observational equivalence can be tested in polynomial time as defined in ccs observational equivalence is the limit of a sequence of successively finer equivalence relations k where 1 is nondeterministic finite automaton equivalence we prove that for each fixed k deciding k is pspace complete we show that strong observational equivalence can be decided in polynomial time by reducing it to generalized partitioning a new combinatorial problem of independent interest finally we demonstrate that testing for failure equivalence is pspace complete even for a very restricted type of process

pushdown module checking with imperfect information the model checking problem for finite state open systems module checking has been extensively studied in the literature both in the context of environments with perfect and imperfect information about the system recently the perfect information case has been extended to infinite state systems pushdown module checking in this paper we extend pushdown module checking to the imperfect information setting i e to the case where the environment has only a partial view of the email protected s control states and pushdown store content we study the complexity of this problem with respect to the branching time temporal logics ctl ctl and the propositional m calculus we show that pushdown module checking which is by itself harder than pushdown model checking becomes undecidable when the environment has imperfect information we also show that undecidability relies on hiding information about the pushdown store indeed we prove that with imperfect information about the control states but a visible pushdown store the problem is decidable and its complexity is 2exptime complete for ctl and the propositional m calculus and 3exptime complete for ctl

some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets we establish some necessary and sufficient conditions for the convergence in w01 p of sequences of convex sets determined by obstacles these conditions are expressed in terms of properties of the p capacities of the level sets of the obstacles

on the power of bounded concurrency i finite automata we investigate the descriptive succinctness of three fundamental notions for modeling concurrency nondeterminism and pure parallelism the two facets of alternation and bounded cooperative concurrency whereby a system configuration consists of a bounded number of cooperating states our results are couched in the general framework of finite state automata but hold for appropriate versions of most concurrent models of computation such as petri nets statecharts or finite state versions of concurrent programming languages we exhibit exhaustive sets of upper and lower bounds on the relative succinctness of these features over s and s establishing that 1 each of the three features represents an exponential saving in succinctness of the representation in a manner that is independent of the other two and additive with respect to them 2 of the three bounded concurrency is the strongest representing a similar exponential saving even when substituted for each of the others for example we prove exponential upper and lower bounds on the simulation of deterministic concurrent automata by afas and triple exponential bounds on the simulation of alternating concurrent automata by dfas

the regular viewpoint on pa processes pa is the process algebra allowing non determinism sequential and parallel compositions and recursion we suggest viewing pa processes astrees and usingtree automata techniques for verification problems on pa our main result is that the set of iterated predecessors of a regular set of pa processes is a regular tree language and similarly for iterated successors furthermore the corresponding tree automata can be built effectively in polynomial time this has many immediate applications to verification problems for pa processes among which a simple and general model checking algorithm

an existence result for optimal obstacles we consider the optimization problem min f g g x wheref g is a variational energy associated to the obstaclegand the classx of admissible obstacles is given byx g r g on g dx c with w1 p0 andc r fixed generally this problem does not have a solution and it may happen that the optimal obstacle is of relaxed form under a monotonicity assumption onf we prove the existence of a non relaxed optimal obstacle in the familyx through a new method based on the notions of andw convergences

alternating finite automata on ω words abstract alternating finite automata on words are introduced as an extension of nondeterministic finite automata which process infinite sequences of symbols the classes of languages defined by alternating finite automata are investigated and characterized under four types of acceptance conditions it is shown that for one type of acceptance condition alternation increases the power in comparison with nondeterminism and for other three acceptance conditions nondeterministic finite automata on words have the same power as alternating ones

an automata theoretic approach to branching time model checking translating linear temporal logic formulas to automata has proven to be an effective approach for implementing linear time model checking and for obtaining many extensions and improvements to this verification method on the other hand for branching temporal logic automata theoretic techniques have long been thought to introduce an exponential penalty making them essentially useless for model checking recently bernholtz and grumberg 1993 have shown that this exponential penalty can be avoided though they did not match the linear complexity of non automata theoretic algorithms in this paper we show that alternating tree automata are the key to a comprehensive automata theoretic framework for branching temporal logics not only can they be used to obtain optimal decision procedures as was shown by muller et al but as we show here they also make it possible to derive optimal model checking algorithms moreover the simple combinatorial structure that emerges from the automata theoretic approach opens up new possibilities for the implementation of branching time model checking and has enabled us to derive improved space complexity bounds for this long standing problem

