ltl over integer periodicity constraints periodicity constraints are used in many logical formalisms in fragments of presburger ltl in calendar logics and in logics for access control to quote a few examples in the paper we introduce the logic pltlmod an extension of linear time temporal logic ltl with past time operators whose atomic formulae are defined from a first order constraint language dealing with periodicity although the underlying constraint language is a fragment of presburger arithmetic shown to admit a pspace complete satisfiability problem we establish that pltlmod model checking and satisfiability problems remain in pspace as plain ltl full presburger ltl is known to be highly undecidable this is particularly interesting for dealing with periodicity constraints since the language of pltlmod has a language more concise than existing languages and the temporalization of our first order language of periodicity constraints has the same worst case complexity as the underlying constraint language finally we show examples of introduction the quantification in the logical language that provide to pltlmod expspace complete problems as another application we establish that the equivalence problem for extended single string automata known to express the equality of time granularities is pspace complete by designing a reduction from qbf and by using our results for pltlmod

approximate abstractions of stochastic hybrid systems we present a constructive procedure for obtaining a finite approximate abstraction of a discrete time stochastic hybrid system the procedure consists of a partition of the state space of the system and depends on a controllable parameter given proper continuity assumptions on the model the approximation errors introduced by the abstraction procedure are explicitly computed and it is shown that they can be tuned through the parameter of the partition the abstraction is interpreted as a markov set chain we show that the enforcement of certain ergodic properties on the stochastic hybrid model implies the existence of a finite abstraction with finite error in time over the concrete model and allows introducing a finite time algorithm that computes the abstraction

testing preorders for probabilistic processes can be characterized by simulations the domestic dog may be exceptionally well suited for behavioral genetic studies owing to its population history and the striking behavior differences among breeds to explore to what extent and ho

enhancing unsatisfiable cores for ltl with information on temporal relevance ltl is frequently used to express specifications in many domains such as embedded systems or business processes witnesses can help to understand why an ltl specification is satisfiable and a number of approaches exist to make understanding a witness easier in the case of unsatisfiable specifications unsatisfiable cores ucs i e parts of an unsatisfiable formula that are themselves unsatisfiable are a well established means for debugging however little work has been done to help understanding a uc of an unsatisfiable ltl formula in this paper we suggest to enhance a uc of an unsatisfiable ltl formula with information about the time points at which the subformulas of the uc are relevant for unsatisfiability in previous work we showed how to obtain a uc in ltl by translating the ltl formula into a clausal normal form applying temporal resolution extracting a clausal uc from the resolution proof and mapping the clausal uc back to a uc in ltl in this paper we extend that method by extracting information at which time points the clauses of a clausal uc are relevant for unsatisfiability from a resolution proof and by transferring that information to a uc in ltl we implement our method in trp and we experimentally evaluate it

an automata theoretic approach to constraint ltl we consider an extension of linear time temporal logic ltl with constraints interpreted over a concrete domain we use a new automata theoretic technique to show pspace decidability of the logic for the constraint systems z and n along the way we give an automata theoretic proof of a result of balbiani and condotta when the constraint system satisfies the completion property our decision procedures extend easily to handle extensions of the logic with past time operators and constants as well as an extension of the temporal language itself to monadic second order logic finally we show that the logic becomes undecidable when one considers constraint systems that allow a counting mechanism

e connections of abstract description systems combining knowledge representation and reasoning formalisms is an important and challenging task it is important because non trivial ai applications often comprise different aspects of the world thus requiring suitable combinations of available formalisms modeling each of these aspects it is challenging because the computational behavior of the resulting hybrids is often much worse than the behavior of their components in this paper we propose a new combination method which is computationally robust in the sense that the combination of decidable formalisms is again decidable and which nonetheless allows non trivial interactions between the combined components the new method called e connection is defined in terms of abstract description systems adss a common generalization of description logics many logics of time and space as well as modal and epistemic logics the basic idea of e connections is that the interpretation domains of n combined systems are disjoint and that these domains are connected by means of n ary link relations we define several natural variants of e connections and study in depth the transfer of decidability from the component systems to their e connections

on probabilistic timed automata we propose a model of probabilistic timed automaton which substitutes for the nondeterminism of an ordinary timed automaton a new one directly drawn from markov decision processes by means of actions which provide a probabilistic distribution over transitions using buchi acceptance conditions timed automata can refer timing properties as during every open time interval of length 1 at least one message is delivered a policy is a mechanism which solves the non determinism by choosing for each finite run an action and the time moment of the next transition step implied by this action we prove that given a probabilistic timed automaton a there exists a markov memoryless policy which maximizes the probability p of the set of accepting runs realized by this policy this policy as well as the maximal value of p are computable in polytime in the size of the region automaton of a this result provides an algorithm of model checking for properties like there is a policy which realizes a correct behavior of the system with probability at least p

qualitative analysis of gene regulatory networks by temporal logic in this article we propose a novel formalism to model and analyse gene regulatory networks using a well established formal verification technique we model the possible behaviours of networks by logical formulae in linear temporal logic ltl by checking the satisfiability of ltl it is possible to check whether some or all behaviours satisfy a given biological property which is difficult in quantitative analyses such as the ordinary differential equation approach owing to the complexity of ltl satisfiability checking analysis of large networks is generally intractable in this method to mitigate this computational difficulty we developed two methods one is a modular checking method where we divide a network into subnetworks check them individually and then integrate them the other is an approximate analysis method in which we specify behaviours in simpler formulae which compress or expand the possible behaviours of networks in the approximate method we focused on network motifs and presented approximate specifications for them we confirmed by experiments that both methods improved the analysis of large networks we propose a novel qualitative method for analysing gene networks based on formal verification technique behaviours and properties of networks are described in temporal logic formulae by checking satisfiability of the formula we can analyse properties of the network to improve the efficiency of analysis we developed the modular and approximate method

