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42
Reasoning about Infinite Computations
 Information and Computation
, 1994
"... 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 ..."
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Cited by 302 (58 self)
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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 PSPACEcomplete. 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. 1 Introduction For many years, logics of programs have been tools for reasoning about the input/output behavior of programs. When dealing with concurrent or nonterminating processes (like operating systems) there is, however, a need to reason about infinite computations. Thus, instead of considering the first and last states of finite computations, we need to consider the infinite sequences of states that the program goes through...
Partial Derivatives of Regular Expressions and Finite Automata Constructions
 Theoretical Computer Science
, 1995
"... . We introduce a notion of a partial derivative of a regular expression. It is a generalization to the nondeterministic case of the known notion of a derivative invented by Brzozowski. We give a constructive definition of partial derivatives, study their properties, and employ them to develop a new ..."
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Cited by 82 (0 self)
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. We introduce a notion of a partial derivative of a regular expression. It is a generalization to the nondeterministic case of the known notion of a derivative invented by Brzozowski. We give a constructive definition of partial derivatives, study their properties, and employ them to develop a new algorithm for turning regular expressions into relatively small NFA and to provide certain improvements to Brzozowski's algorithm constructing DFA. We report on a prototype implementation of our algorithm constructing NFA and present some examples. Introduction In 1964 Janusz Brzozowski introduced word derivatives of regular expressions and suggested an elegant algorithm turning a regular expression r into a deterministic finite automata (DFA); the main point of the algorithm is that the word derivatives of r serve as states of the resulting DFA [5]. In the following years derivatives were recognized as a quite useful and productive tool. Conway [8] uses derivatives to present various comp...
The complexity of set constraints
, 1993
"... Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. We present several results on the computational complexity of solving systems of set constraints. The systems we study form a natural complexity hierar ..."
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Cited by 67 (11 self)
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Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. We present several results on the computational complexity of solving systems of set constraints. The systems we study form a natural complexity hierarchy depending on the form of the constraint language.
Decidability of Systems of Set Constraints with Negative Constraints
 INFORMATION AND COMPUTATION
, 1994
"... Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. Recently, several algorithms for solving general systems of positive set constraints have appeared. In this paper we consider systems of mixed posi ..."
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Cited by 55 (10 self)
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Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. Recently, several algorithms for solving general systems of positive set constraints have appeared. In this paper we consider systems of mixed positive and negative constraints, which are considerably more expressive than positive constraints alone. We show that it is decidable whether a given such system has a solution. The proof involves a reduction to a numbertheoretic decision problem that may be of independent interest.
Antichains: A new algorithm for checking universality of finite automata
 In Proc. of CAV 2006, LNCS 4144
, 2006
"... Abstract. We propose and evaluate a new algorithm for checking the universality of nondeterministic finite automata. In contrast to the standard algorithm, which uses the subset construction to explicitly determinize the automaton, we keep the determinization step implicit. Our algorithm computes th ..."
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Cited by 39 (16 self)
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Abstract. We propose and evaluate a new algorithm for checking the universality of nondeterministic finite automata. In contrast to the standard algorithm, which uses the subset construction to explicitly determinize the automaton, we keep the determinization step implicit. Our algorithm computes the least fixed point of a monotone function on the lattice of antichains of state sets. We evaluate the performance of our algorithm experimentally using the random automaton model recently proposed by Tabakov and Vardi. We show that on the difficult instances of this probabilistic model, the antichain algorithm outperforms the standard one by several orders of magnitude. We also show how variations of the antichain method can be used for solving the languageinclusion problem for nondeterministic finite automata, and the emptiness problem for alternating finite automata. 1
Solving Systems of Set Constraints (Extended Abstract)
 In Seventh Annual IEEE Symposium on Logic in Computer Science
, 1992
"... ) Alexander Aiken Edward L. Wimmers IBM Almaden Research Center 650 Harry Rd. San Jose, CA 95120 phone: 408/9271876 or 9271882 email: lastname@almaden.ibm.com fax: 408/9272100 Abstract Systems of set constraints are a natural formalism for many problems in program analysis. Set constraints ar ..."
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) Alexander Aiken Edward L. Wimmers IBM Almaden Research Center 650 Harry Rd. San Jose, CA 95120 phone: 408/9271876 or 9271882 email: lastname@almaden.ibm.com fax: 408/9272100 Abstract Systems of set constraints are a natural formalism for many problems in program analysis. Set constraints are also a generalization of tree automata. We present an algorithm for solving systems of set constraints built from free variables, constructors, and the set operations of intersection, union, and complement. Furthermore, we show that all solutions of such systems can be finitely represented. 1 1 Introduction Set constraints are a natural formalism for describing relationships between sets of terms of a free algebra. A set constraint has the form X ` Y , where X and Y are set expressions. Examples of set expressions are 0 (the empty set), 1 (the set of all terms), ff (a setvalued variable), c(X; Y ) (a constructor application) , and the union, intersection, or complement of set expressi...
