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Realtime logics: complexity and expressiveness
 INFORMATION AND COMPUTATION
, 1993
"... The theory of the natural numbers with linear order and monadic predicates underlies propositional linear temporal logic. To study temporal logics that are suitable for reasoning about realtime systems, we combine this classical theory of in nite state sequences with a theory of discrete time, via ..."
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Cited by 203 (16 self)
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The theory of the natural numbers with linear order and monadic predicates underlies propositional linear temporal logic. To study temporal logics that are suitable for reasoning about realtime systems, we combine this classical theory of in nite state sequences with a theory of discrete time, via a monotonic function that maps every state to its time. The resulting theory of timed state sequences is shown to be decidable, albeit nonelementary, and its expressive power is characterized by! regular sets. Several more expressive variants are proved to be highly undecidable. This framework allows us to classify a wide variety of realtime logics according to their complexity and expressiveness. Indeed, it follows that most formalisms proposed in the literature cannot be decided. We are, however, able to identify two elementary realtime temporal logics as expressively complete fragments of the theory of timed state sequences, and we present tableaubased decision procedures for checking validity. Consequently, these two formalisms are wellsuited for the speci cation and veri cation of realtime systems.
The Complexity of Firstorder and Monadic Secondorder Logic Revisited
 Annals of Pure and Applied Logic
, 2002
"... The modelchecking problem for a logic L on a class C of structures asks whether a given Lsentence holds in a given structure in C. In this paper, we give superexponential lower bounds for fixedparameter tractable modelchecking problems for firstorder and monadic secondorder logic. We show tha ..."
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Cited by 65 (6 self)
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The modelchecking problem for a logic L on a class C of structures asks whether a given Lsentence holds in a given structure in C. In this paper, we give superexponential lower bounds for fixedparameter tractable modelchecking problems for firstorder and monadic secondorder logic. We show that unless PTIME = NP, the modelchecking problem for monadic secondorder logic on finite words is not solvable in time f(k) · p(n), for any elementary function f and any polynomial p. Here k denotes the size of the input sentence and n the size of the input word. We prove the same result for firstorder logic under a stronger complexity theoretic assumption from parameterized complexity theory. Furthermore, we prove that the modelchecking problems for firstorder logic on structures of degree 2 and of bounded degree d ≥ 3 are not solvable in time 2 2o(k) · p(n) (for degree 2) and 2 22o(k) · p(n) (for degree d), for any polynomial p, again under an assumption from parameterized complexity theory. We match these lower bounds by corresponding upper bounds. 1.
FirstOrder Logic with Two Variables and Unary Temporal Logic
 INF. COMPUT
, 1997
"... We investigate the power of firstorder logic with only two variables over ωwords and finite words, a logic denoted by FO². We prove that FO² can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", ..."
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Cited by 60 (9 self)
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We investigate the power of firstorder logic with only two variables over ωwords and finite words, a logic denoted by FO². We prove that FO² can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", and "sometime in the past", a logic we denote by unaryTL. Moreover, our translation from FO² to unaryTL converts every FO² formula to an equivalent unaryTL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the firstorder formula. We show
Mona Fido: The LogicAutomaton Connection in Practice
, 1998
"... We discuss in this paper how connections, discovered almost forty years ago, between logics and automata can be used in practice. For such logics expressing regular sets, we have developed tools that allow efficient symbolic reasoning not attainable by theorem proving or symbolic model checking. ..."
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Cited by 53 (10 self)
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We discuss in this paper how connections, discovered almost forty years ago, between logics and automata can be used in practice. For such logics expressing regular sets, we have developed tools that allow efficient symbolic reasoning not attainable by theorem proving or symbolic model checking. We explain how the logicautomaton connection is already exploited in a limited way for the case of Quantified Boolean Logic, where Binary Decision Diagrams act as automata. Next, we indicate how BDD data structures and algorithms can be extended to yield a practical decision procedure for a more general logic, namely WS1S, the Weak Secondorder theory of One Successor. Finally, we mention applications of the automatonlogic connection to software engineering and program verification. 1
On the Complexity of Nonrecursive XQuery and Functional Query Languages on Complex Values
 In Proc. PODS’05
"... This article studies the complexity of evaluating functional query languages for complex values such as monad algebra and the recursionfree fragment of XQuery. We show that monad algebra with equality restricted to atomic values is complete for the class TA[2O(n) , O(n)] of problems solvable in lin ..."
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Cited by 40 (1 self)
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This article studies the complexity of evaluating functional query languages for complex values such as monad algebra and the recursionfree fragment of XQuery. We show that monad algebra with equality restricted to atomic values is complete for the class TA[2O(n) , O(n)] of problems solvable in linear exponential time with a linear number of alternations. The monotone fragment of monad algebra with atomic value equality but without negation is complete for nondeterministic exponential time. For monad algebra with deep equality, we establish TA[2O(n) , O(n)] lower and exponentialspace upper bounds. We also study a fragment of XQuery, Core XQuery, that seems to incorporate all the features of a query language on complex values that are traditionally deemed essential. A close connection between monad algebra on lists and Core XQuery (with “child ” as the only axis) is exhibited, and it is shown that these languages are expressively equivalent up to representation issues. We show that Core XQuery is just as hard as monad algebra w.r.t. query and combined complexity, and that it is in TC0 if the query is assumed fixed. As Core XQuery is NEXPTIMEhard, it is commonly believed that any algorithm for evaluating Core XQuery has to require exponential amounts of working memory and doubly exponential time in the worst case. We present a property of queries – the lack of a certain form of composition – that virtually all realworld XQueries have and that allows for query evaluation in singly exponential time and polynomial space. Still, we are able to show for an important special case – Core XQuery with equality testing restricted to atomic values – that the compositionfree language is just as expressive as the language with composition. Thus, under widelyheld complexitytheoretic assumptions, the compositionfree language is an exponentially less succinct version of the language with composition.