concurrent games with tail objectives we study infinite stochastic games played by two players over a finite state space with objectives specified by sets of infinite traces the games are concurrent players make moves simultaneously and independently stochastic the next state is determined by a probability distribution that depends on the current state and chosen moves of the players and infinite proceed for an infinite number of rounds the analysis of concurrent stochastic games can be classified into quantitative analysis analyzing the optimum value of the game and e optimal strategies that ensure values within e of the optimum value and qualitative analysis analyzing the set of states with optimum value 1 and e optimal strategies for the states with optimum value 1 we consider concurrent games with tail objectives i e objectives that are independent of the finite prefix of traces and show that the class of tail objectives is strictly richer than that of the w regular objectives we develop new proof techniques to extend several properties of concurrent games with w regular objectives to concurrent games with tail objectives we prove the positive limit one property for tail objectives the positive limit one property states that for all concurrent games if the optimum value for a player is positive for a tail objective f at some state then there is a state where the optimum value is 1 for the player for the objective f we also show that the optimum values of zero sum strictly conflicting objectives games with tail objectives can be related to equilibrium values of nonzero sum not strictly conflicting objectives games with simpler reachability objectives a consequence of our analysis presents a polynomial time reduction of the quantitative analysis of tail objectives to the qualitative analysis for the subclass of one player stochastic games markov decision processes

global and local views of state fairness abstract in this paper we compare global and local versions of state fairness for systems of concurrent programs and petri nets we then investigate complexity and decidability issues for the fair nontermination problem it turns out that for systems of concurrent boolean programs and 1 bounded petri nets the problem is pspace complete with respect to global state fairness but exptime complete with respect to local state fairness for general systems of concurrent programs both the globally and locally state fair nontermination problems are undecidable in fact they are 1 complete and complete respectively 1 on the other hand the problem is decidable for general petri nets with respect to global state fairness and undecidable 1 complete with respect to local state fairness

stable high order methods for solving elliptic equations with large first order terms abstract stable high order methods are developed for solving elliptic partial differential equations with large first order terms in particular second order and fourth order methods are developed by stabilizing the central difference method the methods can solve second order linear elliptic partial differential equations and are found to be stable and accurate in all tested examples results are presented for the two dimensional convection diffusion equation involving problems with and without boundary layers the methods converged for all values of parameters attempted and the results compared favorably with other methods the methods developed in this paper are accurate stable easy to use and applicable to other problems

satisfiability in many valued sentential logic is np complete abstract cook s np completeness theorem is extended to all many valued sentential logics including the infinite valued calculus of ukasiewicz

unambiguous buchi automata in this paper we introduce a special class of buchi automata called unambiguous in these automata any infinite word labels exactly one path going infinitely often through final states the word is recognized by the automaton if this path starts in an initial state the main result of the paper is that any rational set of infinite words is recognized by such an automaton we give two proofs of this result we also provide several related results

metrics for labelled markov processes the notion of process equivalence of probabilistic processes is sensitive to the exact probabilities of transitions thus a slight change in the transition probabilities will result in two equivalent processes being deemed no longer equivalent this instability is due to the quantitative nature of probabilistic processes in a situation where the process behavior has a quantitative aspect there should be a more robust approach to process equivalence this paper studies a metric between labelled markov processes this metric has the property that processes are at zero distance if and only if they are bisimilar the metric is inspired by earlier work on logics for characterizing bisimulation and is related in spirit to the hutchinson metric

on finite state approximants for probabilistic computation tree logic we generalize the familiar semantics for probabilistic computation tree logic from finite state to infinite state labelled markov chains such that formulas are interpreted as measurable sets then we show how to synthesize finite state abstractions which are sound for full probabilistic computation tree logic and in which measures are approximated by monotone set functions this synthesis of sound finite state approximants also applies to finite state systems and is a probabilistic analogue of predicate abstraction sufficient and always realizable conditions are identified for obtaining optimal such abstractions for probabilistic propositional modal logic

decidability of infinite state timed ccp processes and first order ltl the ntcc process calculus is a timed concurrent constraint programming ccp model equipped with a first order linear temporal logic ltl for expressing process specifications a typical behavioral observation in ccp is the strongest postcondition sp the ntcc sp denotes the set of all infinite output sequences that a given process can exhibit the verification problem is then whether the sequences in the sp of a given process satisfy a given ntcc ltl formula this paper presents new positive decidability results for timed ccp as well as for ltl in particular we shall prove that the following problems are decidable 1 the sp equivalence for the so called locally independent ntcc fragment unlike other fragments for which similar results have been published this fragment can specify infinite state systems 2 verification for locally independent processes and negation free first order formulae of the ntcc ltl 3 implication for such formulae 4 satisfiability for a first order fragment of manna and pnueli s ltl the purpose of the last result is to illustrate the applicability of ccp to well established formalisms for concurrency

quantitative temporal logics over the reals pspace and below in many cases the addition of metric operators to qualitative temporal logics tls increases the complexity of satisfiability by at least one exponential while common qualitative tls are complete for np or pspace their metric extensions are often expspace complete or even undecidable in this paper we exhibit several metric extensions of qualitative tls of the real line that are at most pspace complete and analyze the transition from np to pspace for such logics our first result is that the logic obtained by extending since until logic of the real line with the operators sometime within n time units in the past future is still pspace complete in contrast to existing results we also capture the case where n is coded in binary and the finite variability assumption is not made to establish containment in pspace we use a novel reduction technique that can also be used to prove tight upper complexity bounds for many other metric tls in which the numerical parameters to metric operators are coded in binary we then consider metric tls of the reals that do not offer any qualitative temporal operators in such languages the complexity turns out to depend on whether binary or unary coding of parameters is assumed satisfiability is still pspace complete under binary coding but only np complete under unary coding

