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18
Automatic Structures
 IN PROC. 15TH IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE
, 1999
"... We study definability and complexity issues for automatic and wautomatic structures. These are, in general, infinite structures but they can be finitely presented by a collection of automata. Moreover, they admit effective (in fact automatic) evaluation of all firstorder queries. Therefore, automa ..."
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Cited by 89 (7 self)
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We study definability and complexity issues for automatic and wautomatic structures. These are, in general, infinite structures but they can be finitely presented by a collection of automata. Moreover, they admit effective (in fact automatic) evaluation of all firstorder queries. Therefore, automatic structures provide an interesting framework for extending many algorithmic and logical methods from finite structures to infinite ones. We explain the notion of (w)automatic structures, give examples, and discuss the relationship to automatic groups. We determine the complexity of model checking and query evaluation on automatic structures for fragments of firstorder logic. Further, we study closure properties and definability issues on automatic structures and present a technique for proving that a structure is not automatic. We give modeltheoretic characterisations for automatic structures via interpretations. Finally we discuss the composition theory of automatic structures and pro...
Finite Presentations of Infinite Structures: Automata and Interpretations
 Theory of Computing Systems
, 2002
"... We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations. ..."
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Cited by 42 (3 self)
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We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations.
An effective decision procedure for linear arithmetic with integer and real variables
 ACM Transactions on Computational Logic (TOCL
, 2005
"... This article considers finiteautomatabased algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. T ..."
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Cited by 26 (6 self)
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This article considers finiteautomatabased algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. The contribution of this article is to show, using topological arguments, that only a restricted class of automata on infinite words are necessary for handling real and integer linear arithmetic. This allows the use of substantially simpler algorithms, which have been successfully implemented.
Binary Reachability Analysis of Pushdown Timed Automata with Dense Clocks
 In CAV’01, volume 2102 of LNCS
"... . We consider pushdown timed automata (PTAs) that are timed automata (with dense clocks) augmented with a pushdown stack. A configuration of a PTA includes a control state, dense clock values and a stack word. By using the pattern technique, we give a decidable characterization of the binary reachab ..."
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Cited by 18 (12 self)
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. We consider pushdown timed automata (PTAs) that are timed automata (with dense clocks) augmented with a pushdown stack. A configuration of a PTA includes a control state, dense clock values and a stack word. By using the pattern technique, we give a decidable characterization of the binary reachability (i.e., the set of all pairs of configurations such that one can reach the other) of a PTA. Since a timed automaton can be treated as a PTA without the pushdown stack, we can show that the binary reachability of a timed automaton is definable in the additive theory of reals and integers. The results can be used to verify a class of properties containing linear relations over both dense variables and unbounded discrete variables. The properties previously could not be verified using the classic region technique nor expressed by timed temporal logics for timed automata and CTL for pushdown systems. 1 Introduction A timed automaton [3] can be considered as a finite automaton augmented...
Pushdown Timed Automata: a Binary Reachability Characterization and Safety Verification
 Theoretical Computer Science
, 2003
"... We consider pushdown timed automata (PTAs) that are timed automata (with dense clocks) augmented with a pushdown stack. A configuration of a PTA includes a state, dense clock values and a stack word. By using the pattern technique, we give a decidable characterization of the binary reachability ( ..."
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Cited by 16 (7 self)
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We consider pushdown timed automata (PTAs) that are timed automata (with dense clocks) augmented with a pushdown stack. A configuration of a PTA includes a state, dense clock values and a stack word. By using the pattern technique, we give a decidable characterization of the binary reachability (i.e., the set of all pairs of configurations such that one can reach the other) of a PTA. Since a timed automaton can be treated as a PTA without the pushdown stack, we can show that the binary reachability of a timed automaton is definable in the additive theory of reals and integers. The results can be used to verify a class of properties containing linear relations over both dense variables and unbounded discrete variables. The properties previously could not be verified using the classic region technique nor expressed by timed temporal logics for timed automata and CTL for pushdown systems. The results are also extended to other generalizations of timed automata.
