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105
Visibly pushdown languages
, 2004
"... Abstract. We study congruences on words in order to characterize the class of visibly pushdown languages (Vpl), a subclass of contextfree languages. For any language L, we define a natural congruence on words that resembles the syntactic congruence for regular languages, such that this congruence i ..."
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Cited by 134 (15 self)
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Abstract. We study congruences on words in order to characterize the class of visibly pushdown languages (Vpl), a subclass of contextfree languages. For any language L, we define a natural congruence on words that resembles the syntactic congruence for regular languages, such that this congruence is of finite index if, and only if, L is a Vpl. We then study the problem of finding canonical minimal deterministic automata for Vpls. Though Vpls in general do not have unique minimal automata, we consider a subclass of VPAs called kmodule singleentry VPAs that correspond to programs with recursive procedures without input parameters, and show that the class of wellmatched Vpls do indeed have unique minimal kmodule singleentry automata. We also give a polynomial time algorithm that minimizes such kmodule singleentry VPAs. 1 Introduction The class of visibly pushdown languages (Vpl), introduced in [1], is a subclassof contextfree languages accepted by pushdown automata in which the input letter determines the type of operation permitted on the stack. Visibly pushdown languages are closed under all boolean operations, and problems such as inclusion, that are undecidable for contextfree languages, are decidable for Vpl. Vpls are relevant to several applications that use contextfree languages suchas the modelchecking of software programs using their pushdown models [13]. Recent work has shown applications in other contexts: in modeling semanticsof effects in processing XML streams [4], in game semantics for programming languages [5], and in identifying larger classes of pushdown specifications thatadmit decidable problems for infinite games on pushdown graphs [6].
From nondeterministic Büchi and Streett automata to deterministic parity automata
 In 21st Symposium on Logic in Computer Science (LICS’06
, 2006
"... Determinization and complementation are fundamental notions in computer science. When considering finite automata on finite words determinization gives also a solution to complementation. Given a nondeterministic finite automaton there exists an exponential construction that gives a deterministic au ..."
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Cited by 44 (3 self)
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Determinization and complementation are fundamental notions in computer science. When considering finite automata on finite words determinization gives also a solution to complementation. Given a nondeterministic finite automaton there exists an exponential construction that gives a deterministic automaton for the same language. Dualizing the set of accepting states gives an automaton for the complement language. In the theory of automata on infinite words, determinization and complementation are much more involved. Safra provides determinization constructions for Büchi and Streett automata that result in deterministic Rabin automata. For a Büchi automaton with n states, Safra constructs a deterministic Rabin automaton with n O(n) states and n pairs. For a Streett automaton with n states and k pairs, Safra constructs a deterministic Rabin automaton with (nk) O(nk) states and n(k + 1) pairs. Here, we reconsider Safra’s determinization constructions. We show how to construct automata with fewer states and, most importantly, parity acceptance condition. Specifically, starting from a nondeterministic Büchi automaton with n states our construction yields a deterministic parity automaton with n 2n+2 states and index 2n (instead of a Rabin automaton with (12) n n 2n states and n pairs). Starting from a nondeterministic Streett automaton with n states and k pairs our construction yields a deterministic parity automaton with n n(k+2)+2 (k+1) 2n(k+1) states and index 2n(k + 1) (instead of a Rabin automaton with (12) n(k+1) n n(k+2) (k+1) 2n(k+1) states and n(k+1) pairs). The parity condition is much simpler than the Rabin condition. In applications such as solving games and emptiness of tree automata handling the Rabin condition involves an additional multiplier of n 2 n! (or (n(k + 1)) 2 (n(k + 1))! in the case of Streett) which is saved using our construction.
Quantitative Solution of OmegaRegular Games
"... We consider twoplayer games played for an infinite number of rounds, with ωregular winning conditions. The games may be concurrent, in that the players choose their moves simultaneously and independently, and probabilistic, in that the moves determine a probability distribution for the successor s ..."
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Cited by 38 (13 self)
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We consider twoplayer games played for an infinite number of rounds, with ωregular winning conditions. The games may be concurrent, in that the players choose their moves simultaneously and independently, and probabilistic, in that the moves determine a probability distribution for the successor state. We introduce quantitative game µcalculus, and we show that the maximal probability of winning such games can be expressed as the fixpoint formulas in this calculus. We develop the arguments both for deterministic and for probabilistic concurrent games; as a special case, we solve probabilistic turnbased games with ωregular winning conditions, which was also open. We also characterize the optimality, and the memory requirements, of the winning strategies. In particular, we show that while memoryless strategies suffice for winning games with safety and reachability conditions, Büchi conditions require the use of strategies with infinite memory. The existence of optimal strategies, as opposed to εoptimal, is only guaranteed in games with safety winning conditions.
Optimizations for LTL synthesis
 IN 6TH CONFERENCE ON FORMAL METHODS IN COMPUTER AIDED DESIGN (FMCAD’06
, 2006
"... We present an approach to automatic synthesis of specifications given in Linear Time Logic. The approach is based on a translation through universal coBüchi tree automata and alternating weak tree automata [1]. By careful optimization of all intermediate automata, we achieve a major improvement i ..."
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Cited by 35 (9 self)
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We present an approach to automatic synthesis of specifications given in Linear Time Logic. The approach is based on a translation through universal coBüchi tree automata and alternating weak tree automata [1]. By careful optimization of all intermediate automata, we achieve a major improvement in performance. We present several optimization techniques for alternating tree automata, including a gamebased approximation to language emptiness and a simulationbased optimization. Furthermore, we use an incremental algorithm to compute the emptiness of nondeterministic Büchi tree automata. All our optimizations are computed in time polynomial in the size of the automaton on which they are computed. We have applied our implementation to several examples and show a significant improvement over the straightforward implementation. Although our examples are still small, this work constitutes the first implementation of a synthesis algorithm for full LTL. We believe that the optimizations discussed here form an important step towards making LTL synthesis practical.
