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Temporal and modal logic
 HANDBOOK OF THEORETICAL COMPUTER SCIENCE
, 1995
"... We give a comprehensive and unifying survey of the theoretical aspects of Temporal and modal logic. ..."
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Cited by 1107 (16 self)
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We give a comprehensive and unifying survey of the theoretical aspects of Temporal and modal logic.
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 202 (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.
On Uniformity within NC¹
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by ..."
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Cited by 127 (19 self)
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In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by firstorder formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic logtime reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for lowlevel circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].
TwoDimensional Languages
, 1997
"... this paper, much work have been done in studying properties of picture languages recognized by finitestate machines and several other models have been designed. A survey on this subject can be found in [21]. An intersting model of twodimensional tape acceptor is the twodimensional online tessell ..."
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Cited by 56 (3 self)
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this paper, much work have been done in studying properties of picture languages recognized by finitestate machines and several other models have been designed. A survey on this subject can be found in [21]. An intersting model of twodimensional tape acceptor is the twodimensional online tessellation automaton introduced by K. Inoue and A. Nakamura in [18]. This is defined as an infinite twodimensional array of identical conventional finitestate automata and it is a special type of cellular automaton. Despite it is not evident that it is a generalization of a onedimensional model, it can be easily 2 identified to a conventional automaton when restricted to onerow (or onecolumn) pictures. Moreover, the family of picture languages recognized by this model of automaton satisfy many important properties. Different systems to generate pictures using grammars have been also explored (cf. [31, 32, 33, 35, 34, 36, 29, 30, 39]). However, in the finite state case, this approach is shown to be less powerful than others. Another possible generalization is to describe picture languages by logic formulas. Recently, W. Thomas gave a general formalism to describe graphs (and, in particular, pictures) as model theoretical structures and showed as "recognizability" corresponds to the notions of definability on existential monadic second order logic (cf. [38]). This is coherent with the string language recognizability theory where Buchi's Theorem holds. In a recent proposal (cf. [13, 14]) a notion of recognizability of a set of pictures in terms of tiling systems is introduced. The underlying idea is to define recognizability by "projection of local properties". Informally, recognition in a tiling system is defined in terms of a finite set of square pictures of side two which c...
Over Words, Two Variables Are as Powerful as One Quantifier Alternation: FO²=Sigma_2\cap Pi_2
, 1998
"... . We show a property of strings is expressible in the twovariable fragment of firstorder logic if and only if it is expressible by both a \Sigma 2 and a \Pi 2 sentence. We thereby establish: UTL = FO 2 = \Sigma 2 " \Pi 2 = UL ; where UTL stands for the string properties expressible in the tempor ..."
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Cited by 42 (8 self)
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. We show a property of strings is expressible in the twovariable fragment of firstorder logic if and only if it is expressible by both a \Sigma 2 and a \Pi 2 sentence. We thereby establish: UTL = FO 2 = \Sigma 2 " \Pi 2 = UL ; where UTL stands for the string properties expressible in the temporal logic with `eventually in the future' and `eventually in the past' as the only temporal operators and UL stands for the class of unambiguous languages. This enables us to show that the problem of determining whether or not a given temporal string property belongs to UTL is decidable (in exponential space), which settles a hitherto open problem. Our proof of \Sigma 2 " \Pi 2 = FO 2 involves a new combinatorial characterization of these two classes and introduces a new method of playing EhrenfeuchtFraiss'e games to verify identities in semigroups. While the number of variables required to express a certain graph property in firstorder logic is an important measure for the descriptional...
Dynamic Linear Time Temporal Logic
 IN ANNALS OF PURE AND APPLIED LOGIC
, 1997
"... A simple extension of the propositional temporal logic of linear time is proposed. The extension consists of strengthening the until operator by indexing it with the regular programs of propositional dynamic logic (PDL). It is shown that DLTL, the resulting logic, is expressively equivalent to S ..."
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Cited by 42 (3 self)
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A simple extension of the propositional temporal logic of linear time is proposed. The extension consists of strengthening the until operator by indexing it with the regular programs of propositional dynamic logic (PDL). It is shown that DLTL, the resulting logic, is expressively equivalent to S1S, the monadic secondorder theory of !sequences. In fact a sublogic of DLTL which corresponds to propositional dynamic logic with a linear time semantics is already as expressive as S1S. We pin down in an obvious manner the sublogic of DLTL which correponds to the first order fragment of S1S. We show that DLTL has an exponential time decision procedure. We also obtain an axiomatization of DLTL. Finally, we point to some natural extensions of the approach presented here for bringing together propositional dynamic and temporal logics in a linear time setting.
On the Expressive Power of Temporal Logic
 J. COMPUT. SYSTEM SCI
, 1993
"... We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective ..."
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Cited by 42 (4 self)
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We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the "restricted" temporal logic (RTL), obtained by discarding the "until" operator. A formal language is RTLexpressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads
Automated Temporal Reasoning about Reactive Systems
, 1996
"... . There is a growing need for reliable methods of designing correct reactive systems such as computer operating systems and air traffic control systems. It is widely agreed that certain formalisms such as temporal logic, when coupled with automated reasoning support, provide the most effective a ..."
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Cited by 39 (2 self)
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. There is a growing need for reliable methods of designing correct reactive systems such as computer operating systems and air traffic control systems. It is widely agreed that certain formalisms such as temporal logic, when coupled with automated reasoning support, provide the most effective and reliable means of specifying and ensuring correct behavior of such systems. This paper discusses known complexity and expressiveness results for a number of such logics in common use and describes key technical tools for obtaining essentially optimal mechanical reasoning algorithms. However, the emphasis is on underlying intuitions and broad themes rather than technical intricacies. 1 Introduction There is a growing need for reliable methods of designing correct reactive systems. These systems are characterized by ongoing, typically nonterminating and highly nondeterministic behavior. Examples include operating systems, network protocols, and air traffic control systems. There is w...
Weighted automata and weighted logics
 In Automata, Languages and Programming – 32nd International Colloquium, ICALP 2005
, 2005
"... Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speechtotext processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We g ..."
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Cited by 39 (7 self)
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Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speechtotext processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We generalize Büchi’s and Elgot’s fundamental theorems to this quantitative setting. We introduce a weighted version of MSO logic and prove that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with our weighted logic. We also consider weighted firstorder logic and show that aperiodic series coincide with the firstorder definable ones, if the semiring is locally finite, commutative and has some aperiodicity property. 1
Classifying Discrete Temporal Properties
 Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS 99), Lecture Notes in Computer Science 1563, SpringerVerlag
, 1999
"... This paper surveys recent results on the classification of discrete temporal properties, gives an introduction to the methods that have been developed to obtain them, and explains the connections to the theory of finite automata, the theory of nite semigroups, and to firstorder logic. ..."
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Cited by 38 (0 self)
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This paper surveys recent results on the classification of discrete temporal properties, gives an introduction to the methods that have been developed to obtain them, and explains the connections to the theory of finite automata, the theory of nite semigroups, and to firstorder logic.