an algebraic characterization of transition system equivalences abstract i castellani 1987 j comput system sci 34 210 235 has shown that observation equivalence of transition systems could be characterized by particular reductions systems are equivalent if and only if they can be reduced to the same form moreover every transition system has a minimal reduced form we extend these results to logical equivalence by an algebraic interpretation of temporal logics we characterize logical equivalence of transition systems by particular reductions saturating quasi homomorphisms or their power algebras of sets of states and paths and prove that every power algebra has a minimal reduced form we then offer alternative proofs for logical characterizations of observation equivalence in particular we apply our method to prove m hennessy and c stirling s 1984 lecture notes in comput sci vol 176 pp 301 311 springer verlag new york berlin result that future perfect logic characterizes observation equivalence of generalized transition systems i e systems whose infinite behaviours are restricted by arbitrary fairness constraints

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

entanglement and the complexity of directed graphs entanglement is a parameter for the complexity of finite directed graphs that measures to what extent the cycles of the graph are intertwined it is defined by way of a game similar in spirit to the cops and robber games used to describe treewidth directed treewidth and hypertree width nevertheless on many classes of graphs there are significant differences between entanglement and the various incarnations of treewidth entanglement is intimately related with the computational and descriptive complexity of the modal m calculus the number of fixed point variables needed to describe a finite graph up to bisimulation is captured by its entanglement this plays a crucial role in the proof that the variable hierarchy of the m calculus is strict we study complexity issues for entanglement and compare it to other structural parameters of directed graphs one of our main results is that parity games of bounded entanglement can be solved in polynomial time specifically we establish that the complexity of solving a parity game can be parametrised in terms of the minimal entanglement of subgames induced by a winning strategy furthermore we discuss the case of graphs of entanglement two while graphs of entanglement zero and one are very simple graphs of entanglement two allow arbitrary nesting of cycles and they form a sufficiently rich class for modelling relevant classes of structured systems we provide characterisations of this class and propose decomposition notions similar to the ones for treewidth dag width and kelly width

logical vs behavioural specifications abstract there are two fundamentally different approaches for specifying and verifying properties of systems the logical approach makes use of specifications given as formulae of temporal or modal logics and relies on efficient model checking algorithms the behavioural approach exploits various equivalence or refinement checking methods provided the specifications are given in the same formalism as implementations in this paper we provide translations between the logical formalism of calculus and the behavioural formalism of disjunctive modal transition systems the translations preserve structural properties of the input specification and allow us to perform logical operations on the behavioural specifications as well as behavioural compositions on logical formulae the unification of both approaches provides additional methods for component based stepwise design

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

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

algorithmic correspondence for intuitionistic modal mu calculus in the present paper the algorithmic correspondence theory developed in conradie and palmigiano 9 is extended to mu calculi with a non classical base we focus in particular on the language of bi intuitionistic modal mu calculus we enhance the algorithm alba introduced in conradie and palmigiano 9 so as to guarantee its success on the class of recursive mu inequalities which we introduce in this paper key to the soundness of this enhancement are the order theoretic properties of the algebraic interpretation of the fixed point operators we show that when restricted to the boolean setting the recursive mu inequalities coincide with the sahlqvist mu formulas defined in van benthem bezhanishvili and hodkinson 22

compositionality of hennessy milner logic by structural operational semantics this paper presents a method for the decomposition of hml formulas it can be used to decide whether a process algebra term satisfies a hml formula by checking whether subterms satisfy certain formulas obtained by decomposing the original formula the method uses the structural operational semantics of the process algebra the main contribution of this paper is the extension of an earlier decomposition method for the de simone format from the ph d thesis of larsen in 1986 to more general formats

the finite graph problem for two way alternating automata two way alternating automata on infinite trees were introduced by vardi reasoning about the part with two way automata lecture notes in computer science vol 11 springer berlin 1998 pp 628 641 here we consider alternating two way automata on graphs and show the decidability of the following problem does a given automaton with the buchi condition accept any finite graph using this result we demonstrate the decidability of the finite model problem for a certain fragment of the modal calculus with backward modalities