breather and soliton wave families for the sine gordon equation the approach of steuerwaldlamb for obtaining exact solutions of the sinegordon equation is further developed through transformations by means of which complicated solutions of the corresponding odes are obtained without any explicit integration polynomial and nonpolynomial transformations are discussed and several of them are used to obtain breather and soliton solutions of the sinegordon equation the obtained solutions are discussed from the point of view of the electrodynamics of the long josephson junction

high order compact exponential finite difference methods for convection diffusion type problems a class of high order compact hoc exponential finite difference fd methods is proposed for solving one and two dimensional steady state convection diffusion problems the newly proposed hoc exponential fd schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection dominated problems the o h 4 compact exponential fd schemes developed for the one dimensional 1d problems produce diagonally dominant tri diagonal system of equations which can be solved by applying the tridiagonal thomas algorithm for the two dimensional 2d problems o h 4 k 4 compact exponential fd schemes are formulated on the nine point 2d stencil and the line iterative approach with alternating direction implicit adi procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one dimensional tridiagonal thomas algorithm with a considerable saving in computing time to validate the present hoc exponential fd methods three linear and nonlinear problems mostly with boundary or internal layers where sharp gradients may appear due to high peclet or reynolds numbers are numerically solved comparisons are made between analytical solutions and numerical results for the currently proposed hoc exponential fd methods and some previously published hoc methods the present hoc exponential fd methods produce excellent results for all test problems it is shown that besides including the excellent performances in computational accuracy efficiency and stability the present method has the advantage of better scale resolution the method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2d incompressible navier stokes equations using the stream function vorticity formulation

recursive markov decision processes and recursive stochastic games we introduce recursive markov decision processes rmdps and recursive simple stochastic games rssgs which are classes of finitely presented countable state mdps and zero sum turn based perfect information stochastic games they extend standard finite state mdps and stochastic games with a recursion feature we study the decidability and computational complexity of these games under termination objectives for the two players one player s goal is to maximize the probability of termination at a given exit while the other player s goal is to minimize this probability in the quantitative termination problems given an rmdp or rssg and probability p we wish to decide whether the value of such a termination game is at least p or at most p in the qualitative termination problem we wish to decide whether the value is 1 the important 1 exit subclasses of these models 1 rmdps and 1 rssgs correspond in a precise sense to controlled and game versions of classic stochastic models including multitype branching processes and stochastic context free grammars where the objective of the players is to maximize or minimize the probability of termination extinction we provide a number of upper and lower bounds for qualitative and quantitative termination problems for rmdps and rssgs we show both problems are undecidable for multi exit rmdps but are decidable for 1 rmdps and 1 rssgs specifically the quantitative termination problem is decidable in pspace for both 1 rmdps and 1 rssgs and is at least as hard as the square root sum problem a well known open problem in numerical computation we show that the qualitative termination problem for 1 rmdps i e a controlled version of branching processes can be solved in polynomial time both for maximizing and minimizing 1 rmdps the qualitative problem for 1 rssgs is in np conp and is at least as hard as the quantitative termination problem for condon s finite state simple stochastic games whose complexity remains a well known open problem finally we show that even for 1 rmdps more general qualitative and quantitative model checking problems with respect to linear time temporal properties are undecidable even for a fixed property

the propositional dynamic logic of deterministic well structured programs abstract we consider a restricted propositional dynamic logic strict deterministic propositional dynamic logic sdpdl which is appropriate for reasoning about deterministic well structured programs in contrast to pdl for which the validity problem is known to be complete in deterministic exponential time the validity problem for sdpdl is shown to be polynomial space complete we also show that sdpdl is less expressive than pdl and give a complete axiomatization for it the results rely on structure theorems for models of satisfiable sdpdl formulas and the proofs give insight into the effects of nondeterminism on intractability and expressiveness in program logics

iterated boolean games iterated games are well known in the game theory literature we study iterated boolean games these are games in which players repeatedly choose truth values for boolean variables they have control over our model of iterated boolean games assumes that players have goals given by formulae of linear temporal logic ltl a formalism for expressing properties of state sequences in order to represent the strategies of players in such games we use a finite state machine model after introducing and formally defining iterated boolean games we investigate the computational complexity of their associated game theoretic decision problems as well as semantic conditions characterising classes of ltl properties that are preserved by equilibrium points pure strategy nash equilibria whenever they exist

an incremental block line gauss seidel method for the navier stokes equations a block line gauss seidel lgs method is developed for solving the incompressible and compressible navier stokes equations in two dimensions the method requires only one block tridiagonal solution process per iteration and is consequently faster per step than the linearized block adi methods results are presented for both incompressible and compressible separated flows in all cases the proposed block lgs method is more efficient than the block adi methods furthermore for high reynolds number weakly separated incompressible flow in a channel which proved to be an impossible task for a block adi method solutions have been obtained very efficiently by the new scheme

bounded model checking of strategy ability with perfect recall the paper works with a logic which has the expressiveness to quantify over strategies of bounded length the semantics of the logic is based on systems with multiple agents agents have incomplete information about the underlying system state and their strategies are based on perfect recall memory over observations and local actions the computational complexity of model checking is shown to be pspace complete we give two bdd based model checking algorithms the algorithms are implemented in a model checker and experimental results are reported to show their applications

fair petri nets and structural induction for rings of processes abstract we present a structural induction theorem for rings consisting of an arbitrary number of identical components the components of a ring are modeled using a fair petri net in which the firing of a prespecified set of transitions is assumed to occur fairly i e any of these transitions that becomes firable infinitely often must fire infinitely often specifically we introduce the concept of similarity between rings of different sizes and give a condition under which the similarity between the rings of sizes two and three guarantees the similarity among the rings of all sizes so if the given condition is satisfied then the correctness of a ring of any large size can be inferred from the correctness of a ring having only a few components the usefulness of the theorem is demonstrated using the examples of token passing mutual exclusion and a simple producer consumer system

