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26
Parametric realtime reasoning
 IN PROCEEDINGS OF THE 25TH ANNUAL SYMPOSIUM ON THEORY OF COMPUTING
, 1993
"... Traditional approaches to the algorithmic verification of realtime systems are limited to checking program correctness with respect to concrete timing properties (e.g., "message delivery within 10 milliseconds"). We address the more realistic and more ambitious problem of deriving symboli ..."
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Cited by 140 (6 self)
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Traditional approaches to the algorithmic verification of realtime systems are limited to checking program correctness with respect to concrete timing properties (e.g., "message delivery within 10 milliseconds"). We address the more realistic and more ambitious problem of deriving symbolic constraints on the timing properties required of realtime systems (e.g., "message delivery within the time it takes to execute two assignment statements"). To model this problem, we introduce parametric timed automata  finitestate machines whose transitions are constrained with parametric timing requirements. The emptiness question for parametric timed automata is central to the verification problem. On the negative side, we show that in general this question is undecidable. On the positive side, we provide algorithms for checking the emptiness of restricted classes of parametric timed automata. The practical relevance of these classes is illustrated with several verification examples. There remains a gap between the automata classes for which we know that emptiness is decidable and undecidable, respectively, and this gap is related to various hard and open problems of logic and automata theory.
Mixed RealInteger Linear Quantifier Elimination
, 1999
"... Consider the elementary theory T of the real numbers in the language L having 0, 1 as constants, addition and subtraction and integer part as operations, and equality, order and congruences modulo natural number constants as relations. We show that T admits an effective quantifier elimination proced ..."
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Cited by 31 (1 self)
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Consider the elementary theory T of the real numbers in the language L having 0, 1 as constants, addition and subtraction and integer part as operations, and equality, order and congruences modulo natural number constants as relations. We show that T admits an effective quantifier elimination procedure and is decidable. Moreover this procedure provides sample answers for existentially quantified variables. The procedure comprises as special cases linear elimination for the reals and for Presburger arithmetic. We provide closely matching upper and lower bounds for the complexity of the quantifier elimination and decision problem for T . Applications include a characterization of T definable subsets of the real line, and the modeling of parametric mixed integer linear optimization, of continuous phenomena with periodicity, and the simulation and analysis of hybrid control systems. We also consider the elementary theory of reals in variations of this language in view of quantifier elimination...
Topology of Diophantine sets: remarks on Mazur’s conjectures. In Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent
 of Contemp. Math
, 1999
"... Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S ..."
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Cited by 16 (1 self)
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Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S, P on D whose graphs are diophantine in Q 3i (via the inclusion D 3 ⊂ Q 3i), and such that for two specific elements d0, d1 ∈ D the structure (D, S, P, d0, d1) is a model for integer arithmetic (Z,+, ·,0, 1). Using a construction of Pheidas, we give a counterexample to the analogue of Mazur’s conjecture over a global function field, and prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over a finite field. 1.
Complexity and Uniformity of Elimination in Presburger Arithmetic
 UNIVERSITAT PASSAU
, 1997
"... The decision complexity of Presburger Arithmetic PA and its variants has received much attention in the literature. We investigate the complexity of quantifier elimination procedures for PA  a topic that is even more relevant for applications. First we show that the the author's triply expone ..."
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Cited by 15 (3 self)
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The decision complexity of Presburger Arithmetic PA and its variants has received much attention in the literature. We investigate the complexity of quantifier elimination procedures for PA  a topic that is even more relevant for applications. First we show that the the author's triply exponential upper bound is essentially tight. This fact seems to preclude practical applications. By weakening the concept of quantifier elimination slightly to bounded quantifier elimination, we show, however, that the upper and lower bound for quantifier elimination in PA can be lowered by exactly one exponential. Moreover we gain uniformity in the coefficients, a property that we prove to be impossible for complete quantifier elimination in PA. Thus we have tight upper and lower complexity bounds for elimination theory in PA and uniform PA. The results are inspired by experimental implementations of bounded quantifier elimination that have solved nontrivial application problems e.g. in parametric i...