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

an initial semantics for the m calculus on trees and rabin s complementation lemma in this paper we show that the function associated with any closed or nonclosed term of the calculus on trees can be represented by a recognizable set of trees whose nodes are labeled by letters and by sets of variables rabin s complementation lemma is an immediate consequence of this result

completeness of kozen s axiomatisation of the propositional m calculus propositional calculus is an extension of the propositional modal logic with the least fixpoint operator in the paper introducing the logic kozen posed a question about completeness of the axiomatisation which is a small extension of the axiomatisation of the model system k it is shown that this axiomatisation is complete

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

forgetting in multi agent modal logics abstract in the past decades forgetting has been investigated for many logics and has found many applications in knowledge representation and reasoning in this paper we study forgetting in multi agent modal logics we adopt the semantic definition of existential bisimulation quantifiers as that of forgetting we resort to canonical formulas of modal logics introduced by moss an arbitrary modal formula is equivalent to the disjunction of a unique set of satisfiable canonical formulas we show that for the logics of k n d n t n k 45 n kd 45 n and s 5 n the result of forgetting an atom from a satisfiable canonical formula can be computed by simply substituting the literals of the atom with thus we show that these logics are closed under forgetting and hence have uniform interpolation finally we generalize the above results to include common knowledge of propositional formulas

characterization definability and separation via saturated models abstract three important results about the expressivity of a modal logic l are the characterization theorem that identifies a modal logic l as a fragment of a better known logic the definability theorem that provides conditions under which a class of l models can be defined by a formula or a set of formulas of l and the separation theorem that provides conditions under which two disjoint classes of l models can be separated by a class definable in l we provide general conditions under which these results can be established for a given choice of model class and modal language whose expressivity is below first order logic besides some basic constraints that most modal logics easily satisfy the fundamental condition that we require is that the class of saturated models in question has the hennessy milner property with respect to the notion of observational equivalence under consideration given that the characterization definability and separation theorems are among the cornerstones in the model theory of l this property can be seen as a test that identifies the adequate notion of observational equivalence for a particular modal logic

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

three logics for branching bisimulation three temporal logics are introduced that induce on labeled transition systems the same identifications as branching bisimulation a behavioral equivalence that aims at ignoring invisible transitions while preserving the branching structure of systems the first logic is an extension of hennessy milner logic with an until operator the second one is another extension of hennessy milner logic which exploits the power of backward modalities the third logic is ctl without the next time operator a relevant side effect of the last characterization is that it sets a bridge between the state and action based approaches to the semantics of concurrent systems

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

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

complete proof systems for weighted modal logic the weighted transition systems wts considered in this paper are transition systems having both states and transitions labeled with real numbers the state labels denote quantitative resources while the transition labels denote costs of transitions in terms of resources weighted modal logic wml is a multi modal logic that expresses qualitative and quantitative properties of wtss while wml has been studied in various contexts and for various application domains no proof system has been developed for it in this paper we solve this open problem and propose both weak complete and strong complete axiomatizations for wml against wtss we prove a series of metatheorems including the finite model property and the existence of canonical models we show how the proof system can be used in the context of a process algebra semantics for wml to convert a model checking problem into a theorem proving problem this work emphasizes a series of similarities between wml and the probabilistic stochastic modal logics for markov processes and harsanyi type spaces such as the use of particular infinitary rules to guarantee the strong completeness

logical definability of fixed points note abstract we define a class of monotonic functions whose least fixed points are weakly definable in the sense of rabin we use this property to give a new proof that v definable sets of trees are recognized by rabin s special automata

the alternation hierarchy in fixpoint logic with chop is strict too fixpoint logic with chop extends the modal calculus with a sequential composition operator which results in an increase in expressive power we develop a game theoretic characterisation of its model checking problem and use these games to show that the alternation hierarchy in this logic is strict the structure of this result follows the lines of arnold s proof showing that the alternation hierarchy in the modal calculus is strict over the class of binary trees

maximal traces and path based coalgebraic temporal logics this paper gives a general coalgebraic account of temporal logics whose semantics involves a notion of computation path examples of such logics include the logic ctl for transition systems and the logic pctl for probabilistic transition systems our path based temporal logics are interpreted over coalgebras of endofunctors obtained as the composition of a computation type e g non deterministic or stochastic with a general transition type the semantics of such logics relies on the existence of execution maps similar to the trace maps introduced by jacobs and co authors as part of the coalgebraic theory of finite traces hasuo et al 2007 1 we consider finite execution maps derived from the theory of finite traces and a new notion of maximal execution map that accounts for maximal possibly infinite executions the latter is needed to recover the logics ctl and pctl as specific path based logics