some tractable supervisory control problems for discrete event systems modeled by buchi automata discrete event systems des are modeled by buchi automata together with a means of online control in this setting the concept of a controllable language is extended to infinite strings and conditions for the existence of a supervisor controller to implement a prescribed closed loop behavior are derived the focus is on a class of des called product systems these are des composed of a finite set of asynchronous components a control problem for such a system typically requires the synthesis of an online controller so as to achieve some prescribed coordinated behavior of the component subsystems one of the principal difficulties in this task is that the size of the state space increases exponentially with the number of components it is shown that despite this fact several interesting control synthesis problems for such systems are computationally feasible and algorithms are developed for solution

measuring and synthesizing systems in probabilistic environments the traditional synthesis question given a specification asks for the automatic construction of a system that satisfies the specification whereas often there exists a preference order among the different systems that satisfy the given specification under a probabilistic assumption about the possible inputs such a preference order is naturally expressed by a weighted automaton which assigns to each word a value such that a system is preferred if it generates a higher expected value we solve the following optimal synthesis problem given an omega regular specification a markov chain that describes the distribution of inputs and a weighted automaton that measures how well a system satisfies the given specification under the input assumption synthesize a system that optimizes the measured value for safety specifications and quantitative measures that are defined by mean payoff automata the optimal synthesis problem reduces to finding a strategy in a markov decision process mdp that is optimal for a long run average reward objective which can be achieved in polynomial time for general omega regular specifications along with mean payoff automata the solution rests on a new polynomial time algorithm for computing optimal strategies in mdps with mean payoff parity objectives our algorithm constructs optimal strategies that consist of two memoryless strategies and a counter the counter is in general not bounded to obtain a finite state system we show how to construct an e optimal strategy with a bounded counter for all e 0 furthermore we show how to decide in polynomial time if it is possible to construct an optimal finite state system i e a system without a counter for a given specification we have implemented our approach and the underlying algorithms in a tool that takes qualitative and quantitative specifications and automatically constructs a system that satisfies the qualitative specification and optimizes the quantitative specification if such a system exists we present some experimental results showing optimal systems that were automatically generated in this way

the complementation problem for bu chi automata with applications to temporal logic the problem of complementing buchi automata arises when developing procedures for temporal logics of programs unfortunately previously known constructions for complementing buchi automata involve a doubly exponential blow up in the size of the automaton we present a construction that involves only an exponential blow up we use this construction to prove a polynomial space upper bound for the propositional temporal logic of regular events and to prove a complexity hierarchy result for quantified propositional temporal logic

model checking for fragments of the interval temporal logic hs at the low levels of the polynomial time hierarchy abstract some temporal properties of reactive systems such as actions with duration and temporal aggregations which are inherently interval based can not be properly expressed by the standard point based temporal logics ltl ctl and ctl as they give a state by state account of system evolution conversely interval temporal logics which feature intervals instead of points as their primitive entities naturally express them we study the model checking mc problem for halpern and shoham s modal logic of time intervals hs interpreted on kripke structures under the homogeneity assumption hs is the best known interval based temporal logic which has one modality for each of the 13 ordering relations between pairs of intervals allen s relations apart from equality we focus on mc for some hs fragments featuring modalities for a subset of allen s relations meet met by started by and finished by showing that it is in p np additionally we provide some complexity lower bounds to the problem

nonclausal deduction in first order temporal logic this paper presents a proof system for first order temporal logic the system extends the nonclausal resolution method for ordinary first order logic with equality to handle quantifiers and temporal operators soundness and completeness issues are considered the use of the system for verifying concurrent programs is discussed and variants of the system for other modal logics are also described

on deciding trace equivalences for processes abstract in this paper we investigate the complexity of deciding trace maximal trace and trace equivalences for finite state processes and recursively defined processes specified by normed context free grammars cfgs in greibach normal form gnf the main results are as follows 1 1 trace maximal trace and trace equivalences for processes specified by normed gnf cfgs are all undecidable for this class of processes the regularity problem with respect to trace maximal trace or trace equivalence is also undecidable moreover all these undecidability results hold even for locally unary processes for processes specified by unary gnf cfgs the maximal trace equivalence is 2 p complete while the trace equivalence is nl complete and the trace equivalence is decidable in polynomial time by a dynamic programming algorithm 2 2 trace maximal trace and trace equivalences for finite state processes are pspace complete this holds even for locally unary finite state processes for unary finite state processes the maximal trace equivalence is co np complete while trace and trace equivalences are nl complete

on the expressiveness of tptl and mtl abstract tptl and mtl are two classical timed extensions of ltl in this paper we prove the 20 year old conjecture that tptl is strictly more expressive than mtl but we show that surprisingly the tptl formula proposed by alur and henzinger for witnessing this conjecture it can be expressed in mtl more generally we show that tptl formulae using only modality f can be translated into mtl

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

the complexity of probabilistic verification we determine the complexity of testing whether a finite state sequential or concurrent probabilistic program satisfies its specification expressed in linear time temporal logic for sequential programs we present an algorithm that runs in time linear in the program and exponential in the specification and also show that the problem is in pspace matching the known lower bound for concurrent programs we show that the problem can be solved in time polynomial in the program and doubly exponential in the specification and prove that it is complete for double exponential time we also address these questions for specifications described by automata or formulas in extended temporal logic

event clock automata a determinizable class of timed automata we introduce event recording automata an event recording automaton is a timed automaton that contains for every event a a clock that records the time of the last occurrence of a the class of event recording automata is on one hand expressive enough to model finite timed transition systems and on the other hand determinizable and closed under all boolean operations as a result the language inclusion problem is decidable for event recording automata we present a translation from timed transition systems to event recording automata which leads to an algorithm for checking if two timed transition systems have the same set of timed behaviors we also consider event predicting automata which contain clocks that predict the time of the next occurrence of an event the class of event clock automata which contain both event recording and event predicting clocks is a suitable specification language for real time properties we provide an algorithm for checking if a timed automaton meets a specification that is given as an event clock automaton