isomorphism of k contractible graphs a generalization of bounded valence and bounded genus a polynomial time isomorphism test for graphs called k contractible graphs for fixed k is included the class of k contractible graphs includes the graphs of bounded valence and the graphs of bounded genus the algorithm uses several new ideas including 1 it removes portions of the graph and replaces them with groups which are used to keep track of the symmetries of these portions 2 it maintains with each group a tower of equivalence relations which allow a decomposition of the group these towers are called towers of k actions

pushdown processes a pushdown game is a two player perfect information infinite game on a transition graph of a pushdown automaton a winning condition in such a game is defined in terms of states appearing infinitely often in the play it is shown that if there is a winning strategy in a pushdown game then there is a winning strategy realized by a pushdown automaton an exptime procedure for finding a winner in a pushdown game is presented the procedure is then used to solve the model checking problem for the pushdown processes and the propositional calculus the problem is shown to be dexptime complete

uniform strategies rational relations and jumping automata a general concept of uniform strategies has recently been proposed as a relevant notion in game theory for computer science which subsumes various notions from the literature it relies on properties involving sets of plays in two player turn based arenas equipped with arbitrary binary relations between plays these properties are expressed in a language based on ctl with a quantifier over related plays there are two semantics for our quantifier a strict one and a full one that we study separately regarding the strict semantics the existence of a uniform strategy is undecidable for rational binary relations but introducing jumping tree automata and restricting attention to recognizable relations allows us to establish a 2 exptime complete complexity and still capture a class of two player imperfect information games with epistemic temporal objectives regarding the full semantics relying on information set automata we establish that the existence of a uniform strategy is decidable for rational relations and we provide a nonelementary synthesis procedure we also exhibit an essentially optimal subclass of rational relations for which the problem becomes 2 exptime complete considering rich classes of relations makes the theory of uniform strategies powerful it directly entails various results in logics of knowledge and time some of them already known and others new

on effective construction of the greatest solution of language inequality xa bx in this paper we consider effective constructions of the greatest solution of the language inequality xa bx it has been proved by kunc in 2005 that the greatest solution of xa bx is regular provided that b is regular no matter what a is however this proof is based on kruskal s tree theorem and does not provide any effective way to construct the greatest solution we focus on this gap in this paper we give an effective construction of the greatest solution for the following two cases i a b are regular and there exists k 1 such that pref b a k b kpref b a where pref b is the set of prefixes of words in b ii a b are regular and b is a code with finite decoding delay our construction takes the point of view of games as shown by kunc in his regularity proof the construction of the greatest solution can be reduced to the construction of the winning region of a two player game our contribution is to show that the winning regions of the two player game for the two cases can be constructed effectively the main ingredient of the construction for the first case is a shrinking lemma for the words on which one of the players has a winning strategy while the construction for the second case is based on the observation that the two player game can be reduced to a two player reachability game played on the transition graph of a one counter machine

an existence result for a class of shape optimization problems given a bounded open subset of rn we prove the existence of a minimum point for a functional f defined on the family a of all quasiopen subsets of under the assumption that f is decreasing with respect to set inclusion and that f is lower semicontinuous on a with respect to a suitable topology related to the resolvents of the laplace operator with dirichlet boundary condition applications are given to the existence of sets of prescribed volume with minimal kth eigenvalue or with minimal capacity with respect to a given elliptic operator

borel selectors for upper semi continuous multi valued functions on presente un theoreme de selection plus fin qui s applique dans plusieurs espaces de banach non separables