Rewriting Extended Regular Expressions
, 1993
"... We concider an extened algebra of regular events (languages) with intersection besides the usual operations. This algebra has the structure of a distributive lattice with monotonic operations; the latter property is crucial for some applications. We give a new complete Horn equational axiomatiztion ..."
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Cited by 20 (1 self)
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We concider an extened algebra of regular events (languages) with intersection besides the usual operations. This algebra has the structure of a distributive lattice with monotonic operations; the latter property is crucial for some applications. We give a new complete Horn equational axiomatiztion of the algebra and develop some termrewriting techniques for constructing logical inferences of valid equations. A shorter version of this paper is to appear in the proceedings of Developments in Language Theory, Univ. of Turku, July 1993, published by World Scientific. The present version has been submitted for publication elsewhere. 1 Introduction In this paper we consider an extended algebra of regular events (languages) on a given alphabet with intersection besides the usual operations (union, concatenation, Kleene star, empty, and the regular unit). This algebra has the structure of a distributive lattice (join is union, meet is intersection) with only monotonic operations. The latte...
Recognizing Regular Expressions by means of Dataflow Networks
 In proc. of the 23rd International Colloquium on Automata, Languages, and Programming, (ICALP'96
, 1996
"... . This paper addresses the problem of building a Boolean dataflow network (sequential circuit) recognizing the language described by a regular expression. The main result is that both the construction time and the size of the resulting network are linear with respect to the size of the regular expre ..."
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Cited by 18 (3 self)
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. This paper addresses the problem of building a Boolean dataflow network (sequential circuit) recognizing the language described by a regular expression. The main result is that both the construction time and the size of the resulting network are linear with respect to the size of the regular expression. Introduction "Grep" machine: Let \Sigma be a vocabulary, L be a regular language on \Sigma . A "grep" machine is a machine receiving a sequence s 0 ; s 1 ; : : : ; s n ; : : : of symbols (s i 2 \Sigma ) and computing a sequence b 0 ; b 1 ; : : : ; b n ; : : : of Booleans, such that b n is true if and only if the word s 0 s 1 : : : s n belongs to L 2 . This paper addresses the problem of building a "grep" machine for languages described by regular expressions. This problem is rather classical [4, 11, 10, 3, 1, 2]. We propose a solution which, to our knowledge, is new: Informally, it consists of building, from a regular expression E, a "circuit" (or Boolean dataflow network) explori...
Towards Nominal Computation
, 2012
"... Nominal sets are a different kind of set theory, with a more relaxed notion of finiteness. They offer an elegant formalism for describing λterms modulo αconversion, or automata on data words. This paper is an attempt at defining computation in nominal sets. We present a rudimentary programming lan ..."
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Cited by 16 (4 self)
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Nominal sets are a different kind of set theory, with a more relaxed notion of finiteness. They offer an elegant formalism for describing λterms modulo αconversion, or automata on data words. This paper is an attempt at defining computation in nominal sets. We present a rudimentary programming language, called Nλ. The key idea is that it includes a native type for finite sets in the nominal sense. To illustrate the power of our language, we write short programs that process automata on data words.
Symbolic backwardsreachability analysis for higherorder pushdown systems
 IN FOSSACS
, 2007
"... Higherorder pushdown systems (PDSs) generalise pushdown systems through the use of higherorder stacks; that is, a nested “stack of stacks ” structure. These systems may be used to model higherorder programs and are closely related to the Caucal hierarchy of infinite graphs and safe higherorder ..."
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Cited by 15 (5 self)
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Higherorder pushdown systems (PDSs) generalise pushdown systems through the use of higherorder stacks; that is, a nested “stack of stacks ” structure. These systems may be used to model higherorder programs and are closely related to the Caucal hierarchy of infinite graphs and safe higherorder recursion schemes. We generalise higherorder PDSs to higherorder Alternating PDSs (APDSs) and consider the backwardsreachability problem over these systems. This builds on and extends previous work into pushdown systems and contextfree higherorder processes in a nontrivial manner. In particular, we show that the set of configurations from which a regular set of higherorder APDS configurations is reachable is regular and computable in nEXPTIME. In fact, the problem is nEXPTIMEcomplete. We show that this work has several applications in the verification of higherorder PDSs, such as lineartime modelchecking, alternationfree µcalculus modelchecking and the computation of winning regions of reachability games.