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Showing papers on "Undecidable problem published in 2002"


Journal ArticleDOI
TL;DR: This article investigates XML document specifications with DTDs and integrity constraints, such as keys and foreign keys, and establishes complexity bounds on the implication problem, which is shown to be coNP-complete for unary keys andforeign keys.
Abstract: The article investigates XML document specifications with DTDs and integrity constraints, such as keys and foreign keys. We study the consistency problem of checking whether a given specification is meaningful: that is, whether there exists an XML document that both conforms to the DTD and satisfies the constraints. We show that DTDs interact with constraints in a highly intricate way and as a result, the consistency problem in general is undecidable. When it comes to unary keys and foreign keys, the consistency problem is shown to be NP-complete. This is done by coding DTDs and integrity constraints with linear constraints on the integers. We consider the variations of the problem (by both restricting and enlarging the class of constraints), and identify a number of tractable cases, as well as a number of additional NP-complete ones. By incorporating negations of constraints, we establish complexity bounds on the implication problem, which is shown to be coNP-complete for unary keys and foreign keys.

167 citations


Posted Content
TL;DR: In this article, a quantified inequality constraint over the reals is a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are $\leq$ and $<$.
Abstract: Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are $\leq$ and $<$. Solving such constraints is an undecidable problem when allowing function symbols such $\sin$ or $\cos$. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques.

94 citations


Proceedings Article
23 Apr 2002
TL;DR: It is shown that the introduction of numerical state variables makes the planning problem undecidable in the general case and many restrictions thereof and identified special cases for which the author can provide decidability results.
Abstract: These days, propositional planning can be considered a quite well-understood problem. Good algorithms are known that can solve a wealth of very different and sometimes challenging planning tasks, and theoretical computational properties of both general STRIPS-style planning and the best-known benchmark problems have been established. However, propositional planning has a major drawback: The formalism is too weak to allow for the easy encoding of many genuinely interesting planning problems, specifically those involving numbers. A recent effort to enhance the PDDL planning language to cope with (among other additions) numerical state variables, to be used at the third international planning competition, has increased interest in these issues. In this contribution, we analyze "STRIPS with numbers" from a theoretical point of view. Specifically, we show that the introduction of numerical state variables makes the planning problem undecidable in the general case and many restrictions thereof and identify special cases for which we can provide decidability results.

94 citations


Book ChapterDOI
08 Apr 2002
TL;DR: This paper shows how the problem of job-shop scheduling where the jobs are preemptible can be modeled naturally as a shortest path problem defined on an extension of timed automata, namely stopwatch automata where some of the clocks might be freezed at certain states.
Abstract: In this paper we show how the problem of job-shop scheduling where the jobs are preemptible can be modeled naturally as a shortest path problem defined on an extension of timed automata, namely stopwatch automata where some of the clocks might be freezed at certain states. Although general verification problems on stopwatch automata are known to be undecidable, we show that due to particular properties of optimal schedules, the shortest path in the automaton belongs to a finite subset of the set of acyclic paths and hence the problem is solvable. We present several algorithms and heuristics for finding the shortest paths in such automata and test their implementation on numerous benchmark examples.

83 citations


Proceedings ArticleDOI
03 Jun 2002
TL;DR: It is shown that in the presence of foreign keys, compile-time verification of consistency is usually infeasible, and a number of restricted decidable cases are established.
Abstract: XML specifications often consist of a type definition (typically, a DTD) and a set of integrity constraints. It has been shown previously that such specifications can be inconsistent, and thus it is often desirable to check consistency at compile-time. It is known that for general keys and foreign keys, and DTDs, the consistency problem is undecidable; however, it becomes NP-complete when all keys are one-attribute (unary), and tractable, if no foreign keys are used.In this paper, we consider a variety of constraints for XML data, and study the complexity of the consistency problem. Our main conclusion is that in the presence of foreign keys, compile-time verification of consistency is usually infeasible. We look at two types of constraints: absolute (that hold in the entire document), and relative (that only hold in a part of the document). For absolute constraints, we extend earlier decidability results to the case of multi-attribute keys and unary foreign keys, and to the case of constraints involving regular expressions, providing lower and upper bounds in both cases. For relative constraints, we show that even for unary constraints, the consistency problem is undecidable. We also establish a number of restricted decidable cases.

70 citations


Journal ArticleDOI
TL;DR: If the languages defined by M are effectively semilinear, then so are the languagesdefined by Mc, and, hence, their emptiness problem is decidable, and this work proves a surprising undecidability result for commutation of languages.

69 citations


Book ChapterDOI
20 Aug 2002
TL;DR: This work considers linear time specifications and provides a minimal set of restrictions under which this problem of synthesizing controllers in a natural distributed asynchronous setting is effectively solvable and shows that the controller-synthesis problem is decidable while the problem becomes undecidable if any one or more of these three restrictions are dropped.
Abstract: We study the problem of synthesizing controllers in a natural distributed asynchronous setting: a finite set of plants interact with their local environments and communicate with each other by synchronizing on common actions. The controller-synthesis problem is to come up with a local strategy for each plant such that the controlled behaviour of the network meets a specification. We consider linear time specifications and provide, in some sense, a minimal set of restrictions under which this problem is effectively solvable: we show that the controller-synthesis problem under these restrictions is decidable while the problem becomes undecidable if any one or more of these three restrictions are dropped.