From Verification to Control: Dynamic Programs for Omegaregular Objectives
, 2001
"... Dynamic programs, or fixpoint iteration schemes, are useful for solving many problems on state spaces, including model checking on Kripke structures ("verification"), computing shortest paths on weighted graphs ("optimization"), computing the value of games played on game graphs ..."
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Cited by 23 (4 self)
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Dynamic programs, or fixpoint iteration schemes, are useful for solving many problems on state spaces, including model checking on Kripke structures ("verification"), computing shortest paths on weighted graphs ("optimization"), computing the value of games played on game graphs ("control"). For Kripke structures, a rich fixpoint theory is available in the form of the calculus. Yet few connections have been made between different interpretations of fixpoint algorithms. We study the question of when a particular fixpoint iteration scheme ' for verifying an !regular property on a Kripke structure can be used also for solving a twoplayer game on a game graph with winning objective. We provide a sufficient and necessary criterion for the answer to be a rmative in the form of an extremalmodel theorem for games: under a game interpretation, the dynamic program' solves the game with objective if and only if both (1) under an existential interpretation on Kripke structures,' is equivalent to 9, and (2) under a universal interpretation on Kripke structures,' is equivalent to 8. In other words,' is correct on all twoplayer game graphs i it is correct on all extremal game graphs, where one or the other player has no choice of moves. The theorem generalizes to quantitative interpretations, where it connects twoplayer games with costs to weighted graphs. While the standard translations from !regular properties to thecalculus violate (1) or (2), we give a translation that satisfies both conditions. Our construction, therefore, yields fixpoint iteration schemes that can be uniformly applied on Kripke structures, weighted graphs, game graphs, and game graphs with costs, in order to meet or optimize a given !regular objective.
Lattice automata
 In Proc. 8th International Conference on Verification, Model Checking, and Abstract Interpretation
, 2007
"... Abstract. Several verification methods involve reasoning about multivalued systems, in which an atomic proposition is interpreted at a state as a lattice element, rather than a Boolean value. The automatatheoretic approach for reasoning about Booleanvalued systems has proven to be very useful and ..."
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Cited by 21 (7 self)
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Abstract. Several verification methods involve reasoning about multivalued systems, in which an atomic proposition is interpreted at a state as a lattice element, rather than a Boolean value. The automatatheoretic approach for reasoning about Booleanvalued systems has proven to be very useful and powerful. We develop an automatatheoretic framework for reasoning about multivalued objects, and describe its application. The basis to our framework are lattice automata on finite and infinite words, which assign to each input word a lattice element. We study the expressive power of lattice automata, their closure properties, the blowup involved in related constructions, and decision problems for them. Our framework and results are different and stronger then those known for semiring and weighted automata. Lattice automata exhibit interesting features from a theoretical point of view. In particular, we study the complexity of constructions and decision problems for lattice automata in terms of the size of both the automaton and the underlying lattice. For example, we show that while determinization of lattice automata involves a blow up that depends on the size of the lattice, such a blow up can be avoided when we complement lattice automata. Thus, complementation is easier than determinization. In addition to studying the theoretical aspects of lattice automata, we describe how they can be used for an efficient reasoning about a multivalued extension of LTL. 1
On Complementing Nondeterministic Büchi Automata
, 2003
"... Several optimal algorithms have been proposed for the complementation of nondeterministic B uchi word automata. Due to the intricacy of the problem and the exponential blowup that complementation involves, these algorithms have never been used in practice, even though an effective complementatio ..."
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Cited by 20 (8 self)
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Several optimal algorithms have been proposed for the complementation of nondeterministic B uchi word automata. Due to the intricacy of the problem and the exponential blowup that complementation involves, these algorithms have never been used in practice, even though an effective complementation construction would be of significant practical value. Recently, Kupferman and Vardi described a complementation algorithm that goes through weak alternating automata and that seems simpler than previous algorithms. We combine their algorithm with known and new minimization techniques. Our approach is based on optimizations of both the intermediate weak alternating automaton and the final nondeterministic automaton, and involves techniques of rank and height reductions, as well as direct and fair simulation.
Safraless compositional synthesis
 In CAV
, 2006
"... Abstract. In automated synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. In spite of the rich theory developed for system synthesis, little of this theory has been reduced to practice. This is in contrast with of modelchecking theory, which has l ..."
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Cited by 17 (6 self)
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Abstract. In automated synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. In spite of the rich theory developed for system synthesis, little of this theory has been reduced to practice. This is in contrast with of modelchecking theory, which has led to industrial development and use of formal verification tools. We see two main reasons for the lack of practical impact of synthesis. The first is algorithmic: synthesis involves Safra’s determinization of automata on infinite words, and a solution of parity games with highly complex state spaces; both problems have been notoriously resistant to efficient implementation. The second is methodological: current theory of synthesis assumes a single comprehensive specification. In practice, however, the specification is composed of a set of properties, which is typically evolving – properties may be added, deleted, or modified. In this work we address both issues. We extend the Safraless synthesis algorithm of Kupferman and Vardi so that it handles LTL formulas by translating them to nondeterministic generalized Büchi automata. This leads to an exponential improvement in the complexity of the algorithm. Technically, our algorithm reduces the synthesis problem to the emptiness problem of a nondeterministic Büchi tree automaton A. The generation of A avoids determinization, avoids the parity acceptance condition, and is based on an analysis of runs of universal generalized coBüchi tree automata. The clean and simple structure of A enables optimizations and a symbolic implementation. In addition, it makes it possible to use information gathered during the synthesis process of properties in the process of synthesizing their conjunction. 1