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

measure properties of regular sets of trees abstract we investigate measure theoretic properties of regular sets of infinite trees as a first result we prove that every regular set is universally measurable and that every borel measure on the polish space of trees is continuous with respect to a natural transfinite stratification of regular sets into 1 ranks we also expose a connection between regular sets and the algebra of r sets introduced by a kolmogorov in 1928 as a foundation for measure theory we show that the game tree languages w i k are wadge complete for the finite levels of the hierarchy of r sets we apply these results to answer positively an open problem regarding the game interpretation of the probabilistic calculus

towards the hierarchical verification of reactive systems the hierarchical design approach for action based systems that is known as action refinement has been studied in the literature extensively in a paper of m huhn published in concur 1996 a refinement operator on a linear time logic is presented that mimics precisely a semantic action refinement on synchronisation structures we present here an alternative approach where our starting point is a process algebraic setting with a syntactic action refinement we present a refinement operator on the modal mu calculus that conforms with the process algebraic refinement in the following sense provided some reasonable conditions are met the transition system induced by a process term p satisfies a modal mu calculus specification if and only if the system which is induced by a refinement of p satisfies a particular refinement of alleviating these conditions we show that each of the two implications in the equivalence assertion above can be separately proven valid for a particular fragment of the modal mu calculus we demonstrate that the obtained results can indeed be used as a hierarchical verification technique as a further application of our results we explain how they can be employed as an abstraction technique in order to enhance model checking techniques

the µ calculus alternation hierarchy collapses over structures with restricted connectivity the alternation hierarchy of least and greatest fixpoint operators in the calculus is strict however the strictness of the hierarchy does not necessarily carry over when considering restricted classes of structures for instance over the class of infinite words the alternation free fragment of the calculus is already as expressive as the full logic our current understanding of when and why the calculus alternation hierarchy is and is not strict is limited this article makes progress in answering these questions by showing that the alternation hierarchy of the calculus collapses to the alternation free fragment over some classes of structures including infinite nested words and finite graphs with feedback vertex sets of a bounded size common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structure s graph decomposes it into graphs in which all paths are of finite length the collapse results herein are obtained in an automata theoretic setting they subsume generalize and strengthen several prior results on the expressivity of the calculus over restricted classes of structures

an algebraic and algorithmic method for analysing transition systems to describe significant properties of concurrent processes a formal algebra is defined whose terms are interpreted in transition systems as effectively computable functions of power algebras of sets of states and transitions we study the expressive power of this formal system and compare it with temporal logics

the μ calculus alternation depth hierarchy is infinite over finite planar graphs abstract we prove that the alternation hierarchy of the modal calculus is infinite over finite planar graphs

logical characterization of branching metrics for nondeterministic probabilistic transition systems abstract in this paper we propose a logical characterization of bi simulation metrics obtained by a probabilistic variant of hennessy milner logic enriched with variables whose semantics is defined following the equational calculus approach our characterization is based on the novel notions of mimicking formulae and syntactical distance on formulae the former ones are the quantitative analogous to characteristic formulae the latter is a 1 bounded pseudometric on formulae measuring their syntactical disparities the characterization is obtained by showing that the bi simulation distance between processes corresponds to the syntactical distance between their mimicking formulae we also discuss the expressive power of mimicking formulae with respect to probabilistic bi simulations we show that two processes are bisimilar if and only if their mimicking formulae are syntactically equivalent moreover we obtain that mimicking formulae of processes coincide with their characteristic formulae for ready simulation and that negation free mimicking formulae coincide with the characteristic formulae for simulation

modular construction of complete coalgebraic logics we present a modular approach to defining logics for a wide variety of state based systems the systems are modelled as coalgebras and we use modal logics to specify their observable properties we show that the syntax semantics and proof systems associated with such logics can all be derived in a modular fashion moreover we show that the logics thus obtained inherit soundness completeness and expressiveness properties from their building blocks we apply these techniques to derive sound complete and expressive logics for a wide variety of probabilistic systems for which no complete axiomatisation has been obtained so far