57 citations


Proceedings ArticleDOI
22 Jul 2002
TL;DR: An exponential-time algorithm to check for the existence of a winning strategy for TCTL games where equality is not allowed in the timing constraints, thus reducing the considered decision problem to the emptiness problem for this class of automata.
Abstract: The rapid development of complex and safety-critical systems requires the use of reliable verification methods and tools for system design (synthesis). Many systems of interest are reactive, in the sense that their behavior depends on the interaction with the environment. A natural framework to model them is a two-player game: the system versus the environment. In this context, the central problem is to determine the existence of a winning strategy according to a given winning condition. We focus on real-time systems, and choose to model the related game as a nondeterministic timed automaton. We express winning conditions by formulas of the branching-time temporal logic TCTL. While timed games have been studied in the literature, timed games with dense-time winning conditions constitute a new research topic. The main result of this paper is an exponential-time algorithm to check for the existence of a winning strategy for TCTL games where equality is not allowed in the timing constraints. Our approach consists on translating to timed tree automata both the game graph and the winning condition, thus reducing the considered decision problem to the emptiness problem for this class of automata. The proposed algorithm matches the known lower bound on timed games. Moreover, if we relax the limitation we have placed on the timing constraints, the problem becomes undecidable.

55 citations


Journal ArticleDOI
TL;DR: An approximation method based on interval arithmetic that uses a generalization of the notion of cylindrical decomposition—as introduced by G. Collins to efficiently give approximate information on problems that are too hard for current exact methods.
Abstract: This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a first-order formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols <, = and function symbols +, ×, are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition—as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantified constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.

47 citations


Book ChapterDOI
17 Sep 2002
TL;DR: This work presents a generic algorithm for constant propagation via a symbolic weakest precondition computation and shows how this generic algorithm can be instantiated for polynomial constant propagation by exploiting techniques from computable ring theory.
Abstract: Constant propagation aims at identifying expressions that always yield a unique constant value at run-time. It is well-known that constant propagation is undecidable for programs working on integers even if guards are ignored as in non-deterministic flow graphs. We show that polynomial constants are decidable in non-deterministic flow graphs. In polynomial constant propagation, assignment statements that use the operators +,-,* are interpreted exactly but all assignments that use other operators are conservatively interpreted as non-deterministic assignments.We present a generic algorithm for constant propagation via a symbolic weakest precondition computation and show how this generic algorithm can be instantiated for polynomial constant propagation by exploiting techniques from computable ring theory.

46 citations


Journal Article
TL;DR: It is proved that name generation with unique receiver and bounded input (a condition weaker than bounded control) is decidable by reduction to the coverability problem for Petri nets with transfer (and back).
Abstract: We study the decidability of the control reachability problem for various fragments of the asynchronous π-calculus. We consider the combination of three main features: name generation, name mobility, and unbounded control. We show that the combination of name generation with either name mobility or unbounded control leads to an undecidable fragment. On the other hand, we prove that name generation with unique receiver and bounded input (a condition weaker than bounded control) is decidable by reduction to the coverability problem for Petri nets with transfer (and back).

Proceedings Article
27 Aug 2002
TL;DR: It is proved that in the general case, the problem is undecidable, but there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations.
Abstract: This paper deals with the problem of deciding whether two Systems of Affine Recurrence Equations are equivalent or not. A solution to this problem would be a step toward algorithm recognition, an important tool in program analysis, optimization and parallelization. We first prove that in the general case, the problem is undecidable. We then show that there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations. This is a non-effective process which has been extensively studied. Many partial solutions are known. We then report on a pilot implementation of the algorithm, describe its limitations, and point to unsolved problems.

Journal ArticleDOI
18 Mar 2002
TL;DR: This work uses a class of labelled transition systems to model both plants and specifications and proves that in the case of simulations, the problem of checking for the existence of a controller is undecidable in a natural concurrent setting.
Abstract: We study the problem of synthesizing controllers for discrete event systems in a branching time framework. We use a class of labelled transition systems to model both plants and specifications. We use first simulations and later bisimulations to capture the role of a controller; the controlled behaviour of the plant should be related via a simulation (bisimulation) to the specification. For both simulations and bisimulations we show that the problem of checking if a pair of finite transition systems - one modelling the plant and the other the specification - admits a controller is decidable in polynomial time. We also show that the size of the controller, if one exists, can be bounded by a polynomial in the sizes of the plant and the specification and can be effectively constructed in polynomial time. Finally, we prove that in the case of simulations, the problem of checking for the existence of a controller is undecidable in a natural concurrent setting.