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem

complexity results on branching time pushdown model checking the model checking problem of pushdown systems pmc problem for short against standard branching temporal logics has been intensively studied in the literature in particular for the modal m calculus the most powerful branching temporal logic used for verification the problem is known to be exptime complete even for a fixed formula the problem remains exptime complete also for the logic ctl which corresponds to a fragment of the alternation free modal m calculus for the logic ctl the problem is known to be in 2exptime in this paper we show that the complexity of the pmc problem for ctl is in fact 2exptime complete moreover we give a new optimal algorithm to solve this problem based on automata theoretic techniques finally we prove that the program complexity of the pmc problem against ctl i e the complexity of the problem in terms of the size of the system is exptime complete

an axiomatization of bisimulation quantifiers via the µ calculus this paper deals with bisimulation quantifiers logic bql that is the extension of propositional dynamic logic pdl with the so called bisimulation quantifiers this logic is expressively equivalent to the calculus an extension of modal logic with extremal fixpoints albeit its formulas are easier to understand in this work we provide a complete axiomatization of bql based on certain normal form results for the calculus obtained by janin and walukiewicz

alternating finite automata and star free languages for a given extended regular expression e we construct an equational representation of an alternating finite automaton accepting the language denoted by e for star free extended regular expressions the construction yields a loop free alternating finite automaton also the inclusion in the opposite direction holds and thus we obtain a new characterization for the class of star free languages

model checking mobile ambients we settle the complexity bounds of the model checking problem for the ambient calculus with public names against the ambient logic we show that if either the calculus contains replication or the logic contains the guarantee operator the problem is undecidable in the case of the replication free calculus and guarantee free logic we prove that the problem is pspace complete for the complexity upper bound we devise a new representation of processes that remains of polynomial size during process execution this allows us to keep the model checking procedure in polynomial space moreover we prove pspace hardness of the problem for several quite simple fragments of the calculus and the logic this suggests that there are no interesting fragments with polynomial time model checking algorithms

efficient timed model checking for discrete time systems we consider model checking of timed temporal formulae in durational transition graphs dtgs i e kripke structures where transitions have integer durations two semantics for dtgs are presented and motivated we consider timed versions ofctl where subscripts put quantitative constraints on the time it takes before a property is satisfied we exhibit an important gap between logics where subscripts of the form c exact duration are allowed and simpler logics that only allow subscripts of the form c or c bounded duration without exact durations model checking can be done in polynomial time but with exact durations it becomes 2p complete or pspace complete depending on the considered semantics

characterization for the kuratowski limits of a sequence of sobolev spaces the purpose of this paper is to study the strong lower and the weak upper limits in the sense of kuratowski of a sequence of sobolev spaces w1 p0 n n nand compare them to a fixed spacew1 p0 the results are expressed in terms of the behavior of the local capacity in balls of the complementary sets if these two limits coeincide we obtain necessary and sufficient conditions for the convergence in the sense of mosco of the sobolev spaces hence for the continuity of the solution of a dirichlet problem associated to thep laplacian in terms of the geometric domain variation

model checking higher order programs we propose a novel verification method for higher order functional programs based on higher order model checking or more precisely model checking of higher order recursion schemes recursion schemes for short the most distinguishing feature of our verification method for higher order programs is that it is sound complete and automatic for the simply typed calculus with recursion and finite base types and for various program verification problems such as reachability flow analysis and resource usage verification we first show that a variety of program verification problems can be reduced to model checking problems for recursion schemes by transforming a program into a recursion scheme that generates a tree representing all the interesting possible event sequences of the program we then develop a new type based model checking algorithm for recursion schemes and implement a prototype recursion scheme model checker to our knowledge this is the first implementation of a recursion scheme model checker experiments show that our model checker is reasonably fast despite the worst case time complexity of recursion scheme model checking being hyperexponential in general altogether the results provide a new promising approach to verification of higher order functional programs

a framework for compositional verification of multi valued systems via abstraction refinement we present a framework for fully automated compositional verification of calculus specifications over multi valued systems based on abstraction and refinement in a multi valued model of a system both the system transitions and the state labels are assigned values from a lattice we formalize our framework based on bilattices consisting of a truth lattice and an information lattice formulas are interpreted on the truth lattice the information lattice determines how definite the value is in terms of the concrete system being modeled our compositional approach views each component as an abstraction of the entire system and checks it separately only if all individual checks return indefinite values the parts of the components which are responsible for these values are composed and checked if the latter check is still indefinite a refinement of the multi valued system is needed refinement is aimed at increasing the information level of model details