on the formalization of the modal µ calculus in the calculus of inductive constructions we present a natural deduction proof system for the propositional modal calculus and its formalization in the calculus of inductive constructions we address several problematic issues such as the use of higher order abstract syntax in inductive sets in the presence of recursive constructors and the formalization of modal sequent style rules and of context sensitive grammars the formalization can be used in the system coq providing an experimental computer aided proof environment for the interactive development of error free proofs in the modal calculus the techniques we adopt can be readily ported to other languages and proof systems featuring similar problematic issues

applicability of fair simulation in this paper we compare four notions of fair simulation direct 9 delay 12 game 19 and exists 16 our comparison refers to three main aspects the time complexity of constructing the fair simulation the ability to use it for minimization and the relationship between the fair simulations and universal branching time logics we developed a practical application that is based on this comparison the application is a new implementation for the assume guarantee modular framework presented by grumberg at al in acm transactions on programming languages and systems toplas 16 1994 843 the new implementation significantly improves the complexity of the framework

control of infinite behavior of finite automata a problem in the control of automata on infinite strings is defined and analyzed the key to the investigation is the development of a fixpoint characterization of the controllability subset of a deterministic rabin automaton the set of states from which the automaton can be controlled to the satisfaction of its own acceptance condition the fixpoint representation allows straightforward computation of the controllability subset and the construction of a suitable state feedback control for the automaton the results have applications to control synthesis automaton synthesis and decision procedures for logical satisfiability in particular they represent a direct efficient and natural solution to church s problem the construction of winning strategies for two player zero sum omega regular games of perfect information and the emptiness problem for automata on infinite trees

local model checking for infinite state spaces bradtield j and c stirling local model checking for infinite spaces theoretical computer science 96 1992 157 174 we present a sound and complete tableau proof system for establishing whether a set of elements of an arbitrary transition system model has a property expressed in a slight extension of the modal mu calculus the proof system we believe offers a very general verification method applicable to a wide range of computational systems in the last twenty years many approaches to program verification have been developed hoare s partial correctness logic for simple while programs gave an early sound and relatively complete proof system this approach was subsequently extended to total correctness and to richer classes of programs dynamic logics offered a more abstract view of hoare logics especially in their propositional versions pnueli pioneered the use of propositional temporal logics as more general program logics capable of describing crucial properties of perpetual concurrent systems a variety of temporal logics have been studied particularly branching and linear time many useful decidability and expressiveness results relating logics and automata have been obtained as well as sound and complete axiomatizations of validity a slightly earlier tradition in the study of correctness was given by the work on program schemes where second order logics were advocated especially in the form of the mu calculus due to de bakker de roever and park an elegant generalization of propositional dynamic and temporal logics drawing on this tradition is the propositional modal mu calculus due to pratt and kozen the modal mu calculus has been shown to include propositional

proving properties of dynamic process networks we present the first compositional proof system for checking processes against formulas in the modal calculus which is capable of handling dynamic process networks the proof system is obtained in a systematic way from the operational semantics of the underlying process algebra a non trivial proof example is given and the proof system is shown to be sound in general and complete for finite state processes

an elementary proof of the completeness of pdl abstract we give an elementary proof of the completeness of the segerberg axions for propositional dynamic logic

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

concurrent program schemes and their logics abstract we define and investigate several classes of concurrent program schemes including goto schemes and two versions of structured schemes based on extensions of the regular expressions to trees the schemes are studied on the first order boolean variable and propositional levels we also define and study the dynamic logics based on these classes of schemes including issues of decidability and axiomatization

fixed point characterization of context free languages several concepts of context freeness of sets of finite infinite words are characterized by means of greatest solutions of systems of equations of the form xi gi i 1 n where gi is a not necessarily finite union of monomials consideration of the systems with the components gi context free regular or finite leads to characterizations of the following classes of languages the kleene closure of the family of context free languages algebraic languages infinitely generated by context free grammars in the sense of nivat 1977 and cantor like topological closures of context free languages respectively