refinement modal logic in this paper we present refinement modal logic a refinement is like a bisimulation except that from the three relational requirements only atoms and back need to be satisfied our logic contains a new operator in addition to the standard box modalities for each agent the operator acts as a quantifier over the set of all refinements of a given model as a variation on a bisimulation quantifier this refinement operator or refinement quantifier can be seen as quantifying over a variable not occurring in the formula bound by it the logic combines the simplicity of multi agent modal logic with some powers of monadic second order quantification we present a sound and complete axiomatization of multi agent refinement modal logic we also present an extension of the logic to the modal calculus and an axiomatization for the single agent version of this logic examples and applications are also discussed to software verification and design the set of agents can also be seen as a set of actions and to dynamic epistemic logic we further give detailed results on the complexity of satisfiability and on succinctness

interval logics and their decision procedures part i an interval logic abstract we present an interval logic called future interval logic fil for the specification and verification of concurrent systems interval logics allow reasoning to be carried out at the level of time intervals rather than instants however unlike some other interval logics the primitive objects in our semantic model for fil are not intervals but instants an intervals is formed by identifying its end points which are instants satisfying given properties the logic has an intuitive graphical representation resembling the back of the envelope timing diagrams that designers often draw to reason about concurrent interacting systems the logic is designed to be insensitive to finite stuttering a property that facilitates refinement based multi level correctness proofs and is exactly as expressive as the fragment of propositional temporal logic with until but no next as the main result of this paper we show that fil is elementarily decidable by reduction to the emptiness problem for buchi automata for most other interval logics the decision problem is at best non elementary and often undecidable we cosider an extension of fil with past operators and show that this extension leads to non elementariness in a companion paper we extend the logic to real time and investigate the decision problem for that extension

the complexity of propositional linear temporal logics in simple cases it is well known that model checking and satisfiability for pltl are pspace complete by contrast very little is known about whether there exist some interesting fragments of pltl with a lower worst case complexity such results would help understand why pltl model checkers are successfully used in practice in this article we investigate this issue and consider model checking and satisfiability for all fragments of pltl obtainable by restricting 1 the temporal connectives allowed 2 the number of atomic propositions and 3 the temporal height 2002 elsevier science usa

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

problems concerning fairness and temporal logic for conflict free petri nets abstract in this paper we examine the complexity of the fair nontermination problem for conflict free petri nets under several definitions of fairness for each definition of fairness we are able to show the problem to be complete for either np ptime or nlogspace we then address the question of whether these results extend to the more general model checking problem with respect to the temporal logic for petri nets introduced by suzuki since many of the model checking problems concerning finite state systems can be reduced to a version of the fair nontermination problem it would seem plausible that the model checking problem for conflict free petri nets would be decidable however it turns out that unless the logic is severely restricted model checking is undecidable for conflict free petri nets in particular the problem is undecidable even when formulas are of the form g invariantly where contains no temporal logic operators on the other hand we show that model checking for conflict free petri nets is np complete for l f x the logic restricted to the operators f eventually x next time and with negations allowed only on the predicates

on fairness and randomness we investigate the relation between the behavior of non deterministic systems under fairness constraints and the behavior of probabilistic systems to this end first a framework based on computable stopping strategies is developed that provides a common foundation for describing both fair and probabilistic behavior on the basis of stopping strategies it is then shown that fair behavior corresponds in a precise sense to random behavior in the sense of martin lof s definition of randomness we view probabilistic systems as concrete implementations of more abstract non deterministic systems under this perspective the question is investigated what probabilistic properties are needed in such an implementation to guarantee with probability one certain required fairness properties in the behavior of the probabilistic system generalizing earlier concepts of e bounded transition probabilities we introduce the notion of divergent probabilistic systems which enables an exact characterization of the fairness properties of a probabilistic implementation looking beyond pure fairness properties we also investigate what other qualitative system properties are guaranteed by probabilistic implementations of fair non deterministic behavior this leads to a completeness result which generalizes a well known theorem by pnueli and zuck

the size of bdds and other data structures in temporal logics model checking temporal logic model checking is a verification method in which we describe a system the model and then we verify whether important properties expressed in a temporal logic formula hold in the system many model checking tools employ bdds or some other data structure to represent sets of states it has been empirically observed that the bdds used in these algorithms may grow exponentially as the model and formula increase in size we formally prove that no kind of data structure of polynomial size can represent the set of valid initial states for all models and all formulae this result holds for all data structures where a state can be checked in polynomial time therefore it holds not only for all types of bdds regardless of variable ordering but also for more powerful data structures such as rbcs mtbdds adds and sdds thus the size explosion of bdds is not a limit of these specific data representation structures but is unavoidable every formalism used in the same way would lead to an exponential size blow up

temporal bi proof system semantics and translations abstract the logic bi of bunched implications is a combination of intuitionistic logic and multiplicative intuitionistic linear logic in this paper a temporal extension tbi of bi is introduced and studied a gentzen type sequent calculus gtbi for tbi is introduced and the cut elimination and decidability theorems for gtbi are proved using a theorem for syntactically embedding gtbi into a sequent calculus gbi for bi a semantics for gtbi is introduced extending the grothendieck topological semantics for bi and the completeness theorem with respect to this semantics is proved using a theorem for semantically embedding tbi into bi a semantics for gtbi without additive falsity constant is introduced extending the kripke resource semantics for bi without additive falsity constant and the completeness theorem with respect to this semantics is proved in a similar way moreover an intuitionistic temporal linear logic itll is introduced as a gentzen type sequent calculus and a theorem for embedding gtbi into itll is proved using a temporal extension of the girard translation of intuitionistic logic into intuitionistic linear logic