reachability in recursive markov decision processes we consider a class of infinite state markov decision processes generated by stateless pushdown automata this class corresponds to 112 player games over graphs generated by bpa systems or equivalently 1 exit recursive state machines an extended reachability objective is specified by two sets s and t of safe and terminal stack configurations where the membership to s and t depends just on the top of the stack symbol the question is whether there is a suitable strategy such that the probability of hitting a terminal configuration by a path leading only through safe configurations is equal to or different from a given email protected 0 1 we show that the qualitative extended reachability problem is decidable in polynomial time and that the set of all configurations for which there is a winning strategy is effectively regular more precisely this set can be represented by a deterministic finite state automaton with a fixed number of control states this result is a generalization of a recent theorem by etessami and yannakakis which says that the qualitative termination for 1 exit rmdps which exactly correspond to our 112 player bpa games is decidable in polynomial time interestingly the properties of winning strategies for the extended reachability objectives are quite different from the ones for termination and new observations are needed to obtain the result as an application we derive the exptime completeness of the model checking problem for 112 player bpa games and qualitative pctl formulae

inverse monoids decidability and complexity of algebraic questions this paper investigates the word problem for inverse monoids generated by a set c subject to relations of the form e f where e and f are both idempotents in the free inverse monoid generated by c it is shown that for every fixed monoid of this form the word problem can be solved both in linear time on a ram as well as in deterministic logarithmic space which solves an open problem of margolis and meakin for the uniform word problem where the presentation is part of the input exptime completeness is shown for the cayley graphs of these monoids it is shown that the first order theory with regular path predicates is decidable regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language as a corollary the decidability of the generalized word problem is deduced

three partition refinement algorithms we present improved partition refinement algorithms for three problems lexicographic sorting relational coarsest partition and double lexical ordering our double lexical ordering algorithm uses a new efficient method for unmerging two sorted sets

efficient ctl model checking for pushdown systems abstract pushdown systems pds are well adapted to model sequential programs with possibly recursive procedure calls therefore it is important to have efficient model checking algorithms for pdss we consider in this paper ctl model checking for pdss we consider the standard ctl model checking problem where whether a configuration of a pds satisfies an atomic proposition or not depends only on the control state of the configuration we consider also ctl model checking with regular valuations where the set of configurations in which an atomic proposition holds is a regular language we reduce these problems to the emptiness problem in alternating buchi pushdown systems and we give an algorithm to solve this emptiness problem our algorithms are more efficient than the other existing algorithms for ctl model checking for pdss in the literature we implemented our techniques in a tool and we applied it to different case studies our results are encouraging in particular we were able to confirm the existence of known bugs in linux source code

a rice style theorem for parallel automata we present a general result similar to rice s theorem concerning the complexity of detecting properties on finite automata enriched by bounded cooperative concurrency such as statecharts and abstract parallel automata which we denote by cfas concurrent finite automata on one extreme the complexity of detecting non trivial properties that preserve equivalence of machines i e properties of the accepted language on finite automata can be as little as o 1 on the other extreme rice s theorem states that all such properties on turing machines are undecidable we state that all the non trivial properties of the regular or w regular languages are pspace hard on cfas with e moves and on cfas without e moves accepting infinite words we also extend this result to cfas without e moves accepting finite words that satisfy a condition that holds for many properties

krivine machines and higher order schemes we propose a new approach to analyzing higher order recursive schemes many results in the literature use automata models generalizing pushdown automata most notably higher order pushdown automata with collapse cpda instead we propose to use the krivine machine model compared to cpda this model is closer to lambda calculus and incorporates nicely many invariants of computations as for example the typing information the usefulness of the proposed approach is demonstrated with new proofs of two central results in the field the decidability of the local and global model checking problems for higher order schemes with respect to the mu calculus