Complexity of Automata on Infinite Objects
, 1989
"... We investigate in this thesis problems concerning the complexity of translation among, and decision procedure for, different types of finite automata on infinite words (! automata). An !automaton is the same as usual finite automata over finite strings but it accepts or rejects infinite strings. I ..."
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Cited by 38 (0 self)
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We investigate in this thesis problems concerning the complexity of translation among, and decision procedure for, different types of finite automata on infinite words (! automata). An !automaton is the same as usual finite automata over finite strings but it accepts or rejects infinite strings. It may be either deterministic or nondeterministic, and may have different types of acceptance condition. Our main result is a new, simpler, determinization construction that yields a single exponent upper bound for the translation of any Buchi nondeterministic !automaton into a deterministic !auomaton. This construction is optimal. We also look at the complexity of the complementation problem for different types of !automata, and, among other results, obtain an exponential complementation for Streett !automata. These results can be used to improve the complexity of decision procedures for different logics that use automatatheoretic techniques. Acknowledgement First and foremost, I o...
The Regular RealTime Languages
 In Proc. 25th Int. Coll. Automata, Languages, and Programming (ICALP'98
, 1998
"... . A specification formalism for reactive systems defines a class of !languages. We call a specification formalism fully decidable if it is constructively closed under boolean operations and has a decidable satisfiability (nonemptiness) problem. There are two important, robust classes of !languages ..."
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Cited by 35 (3 self)
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. A specification formalism for reactive systems defines a class of !languages. We call a specification formalism fully decidable if it is constructively closed under boolean operations and has a decidable satisfiability (nonemptiness) problem. There are two important, robust classes of !languages that are definable by fully decidable formalisms. The !regular languages are definable by finite automata, or equivalently, by the Sequential Calculus. The counterfree !regular languages are definable by temporal logic, or equivalently, by the firstorder fragment of the Sequential Calculus. The gap between both classes can be closed by finite counting (using automata connectives), or equivalently, by projection (existential secondorder quantification over letters). A specification formalism for realtime systems defines a class of timed !languages, whose letters have realnumbered time stamps. Two popular ways of specifying timing constraints rely on the use of clocks, and on the use...
CTL^+ Is Exponentially More Succinct Than CTL
, 1999
"... It is proved that CTL + is exponentially more succinct than CTL. More precisely, it is shown that every CTL formula (and every modal ¯calculus formula) equivalent to the CTL + formula E(Fp 0 \Delta \Delta \Delta Fp n\Gamma1 ) is of length at least \Gamma n dn=2e \Delta , which is \Omega\G ..."
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Cited by 35 (0 self)
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It is proved that CTL + is exponentially more succinct than CTL. More precisely, it is shown that every CTL formula (and every modal ¯calculus formula) equivalent to the CTL + formula E(Fp 0 \Delta \Delta \Delta Fp n\Gamma1 ) is of length at least \Gamma n dn=2e \Delta , which is \Omega\Gamma/ n = p n). This matches almost the upper bound provided by Emerson and Halpern, which says that for every CTL + formula of length n there exists an equivalent CTL formula of length at most 2 n log n . It follows that the exponential blowup as incurred in known conversions of nondeterministic Büchi word automata into alternationfree ¯calculus formulas is unavoidable. This answers a question posed by Kupferman and Vardi. The proof of the above lower bound exploits the fact that for every CTL (¯ calculus) formula there exists an equivalent alternating tree automaton of linear size. The core of the proof is an involved cutandpaste argument for alternating tree automata. 1 In...
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Al ..."
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Cited by 30 (3 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
Qualitative SpatioTemporal Representation and Reasoning: A Computational Perspective
 Exploring Artifitial Intelligence in the New Millenium
, 2001
"... this paper argues for the rich world of representation that lies between these two extremes." Levesque and Brachman (1985) 1 Introduction Time and space belong to those few fundamental concepts that always puzzled scholars from almost all scientific disciplines, gave endless themes to science fict ..."
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Cited by 30 (11 self)
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this paper argues for the rich world of representation that lies between these two extremes." Levesque and Brachman (1985) 1 Introduction Time and space belong to those few fundamental concepts that always puzzled scholars from almost all scientific disciplines, gave endless themes to science fiction writers, and were of vital concern to our everyday life and commonsense reasoning. So whatever approach to AI one takes [ Russell and Norvig, 1995 ] , temporal and spatial representation and reasoning will always be among its most important ingredients (cf. [ Hayes, 1985 ] ). Knowledge representation (KR) has been quite successful in dealing separately with both time and space. The spectrum of formalisms in use ranges from relatively simple temporal and spatial databases, in which data are indexed by temporal and/or spatial parameters (see e.g. [ Srefik, 1995; Worboys, 1995 ] ), to much more sophisticated numerical methods developed in computational geom