taming past ltl and flat counter systems reachability and ltl model checking problems for flat counter systems are known to be decidable but whereas the reachability problem can be shown in np the best known complexity upper bound for the latter problem is made of a tower of several exponentials herein we show that this problem is only np complete even if ltl admits past time operators and arithmetical constraints on counters as far as past time operators are concerned their addition to ltl immediately leads to complications and hence an np upper bound cannot be deduced by translating formulae into ltl and studying the problem only for this latter logic we also provide other complexity results obtained by restricting further the class of flat counter systems

complexity and succinctness issues for linear time hybrid logics full linear time hybrid logic hl is a non elementary and equally expressive extension of standard ltl past obtained by adding the well known binder operators 7 and we investigate complexity and succinctness issues for hl in terms of the number of variables and nesting depth of binder modalities first we present direct automata theoretic decision procedures for satisfiability and model checking of hl which require space of exponential height equal to the nesting depth of the binder modalities the proposed algorithms are proved to be asymptotically optimal by providing matching lower bounds second we show that for the one variable fragment of hl the considered problems are elementary and precisely expspace complete finally we show that for all 0 h

the complexity of achievement and maintenance problems in agent based systems we completely classify the computational complexity of the basic achievement and maintenance agent design problems in bounded environments when these problems are parameterized by the number of environment states and the number of agent actions the different problems are p complete np complete co np complete or pspace complete when they are not trivial we also consider alternative achievement and maintenance agent design problems by allowing longer runs in environments that is our environments are bounded but the bounds are more liberal than was the case previously again we obtain a complete classification but so that the different problems are dexptime complete nexptime complete co nexptime complete or nexpspace complete when they are not trivial

an until hierarchy and other applications of an ehrenfeucht fraisse game for temporal logic we prove there is a strict hierarchy of expressive power according to the until depth of linear temporal logic ltl formulas for each k there is a natural property based on quantitative fairness that is not expressible with k nestings of until operators regardless of the number of applications of other operators but is expressible by a formula with until depth k 1 our proof uses a new ehrenfeucht fra sse ef game designed specifically for ltl these properties can all be expressed in first order logic with quantifier depth and size o logk and we use them to observe some interesting relationships between ltl and first order expressibility we note that our until hierarchy proof for ltl carries over to the branching time logics ctl and ctl we then use the ef game in a novel way to effectively characterize 1 the ltl properties expressible without until as well as 2 those expressible without both until and next by playing the game on finite automata we prove that the automata recognizing languages expressible in each of the two fragments have distinctive structural properties the characterization for the first fragment was originally proved by cohen perrin and pin using sophisticated semigroup theoretic techniques they asked whether such a characterization exists for the second fragment the technique we develop is general and can potentially be applied in other contexts

optimal cost almost sure reachability in pomdps we consider partially observable markov decision processes pomdps with a set of target states and an integer cost associated with every transition the optimization objective we study asks to minimize the expected total cost of reaching a state in the target set while ensuring that the target set is reached almost surely with probability 1 we show that for integer costs approximating the optimal cost is undecidable for positive costs our results are as follows i we establish matching lower and upper bounds for the optimal cost both double exponential in the pomdp state space size ii we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for pomdps with finite horizon objectives while the worst case running time of our algorithm is double exponential we also present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest

solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives the method of simplest equation is applied to nonlinear pdes the pdes contain monomials of odd and even order with respect to derivatives solitary wave solutions are studied numerous examples of the methodology are presented we apply the method of simplest equation for obtaining exact solitary traveling wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives we consider first the general case of presence of monomials of the both odd and even grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade the methodology is illustrated by numerous examples

on some solutions of the two dimensional sine gordon equation there exists an approach for finding of exact solutions of the two dimensional sine gordon equation using this approach three classes of solutions have been found one of the classes consists of running wave solutions which are a generalization of the solutions of the one dimensional sine gordon equation this class of solutions is studied here

a unified translation of linear temporal logic to ω automata we present a unified translation of ltl formulas into nondeterministic buchi automata limit deterministic buchi automata ldba and deterministic rabin automata dra the translations yield automata of asymptotically optimal size double or single exponential respectively all three translations are derived from one single master theorem of purely logical nature the master theorem decomposes the language of a formula into a positive boolean combination of languages that can be translated into automata by elementary means all three translations are direct and no intermediate constructions are used

verification of qualitative z constraints we introduce an ltl like logic with atomic formulae built over a constraint language interpreting variables in z the constraint language includes periodicity constraints and comparison constraints of the form x y and x

on the bifurcation of energy in media governed by 2 1 dimensional modified klein gordon equations abstract in the present work we develop a numerical technique to approximate solutions of a general class of two dimensional modified klein gordon equations that appears amongst other applications in the study of lattices consisting of josephson junctions coupled through superconducting wires when dissipative and dispersive effects are taken into account the basic motivation for our study is the need to posses a numerically accurate finite difference scheme to approximate the occurrence of the process of nonlinear supratransmission when the mentioned systems are submitted to dirichlet neumann or more generally boundary conditions of mixed type the computational technique that we present in this paper is nonstandard implicit nonlinear and not only consistent order o t 2 and conditionally stable on the approximation of solutions but it also yields consistent approximations of the local energies of the individual junctions of the total energy of the system and moreover of the rate of change of energy of the system with respect to time making it an ideal method in the analysis of the process of supratransmission generalizations to higher dimensional cases may be induced from the one and two dimensional versions of the method and applications of the technique are given to the determination of the occurrence of nonlinear supratransmission under the presence of several parameters our results show that the transmission of energy in the system starts at a well defined nonnegative lower threshold after which the creation of localized intrinsic modes that move away from the driving boundaries is imminent the study of the effects of nonzero mass and internal and external damping is carried out from a computational perspective