alternating time temporal logic temporal logic comes in two varieties linear time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system branching time temporal logic allows explicit existential and universal quantification over all paths we introduce a third more general variety of temporal logic alternating time temporal logic offers selective quantification over those paths that are possible outcomes of games such as the game in which the system and the environment alternate moves while linear time and branching time logics are natural specification languages for closed systems alternating time logics are natural specification languages for open systems for example by preceding the temporal operator eventually with a selective path quantifier we can specify that in the game between the system and the environment the system has a strategy to reach a certain state the problems of receptiveness realizability and controllability can be formulated as model checking problems for alternating time formulas depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators we obtain the two alternating time temporal logics atl and atl atl and atl are interpreted over concurrent game structures every state transition of a concurrent game structure results from a choice of moves one for each player the players represent individual components and the environment of an open system concurrent game structures can capture various forms of synchronous composition for open systems and if augmented with fairness constraints also asynchronous composition over structures without fairness constraints the model checking complexity of atl is linear in the size of the game structure and length of the formula and the symbolic model checking algorithm for ctl extends with few modifications to atl over structures with weak fairness constraints atl model checking requires the solution of 1 pair rabin games and can be done in polynomial time over structures with strong fairness constraints atl model checking requires the solution of games with boolean combinations of buchi conditions and can be done in pspace in the case of atl the model checking problem is closely related to the synthesis problem for linear time formulas and requires doubly exponential time

when not losing is better than winning abstraction and refinement for the full μ calculus this work presents a novel game based approach to abstraction refinement for the full m calculus interpreted over 3 valued semantics a novel notion of non losing strategy is introduced and exploited for refinement previous works on refinement in the context of 3 valued semantics require a direct algorithm for solving a 3 valued model checking game this was necessary in order to have the information needed for refinement available on one game board in contrast while still considering a 3 valued model checking game here we reduce the problem of solving the game to solving two 2 valued model checking parity games in case the result is indefinite don t know the corresponding non losing strategies when combined hold all the information needed for refinement this approach is beneficial since it can use any solver for 2 valued parity games thus it can take advantage of newly developed such algorithms with improved complexity

predecessor existence problems for finite discrete dynamical systems we study the predecessor existence problem for finite discrete dynamical systems given a finite discrete dynamical system s and a configuration c the predecessor existence or pre problem is to determine whether there is a configuration c such that s has a transition from c to c in addition to the decision version we also study the following variants the predecessor existence or pre problem counting the number of predecessors the unique predecessor existence or upre problem deciding whether there is a unique predecessor and the ambiguous predecessor existence or apre problem given a configuration c and a predecessor c of c deciding whether there is a different predecessor c of c general techniques are presented for simultaneously characterizing the computational complexity of the pre problem and its three variants our hardness results are based on the concept of simultaneous reductions single transformations that can be used to simultaneously prove the hardness of the different variants of the pre problem for their respective complexity classes our easiness results are based on dynamic programming and they extend the previous results on pre problem for one dimensional cellular automata the hardness results together with the easiness results provide a tight separation between easy and hard instances further the results imply similar bounds for other classes of finite discrete dynamical systems including discrete hopfield and recurrent neural networks concurrent state machines systolic networks and one and two dimensional cellular automata our results extend the earlier results of green sutner and orponen on the complexity of the predecessor existence problem and its variants

symbolic model checking for m calculus requires exponential time we investigate the complexity of symbolic model checking for calculus the exponential time lower bound is proved moreover this lower bound holds even for a fixed alternation free formula

the theory of ends pushdown automata and second order logic abstract a class of edge labeled graphs called context free are defined according to their behavior at infinity such graphs are generalizations of cayley graphs of context free groups they are also shown to be definable in a very natural way in terms of push down automata using rabin s theorem on the monadic second order theory of the finite binary tree these graphs are also shown to have a decidable monadic second order theory questions about tiling systems and cellular automata operating on these graphs are decidable even when the analogous questions are not when the systems operate on a two dimensional grid

on model checking for the m calculus and its fragments in this paper we consider the model checking problem for the calculus and show that it is succinctly equivalent to the non emptiness problem of finite state automata on infinite binary trees with the parity acceptance condition we also present efficient model checking algorithms for two rich subclasses of the calculus formulas and relate their expressive power to well known extensions of branching time temporal logics