on the μ calculus over transitive and finite transitive frames we prove that the modal m calculus collapses to first order logic over the class of finite transitive frames the proof is obtained by using some byproducts of a new proof of the collapse of the m calculus to the alternation free fragment over the class of transitive frames moreover we prove that the modal m calculus is buchi and co buchi definable over the class of all models where in a strongly connected component vertexes are distinguishable by means of the propositions they satisfy

tarskian set constraints we investigate set constraints over set expressions with tarskian functional and relational operations unlike the herbrand constructor symbols used in recent set constraint formalisms the meaning of a tarskian function symbol is interpreted in an arbitrary first order structure we show that satisfiability of tarskian set constraints is decidable in nondeterministic doubly exponential time we also give complexity results and open problems for various extensions and restrictions of the language

an improved algorithm for the evaluation of fixpoint expressions many automated finite state verification procedures can be viewed as fixpoint computations over a finite lattice typically the powerset of the set of system states hence fixpoint calculi such as the propositional calculus have proven useful both as ways to describe verification algorithms and as specification formalisms in their own right we consider the problem of evaluating expressions in a fixpoint calculus over a given model a naive algorithm for this task may require time n q where n is the maximum length of a chain in the lattice and q is the depth of fixpoint nesting in 1986 emerson and lei presented a method requiring about n d steps where d is the number of alternations between least and greatest fixpoints more recent algorithms have reduced the exponent by one or two but the complexity has remained at about nd in this paper we present a new algorithm that makes extensive use of monotonicity considerations to solve the problem in about nd 2 steps thus the time required by our method is only about the square root of the time required by the earlier algorithms

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

incompleteness of states w r t traces in model checking cousot and cousot introduced and studied a general past future time specification language called calculus featuring a natural time symmetric trace based semantics the standard state based semantics of the calculus is an abstract interpretation of its trace based semantics which turns out to be incomplete that is trace incomplete even for finite systems as a consequence standard state based model checking of the calculus is incomplete w r t trace based model checking this paper shows that any refinement or abstraction of the domain of sets of states induces a corresponding semantics which is still trace incomplete for any propositional fragment of the calculus this derives from a number of results one for each incomplete logical temporal connective of the calculus that characterize the structure of models i e transition systems whose corresponding state based semantics of the calculus is trace complete lete

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

a note on isoperimetric inequalities for logarithmic potentials abstract we give an inequality for all schatten p norms with p 2 on a class of bounded domains majorized by a given domain we partially prove the ruzhansky suragan s conjecture 12 by showing that for any integer p 3 the square cube is a maximizer of schatten p norm of the logarithmic newtonian potential among all rectangles rectangular parallelepipeds of a given lebesgue measure

a modular approach to defining and characterising notions of simulation we propose a modular approach to defining notions of simulation and modal logics which characterise them we use coalgebras to model state based systems relators to define notions of simulation for such systems and inductive techniques to define the syntax and semantics of modal logics for coalgebras we show that the expressiveness of an inductively defined logic for coalgebras w r t a notion of simulation follows from an expressivity condition involving one step in the definition of the logic and the relator inducing that notion of simulation moreover we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types to a relational level as well as to a logical level we use these results to obtain baltag s logic for coalgebraic simulation as well as notions of simulation and associated logics for a large class of non deterministic and probabilistic systems

a fully abstract semantics for a higher order functional language with nondeterministic computation abstract this paper presents a functional programming language based on moggi s monadic metalanguage in the first part of this paper we show how the language can be regarded as a monad on a category of signatures and that the resulting category of algebras is equivalent to the category of computationally cartesian closed categories in the second part we extend the language to include a nondeterministic operational semantics and show that the lower powerdomain semantics is fully abstract for may testing

proof systems for satisfiability in hennessy milner logic with recursion abstract an extension of hennessy milner logic with recursion is presented a recursively specified formula will have two standard interpretations a minimal one and a maximal one minimal interpretations of formulas are useful for expressing liveness properties of processes whereas maximal interpretations are useful for expressing safety properties we present sound and complete proof systems for both interpretations for when a process satisfies a possibly recursive formula the rules of the proof systems consist of an introduction rule for each possible structure of a formula and are intended to extend the work of stirling and winskel moreover the proof systems may be presented directly in prolog to yield a decision procedure for verifying when a finite state process satisfies a specification given as a possibly recursive formula