running wave solutions of the two dimensional sine gordon equation after the lamb substitution the 2d sine gordon equation was solved three classes of solutions for this equation were found ad hoc the 2d sine gordon equation was reduced to an algebraic system consisting of three equations the solutions given of the 2d sine gordon equation are a generalization of the solutions of the 1d sine gordon equation they are also a generalization of the solutions of the equation phi xx yy sin phi x y

characterizing finite kripke structures in propositional temporal logic abstract we show that if two finite kripke structures can be distinguished by some ctl formula that contains both branching time and linear time operators then the structures can be distinguished by a ctl formula that contains only branching time operators our proof involves showing that for any finite kripke structure m it is possible to construct a ctl formula f m that uniquely characterizes m since one kripke structure may be a trivial unrolling of another we use a notion of equivalence between kripke structures that is similar to the notion of bisimulation studied by milner 15 our first construction of f m requires the use of the nexttime operator we also consider the case in which the nexttime operator is disallowed in ctl formulas the proof in this case requires another notion of equivalence equivalence with respect to stuttering and is much more difficult since it is possible for two inequivalent states to have exactly the same finite behaviors modulo stuttering but different infinite behaviors we also give a polynomial algorithm for determining if two structures are stuttering equivalent and discuss the relevance of our results for temporal logic model checking and synthesis procedures

modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations we discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities as simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of weierstrass and jacobi we show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions this system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation the methodology is illustrated by obtaining solitary wave solutions for the generalized korteweg devries equation and by obtaining solutions of the higher order korteweg devries equation

new class of running wave solutions of the 2 1 dimensional sine gordon equation a new class of running wave solutions of the 2 1 dimensional sine gordon equation is investigated the obtained waves require two spatial dimensions for their propagation i e they generalize solutions of the 2 0 dimensional sine gordon equation the parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency the obtained solutions describe a new class of josephson waves whose velocity is smaller than the swihart velocity if omega 0 the running waves are reduced to the self consistent phase current and magnetic field distributions in a large two dimensional josephson junction the self restriction coefficient for the josephson current corresponding to one of the structures is calculated

nash equilibria in symmetric graph games with partial observation we investigate a model for representing large multiplayer games which satisfy strong symmetry properties this model is made of multiple copies of an arena each player plays in his own arena and can partially observe what the other players do therefore this game has partial information and symmetry constraints which make the computation of nash equilibria difficult we show several undecidability results and for bounded memory strategies we precisely characterize the complexity of computing pure nash equilibria for qualitative objectives in this game model

efficient and dynamic algorithms for alternating buchi games and maximal end component decomposition the computation of the winning set for buchi objectives in alternating games on graphs is a central problem in computer aided verification with a large number of applications the long standing best known upper bound for solving the problem is o n m where n is the number of vertices and m is the number of edges in the graph we are the first to break the o n m boundary by presenting a new technique that reduces the running time to o n2 this bound also leads to o n2 time algorithms for computing the set of almost sure winning vertices for buchi objectives 1 in alternating games with probabilistic transitions improving an earlier bound of o n m 2 in concurrent graph games with constant actions improving an earlier bound of o n3 and 3 in markov decision processes improving for m n4 3 an earlier bound of o m m we then show how to maintain the winning set for buchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in o n amortized time per operation our algorithms are the first dynamic algorithms for this problem we then consider another core graph theoretic problem in verification of probabilistic systems namely computing the maximal end component decomposition of a graph we present two improved static algorithms for the maximal end component decomposition problem our first algorithm is an o m m time algorithm and our second algorithm is an o n2 time algorithm which is obtained using the same technique as for alternating buchi games thus we obtain an o min m m n2 time algorithm improving the long standing o n m time bound finally we show how to maintain the maximal end component decomposition of a graph under a sequence of edge insertions or a sequence of edge deletions in o n amortized time per edge deletion and o m worst case time per edge insertion again our algorithms are the first dynamic algorithms for this problem

on nonlinear population waves we discuss a model system of partial differential equations for description of the spatio temporal dynamics of interacting populations we are interested in the waves caused by migration of the populations we assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations for the particular case of one population and one spatial dimension the general model is reduced to analytically tractable pde with polynomial nonlinearity up to 4th order we investigate this particular case and obtain two kinds of solutions i approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and ii exact solutions which describe nonlinear kink and solitary waves in an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node finally we derive conditions for the asymptotic stability of the obtained solutions

reasoning with time and chance the temporal propositional logic of linear time is generalized to an uncertain world in which random events may occur the formulas do not mention probabilities explicitly i e the only probability appearing explicitly in formulas is probability one this logic is claimed to be useful for stating and proving properties of probabilistic programs it is convenient for proving those properties that do not depend on the specific distribution of probabilities used in the program s random draws the formulas describe properties of execution sequences the models are stochastic systems with state transition probabilities three different axiomatic systems are proposed and shown complete for general models finite models and models with bounded transition probabilities respectively all three systems are decidable by the results of rabin trans amer math soc 141 1969 1 35

profile trees for buchi word automata with application to determinization the determinization of buchi automata is a celebrated problem with applications in synthesis probabilistic verification and multi agent systems since the 1960s there has been a steady progress of constructions by mcnaughton safra piterman schewe and others despite the proliferation of solutions they are all essentially ad hoc constructions with little theory behind them other than proofs of correctness since safra all optimal constructions employ trees as states of the deterministic automaton and transitions between states are defined operationally over these trees the operational nature of these constructions complicates understanding implementing and reasoning about them and should be contrasted with complementation where a solid theory in terms of automata run dags underlies modern constructions in 2010 we described a profile based approach to buchi complementation where a profile is simply the history of visits to accepting states we developed a structural theory of profiles and used it to describe a complementation construction that is deterministic in the limit here we extend the theory of profiles to prove that every run dag contains a profile tree with at most a finite number of infinite branches we then show that this property provides a theoretical grounding for a new determinization construction where macrostates are doubly preordered sets of states in contrast to extant determinization constructions transitions in the new construction are described declaratively rather than operationally