propositional dynamic logic of looping and converse is elementarily decidable propositional dynamic logic is a formal system for reasoning about the before after behavior of regular program schemes an extension of propositional dynamic logic which includes both an infinite looping construct and a converse or backtracking construct is considered and it is proved that the satisfiability problem for this logic is elementarily decidable in order to establish this result deterministic two way automata on infinite trees are defined and it is shown how they can be simulated by nondeterministic one way automata the satisfiability problem for propositional dynamic logic of looping and converse is then reduced to the emptiness problem for these two way automata

the church problem for expansions of n by unary predicates for a two variable formula b x y of monadic logic of order mlo the church synthesis problem concerns the existence and construction of a finite state operator y f x such that b x f x is universally valid over nat buchi and landweber 1969 proved that the church synthesis problem is decidable we investigate a parameterized version of the church synthesis problem in this extended version a formula b and a finite state operator f might contain as a parameter a unary predicate p a large class of predicates p is exhibited such that the church problem with the parameter p is decidable our proofs use composition method and game theoretical techniques

fixed point characterization of infinite behavior of finite state systems abstract infinite behavior of nondeterministic finite state automata running over infinite trees or more generally over elements of an arbitrary algebraic structure is characterized by a calculus of fixed point terms interpreted in powerset algebras these terms involve the least and greatest fixed point operators and disjunction as the only logical operation a tight correspondence is established between a hierarchy of rabin indices of automata and a hierarchy induced by alternation of the least and greatest fixed point operators it is shown that in the powerset algebra of trees constructed from a set of functional symbols the fixed point hierarchy is infinite unless all the symbols are unary i e trees are words it is also shown that an interpretation of a closed fixed point term in any powerset algebra can be factorized through the interpretation of this term in the powerset algebra of trees from which it is deduced that the question whether a term denotes always can be answered in polynomial time

dichotomy results for fixed point counting in boolean dynamical systems we present dichotomy theorems regarding the computational complexity of counting fixed points in boolean discrete dynamical systems i e finite discrete dynamical systems over the domain 0 1 for a class f of boolean functions and a class g of graphs an f g system is a boolean dynamical system with local transitions functions lying in f and graphs in g we show that if local transition functions are given by lookup tables then the following complexity classification holds let f be a class of boolean functions closed under superposition and let g be a graph class closed under taking minors if f contains all min functions all max functions or all self dual and monotone functions and g contains all planar graphs then it is p complete to compute the number of fixed points in an f g system otherwise it is computable in polynomial time we also prove a dichotomy theorem for the case that local transition functions are given by formulas over logical bases this theorem has a significantly more complicated structure than the theorem for lookup tables a corresponding theorem for boolean circuits coincides with the theorem for formulas

the complexity of propositional linear temporal logics the complexity of satisfiability and determination of truth in a particular finite structure are considered for different propositional linear temporal logics it is shown that these problems are np complete for the logic with f and are pspace complete for the logics with f x with u with u s x operators and for the extended logic with regular operators given by wolper

a partial order approach to branching time logic model checking abstract partial order techniques enable reducing the size of the state space used for model checking thus alleviating the state space explosion problem these reductions are based on selecting a subset of the enabled operations from each program state so far these methods have been studied implemented and demonstrated for assertional languages that model the executions of a program as computation sequences in particular the linear temporal logic the present paper shows for the first time how this approach can be applied to languages that model the behavior of a program as a tree we study here partial order reductions for branching temporal logics e g the logics ctl and ctl with the next time operator removed and process algebra logics such as hennesy milner logic with actions conditions on the selection of subset of successors from each state during the state space construction which guarantee reduction that preserves ctl properties are given the experimental results provided show that the reduction is substantial