Efficient symbolic representations for arithmetic constraints in verification
"... In this paper we discuss efficient symbolic representations for infinitestate systems specified using linear arithmetic constraints. We give algorithms for constructing finite automata which represent integer sets that satisfy linear constraints. These automata can represent either signed or unsign ..."
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Cited by 12 (4 self)
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In this paper we discuss efficient symbolic representations for infinitestate systems specified using linear arithmetic constraints. We give algorithms for constructing finite automata which represent integer sets that satisfy linear constraints. These automata can represent either signed or unsigned integers and have a lower number of states compared to other similar approaches. We present efficient storage techniques for the transition function of the automata and extend the construction algorithms to formulas on both boolean and integer variables. We also derive conditions which guarantee that the precondition computations used in symbolic verification algorithms do not cause an exponential increase in the automata size. We experimentally compare different symbolic representations by using them to verify nontrivial concurrent systems. Experimental results show that the symbolic representations based on our construction algorithms outperform the polyhedral representation used in Omega Library, and the automata representation used in LASH.
Omegaregular model checking
 In Proc. 10th TACAS. LNCS
, 2004
"... Checking infinitestate systems is frequently done by encoding infinite sets of states as regular languages. Computing such a regular representation of, say, the set of reachable states of a system requires acceleration techniques that can finitely compute the effect of an unbounded number of transi ..."
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Cited by 11 (3 self)
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Checking infinitestate systems is frequently done by encoding infinite sets of states as regular languages. Computing such a regular representation of, say, the set of reachable states of a system requires acceleration techniques that can finitely compute the effect of an unbounded number of transitions. Among the acceleration techniques that have been proposed, one finds both specific and generic techniques. Specific techniques exploit the particular type of system being analyzed, e.g. a system manipulating queues or integers, whereas generic techniques only assume that the transition relation is represented by a finitestate transducer, which has to be iterated. In this paper, we investigate the possibility of using generic techniques in cases where only specific techniques have been exploited so far. Finding that existing generic techniques are often not applicable in cases easily handled by specific techniques, we have developed a new approach to iterating transducers. This new approach builds on earlier work, but exploits a number of new conceptual and algorithmic ideas, often induced with the help of experiments, that give it a broad scope, as well as good performances.
On the Automata Size for Presburger Arithmetic
 In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004
, 2004
"... Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Pr ..."
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Cited by 10 (2 self)
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Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.
A Generalization of Cobham’s Theorem to Automata over Real Numbers
, 2008
"... This article studies the expressive power of finitestate automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the firstorder additive theory of real and integer variables 〈R, Z, +, < 〉 can all be recognized by weak deterministic Büchi auto ..."
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Cited by 4 (1 self)
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This article studies the expressive power of finitestate automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the firstorder additive theory of real and integer variables 〈R, Z, +, < 〉 can all be recognized by weak deterministic Büchi automata, regardless of the encoding base r> 1. In this article, we prove the reciprocal property, i.e., that a subset of R that is recognizable by weak deterministic automata in every base r> 1 is necessarily definable in 〈R, Z, +, <〉. This result generalizes to real numbers the wellknown Cobham’s theorem on the finitestate recognizability of sets of integers. Our proof gives interesting insight into the internal structure of automata recognizing sets of real numbers, which may lead to efficient data structures for handling these sets.
Dense counter machines and verification problems, in
 CAV, LNCS
"... Abstract. We generalize the traditional definition of a multicounter machine (where the counters, which can only assume nonnegative integer values, can be incremented/decremented by 1 and tested for zero) by allowing the machine the additional ability to increment/decrement the counters by a nondete ..."
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Cited by 2 (0 self)
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Abstract. We generalize the traditional definition of a multicounter machine (where the counters, which can only assume nonnegative integer values, can be incremented/decremented by 1 and tested for zero) by allowing the machine the additional ability to increment/decrement the counters by a nondeterministically chosen fractional amount between 0 and 1 (the may be different at each step). We show that, under some restrictions on counter behavior, the binary reachability set of such a machine is definable in the additive theory of the reals and integers. There are applications of this result in verification, and we give one example in the paper. We also extend the notion of “semilinear language” to “dense semilinear language ” and show its connection to a restricted class of dense multicounter automata.