a high order difference method for the steady state navier stokes equations a stable high order difference method is developed for solving the heated cavity and the driven cavity problems governed by the steady state navier stokes equations the numerical results compared favorably with other methods coordinate transformations were also used to reflect the boundary layer phenonmena without requiring many mesh points the methods used in this paper are accurate stable and applicable to other flow problems governed by the navier stokes equations

optimal bounds in parametric ltl games parameterized linear temporal logics are extensions of linear temporal logic ltl by temporal operators equipped with variables that bound their scope in model checking such specifications were introduced as pltl by alur et al and as prompt ltl by kupferman et al we show how to determine in doubly exponential time whether a player wins a game with pltl winning condition with respect to some infinitely many or all variable valuations hence these problems are not harder than solving ltl games furthermore we present an algorithm with triply exponential running time to determine optimal variable valuations that allow a player to win a game finally we give doubly exponential upper and lower bounds on the values of such optimal variable valuations

on projective and separable properties a language l over the cartesian product of component alphabets is called projective if it is closed under projections that is together with each word e l it contains all the words that have the same projections up to stuttering as we prove that in each of the behavior classes regular regular and star free regular i e definable by linear temporal logic languages the projective languages are precisely the boolean combinations of stuttering closed component languages from the corresponding class languages of these behavior classes can also be seen as properties of various temporal logics some uses of projective properties for specification and verification of programs are studied

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

efficient model checking for ltl with partial order snapshots certain behavioral properties of distributed systems are difficult to express in interleaving semantics whereas they are naturally expressed in terms of partial orders of events or equivalently mazurkiewicz traces two examples of such properties are serializability of a database and global snapshots of concurrent systems recently a modest extension for ltl by an operator that expresses snapshots has been proposed it combines the ease of linear interleaving specification with this useful partial order concept the new construct allows one to assert that a global snapshot appeared in the past perhaps not in the observed execution sequence but possibly in an equivalent one originally a model checking algorithm for this logic that is exponential space in the size of the system was suggested in this paper we provide a model checking algorithm that is in polynomial space in the size of the system our construction can also serve as an efficient polynomial algorithm for detecting conjunctive properties i e conjunction of local process properties in a concurrent history of execution

a taxonomy of fairness and temporal logic problems for petri nets in this paper we define a temporal logic for reasoning about petri nets we show the model checking problem for this logic to be ptime equivalent to the petri net reachability problem using this logic and two refinements we show the fair nontermination problem to be ptime equivalent to reachability for several definitions of fairness for other versions of fairness this problem is shown to be either ptime equivalent to the boundedness problem or highly undecidable in all 24 versions of fairness are examined

recurrence and transience for finite probabilistic tables a finite probabilistic table or fpt consists of a finite state space s an initial distribution on s and a finite set of markov matrices on s labeled by an alphabet s an infinite word on s induces a non homogeneous markov chain nhmc on s in the context of finite homogeneous markov chains a state s is recurrent if with probability one a run initialized on s visits s infinitely often equivalently s is recurrent if with probability one the proportion of time a run initialized on s spends on s converges to a non zero limit in this paper we introduce two natural notions of recurrence for non homogeneous markovian processes a state s is weakly recurrent resp strongly recurrent if with positive probability the process visits s infinitely often resp spends a non zero proportion of time on s these notions do not coincide in the context of nhmcs and we study the related computational problem on fpts given an fpt and a state s is there w s such that s is weakly resp strongly recurrent for the associated nhmc we prove that the strong recurrence problem is pspace complete along with other complexity results which contrast with previous results which showed for instance the undecidability of the weak recurrence problem

defining fairness in reactive and concurrent systems we define when a linear time temporal property is a fairness property with respect to a given system this captures the essence shared by most fairness assumptions that are used in the specification and verification of reactive and concurrent systems such as weak fairness strong fairness k fairness and many others we provide three characterizations of fairness a language theoretic a game theoretic and a topological characterization it turns out that the fairness properties are the sets that are large from a topological point of view that is they are the co meager sets in the natural topology of runs of a given system this insight provides a link to probability theory where a set is large when it has measure 1 while these two notions of largeness are similar they do not coincide in general however we show that they coincide for regular properties and bounded borel measures that is an regular temporal property of a finite state system has measure 1 under a bounded borel measure if and only if it is a fairness property with respect to that system the definition of fairness leads to a generic relaxation of correctness of a system in linear time semantics we define a system to be fairly correct if there exists a fairness assumption under which it satisfies its specification equivalently a system is fairly correct if the set of runs satisfying the specification is topologically large we motivate this notion of correctness and show how it can be verified in a system

btl 2 and the expressive power of ectl we show that ectl the classical extension of ctl with fairness properties is expressively equivalent to btl2 a natural fragment of the monadic logic of order btl2 is the branching time logic with arbitrary quantification over paths and where path formulac are restricted to quantifier depth 2 first order formulae in the monadic logic of order this result linking ectl to a natural fragment of the monadic logic of order provides a characterization that other branching time logics e g ctl lack we then go on to show that ectl and btl2 are not finitely based i e they cannot be defined by a finite set of temporal modalities and that their model checking problems are of the same complexity

a high order finite difference method with richardsonextrapolation for 3d convection diffusion equation in this paper we extend the sun and zhang s 24 work on high order finite difference method which is based on the richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case firstly we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids then we use the richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid finally we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid during this process we use alternating direction implicit adi method to solve the resulting linear systems numerical experiments are conducted to verify the accuracy and effectiveness of the present method

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