alternation and bounded concurrency are reverse equivalent numerous models of concurrency have been considered in the framework of automata among the more interesting concurrency models are classical nondeterminism and pure concurrency the two facets of alternation and the bounded concurrency model bounded concurrency was previously considered to be similar to nondeterminism and pure concurrency in the sense of the succinctness of automata augmented with these features in this paper we show that when viewed more broadly the power of succinctness of bounded concurrency is in fact most similar to the power of alternation our contribution is that just like nondeterminism and pure concurrency are complement equivalent bounded concurrency and alternation are reverse equivalent over finite automata the reverse equivalence is expressed by the existence of polynomial transformations in both directions between bounded concurrency and alternation for the reverse of the language accepted by the other it follows that bounded concurrency is double exponentially more succinct than dfas with respect to reverse while alternation only saves one exponent this is as opposed to the direct case where alternation saves two exponents and bounded concurrency saves only one an immediate corollary is that for languages over a one letter alphabet bounded concurrency and alternation are equivalent we complete the picture of succinctness results for these languages by considering the different combinations of the concurrency models using additional lower bounds

stability and uniqueness for the crack identification problem this paper deals with the identifiability of nonsmooth defects by boundary measurements and the stability of their detection we introduce and analyze a new pointwise regularity concept at the boundary of an open set which turns out to play a crucial role in the identifiability of defects by two boundary measurements as a consequence we prove the unique identifiability for a large class of closed sets including sets with an infinite number of connected components of positive capacity and totally disconnected sets in order to rigorously justify numerical approximation results of defects by optimal design methods we prove a geometric stability result of the defect identification process without any a priori smoothness assumptions

generalising automaticity to modal properties of finite structures we introduce a complexity measure of modal properties of finite structures which generalises the automaticity of languages it is based on graph automata like devices called labelling systems we define a measure of the size of a structure that we call rank and show that any modal property of structures can be approximated up to any fixed rank n by a labelling system the function that takes n to the size of the smallest labelling system doing this is called the labelling index of the property we demonstrate that this is a useful and fine grained measure of complexity and show that it is especially well suited to characterise the expressive power of modal fixed point logics from this we derive several separation results of modal and non modal fixed point logics some of which are already known whereas others are new

alternating time temporal logics with linear past abstract we investigate the succinctness gap between two known equally expressive and different linear past extensions of standard atl we establish by formal non trivial arguments that the memoryful linear past extension the history leading to the current state is taken into account can be exponentially more succinct than the standard local linear past extension the history leading to the current state is forgotten as a second contribution we consider the atl like fragment denoted atl l p of the known memoryful linear past extension of atl we show that atl l p is strictly more expressive than atl and interestingly it can be exponentially more succinct than the more expressive logic atl moreover we prove that both satisfiability and model checking for the logic atl l p are exptime complete

on the expressivity and complexity of quantitative branching time temporal logics we investigate extensions of ctl allowing to express quantitative requirements about an abstract notion of time in a simple discrete time framework and study the expressive power of several relevant logics when only subscripted modalities are used polynomial time model checking is possible even for the largest logic we consider while the introduction of freeze quantifiers leads to a complexity blow up

weak semi continuity of the duality product in sobolev spaces given a weakly convergent sequence of positive functions in w01 p we prove the equivalence between its convergence in the sense of obstacles and the lower semi continuity of the term by term duality product associated to the p laplacian of weakly convergent sequences of p superharmonic functions of w01 p this result implicitly gives new characterizations for both the convergence in the sense of obstacles of a weakly convergent sequence of positive functions and for the weak l s c of the duality product

complete axiomatization and decidability of alternating time temporal logic alternating time temporal logic atl introduced by alur henzinger and kupferman is a logical formalism for the specification and verification of open systems involving multiple autonomous players agents components in particular this logic allows for the explicit expression of coalition abilities in such systems modelled as infinite transition games between the coalition and its complement formally atl is a non normal multi modal extension of ctl regarded as a one player fragment of atl with temporal operators indexed by coalitions of players and thus expressing selective quantification over those paths which can be effected as outcomes of infinite transition games between the coalition and its complement we present a sound and complete axiomatization of the logic atl based on pauly s axiomatization of his coalition logic augmented with axioms and rules for fixed point formulae characterizing the temporal operators the completeness proof is by construction of a bounded branching tree model for each atl consistent formula these models can be folded into finite models thus rendering the finite model property for atl we also describe an automata based decision procedure for atl by translating the satisfiability problem to the nonemptiness problem for alternating automata on infinite trees when considering formulae over a fixed finite set of players the decidability problem is shown to be exptime complete