Reachability in Succinct and Parametric OneCounter Automata
"... Abstract. Onecounter automata are a fundamental and widelystudied class of infinitestate systems. In this paper we consider onecounter automata with counter updates encoded in binary—which we refer to as the succinct encoding. It is easily seen that the reachability problem for this class of mac ..."
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Cited by 14 (6 self)
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Abstract. Onecounter automata are a fundamental and widelystudied class of infinitestate systems. In this paper we consider onecounter automata with counter updates encoded in binary—which we refer to as the succinct encoding. It is easily seen that the reachability problem for this class of machines is in PSpace and is NPhard. One of the main results of this paper is to show that this problem is in fact in NP, and is thus NPcomplete. We also consider parametric onecounter automata, in which counter updates be integervalued parameters. The reachability problem asks whether there are values for the parameters such that a final state can be reached from an initial state. Our second main result shows decidability of the reachability problem for parametric onecounter automata by reduction to existential Presburger arithmetic with divisibility. 1
Solving BitVector Equations
 Formal Methods in ComputerAided Design (FMCAD '98
, 1998
"... This paper is concerned with solving equations on fixed and nonfixed size bitvector terms. We define an equational transformation system for solving equations on terms where all sizes of bitvectors and extraction positions are known. This transformation system suggests a generalization for dealin ..."
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Cited by 9 (0 self)
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This paper is concerned with solving equations on fixed and nonfixed size bitvector terms. We define an equational transformation system for solving equations on terms where all sizes of bitvectors and extraction positions are known. This transformation system suggests a generalization for dealing with bitvectors of unknown size and unknown extraction positions. Both solvers adhere to the principle of splitting bitvectors only on demand, thereby making them quite effective in practice.
On Flat Programs with Lists
"... In this paper we analyze the complexity of checking safety and termination properties, for a very simple, yet nontrivial, class of programs with singlylinked list data structures. Since, in general, programs with lists are knownto have the power of Turing machines, we restrict the control struct ..."
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Cited by 6 (0 self)
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In this paper we analyze the complexity of checking safety and termination properties, for a very simple, yet nontrivial, class of programs with singlylinked list data structures. Since, in general, programs with lists are knownto have the power of Turing machines, we restrict the control structure, by forbidding nested loops and destructive updates. Surprisingly, even with these simplifying conditions, verifying safety and termination for programs working on heaps with more than one cycle are undecidable, whereas decidability can be established when the input heap may have at most one loop. The proofs for both the undecidability and the decidability results rely on nontrivial numbertheoreticresults.
Towards modelchecking programs with lists
 In ILC’07, volume 5489 of LNCS
, 2009
"... Abstract. We aim at checking safety and temporal properties over models representing the behavior of programs manipulating dynamic singlylinked lists. The properties we consider not only allow to perform a classical shape analysis, but we also want to check quantitative aspect on the manipulated m ..."
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Cited by 5 (1 self)
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Abstract. We aim at checking safety and temporal properties over models representing the behavior of programs manipulating dynamic singlylinked lists. The properties we consider not only allow to perform a classical shape analysis, but we also want to check quantitative aspect on the manipulated memory heap. We first explain how a translation of programs into counter systems can be used to check safety problems and temporal properties. We then study the decidability of these two problems considering some restricted classes of programs, namely flat programs without destructive update. We obtain the following results: (1) the modelchecking problem is decidable if the considered program works over acyclic lists (2) the safety problem is decidable for programs without alias test. We finally explain the limit of our decidability results, showing that relaxing one of the hypothesis leads to undecidability results. 1
Automating elementary numbertheoretic proofs using Gröbner bases
"... Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates ..."
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Abstract. We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain numbertheoretic predicates such as ‘divisible by’, ‘congruent ’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings. 1