Proceedings Article
14 Oct 2002
TL;DR: The model-checking algorithm involves a new solution to the nonemptiness problem of nondeterministic pushdown tree automata, where the best known upper bound is improved from a triple-exponential to a single exponential.
Abstract: Traditionally, model checking is applied to finite-state systems and regular specifications. While researchers have successfully extended the applicability of model checking to infinite-state systems, almost all existing work still consider regular specification formalisms. There are, however, many interesting non-regular properties one would like to model check. In this paper we study model checking of pushdown specifications. Our specification formalism is nondeterministic pushdown parity tree automata (PD-NPT). We show that the model-checking problem for regular systems and PD-NPT specifications can be solved in time exponential in the system and the specification. Our model-checking algorithm involves a new solution to the nonemptiness problem of nondeterministic pushdown tree automata, where we improve the best known upper bound from a triple-exponential to a single exponential. We also consider the model-checking problem for context-free systems and PD-NPT specifications and show that it is undecidable.

Book ChapterDOI
20 Aug 2002
TL;DR: It is shown that the reachable question for some two dimensional hybrid systems are undecidable and that for other 2-dim systems this question remains unanswered, showing that it is as hard as the reachability problem for Piecewise Affine Maps, that is a well known open problem.
Abstract: We revisited decidability of the reachability problem for low dimensional hybrid systems. Even though many attempts have been done to draw the boundary between decidable and undecidable hybrid systems there are still many open problems in between. In this paper we show that the reachability question for some two dimensional hybrid systems are undecidable and that for other 2-dim systems this question remains unanswered, showing that it is as hard as the reachability problem for Piecewise Affine Maps, that is a well known open problem.

Book ChapterDOI
27 Aug 2002
TL;DR: In this paper, it was shown that the problem of deciding whether two Systems of Affine Recurrence Equations are equivalent or not is undecidable, but there is a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations.
Abstract: This paper deals with the problem of deciding whether two Systems of Affine Recurrence Equations are equivalent or not. A solution to this problem would be a step toward algorithm recognition, an important tool in program analysis, optimization and parallelization. We first prove that in the general case, the problem is undecidable. We then show that there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations. This is a non-effective process which has been extensively studied. Many partial solutions are known. We then report on a pilot implementation of the algorithm, describe its limitations, and point to unsolved problems.

Journal ArticleDOI
TL;DR: In this article, it was shown that every polynomial-time predicate can be defined by an (unstratified) local rule set, and a new machine-recognizable subclass of the local rule sets is identified.
Abstract: We consider the concept of a local set of inference rules. A local rule set can be automatically transformed into a rule set for which bottom-up evaluation terminates in polynomial time. The local-rule-set transformation gives polynomial-time evaluation strategies for a large variety of rule sets that cannot be given terminating evaluation strategies by any other known automatic technique. This article discusses three new results. First, it is shown that every polynomial-time predicate can be defined by an (unstratified) local rule set. Second, a new machine-recognizable subclass of the local rule sets is identified. Finally, we show that locality, as a property of rule sets, is undecidable in general.

Journal Article
TL;DR: In this paper, it was shown that the reachability question for some two-dimensional hybrid systems is undecidable and that for other 2-dim systems this question remains unanswered, showing that it is as hard as reachability problem for piecewise affine maps, that is a well known open problem.
Abstract: We revisited decidability of the reachability problem for low dimensional hybrid systems. Even though many attempts have been done to draw the boundary between decidable and undecidable hybrid systems there are still many open problems in between. In this paper we show that the reachability question for some two dimensional hybrid systems are undecidable and that for other 2-dim systems this question remains unanswered, showing that it is as hard as the reachability problem for Piecewise Affine Maps, that is a well known open problem.

Journal ArticleDOI
TL;DR: It is proved that simulation preorder (in both directions) and simulation equivalence are interact able between all classes of infinite-state systems (in the hierarchy of process rewrite systems) and finite-state ones.
Abstract: We consider the problem of simulation preorder/equivalence between infinite-state process and finite-state ones. First, we describe a general method how to utilize the decidability of bisimulation problems to solve (certain instances of) the corresponding simulation problems. For certain process classes, the method allows us to design effective reductions of simulation problems to their bisimulation counterparts and some new decidability border for the problem of simulation preorder/equivalence are decidable in EXPTIME between pushdown and finite-state ones. On the other hand, simulation preorder is undecidable between PA and finite-state process in both directions. These results also hold for those PA and finite-state process which are deterministic and normed, and trace equivalence is also shown to be undecidable for PA. Finally, we prove that simulation preorder (in both directions) and simulation equivalence are interact able between all classes of infinite-state systems (in the hierarchy of process rewrite systems) and finite-state ones. This result is obtained by showing that the problem whether a BPA (or BPP) process simulation a finite-state one is PSPACE- hard and the other direction is co-NP-hard; consequently, simulation equivalence between BPA (or BPP) and finite-state processes is also co-NP-hard.

Proceedings ArticleDOI
01 Jan 2002
TL;DR: It is shown that in the case where all constructors are either unary or nullary, the first-order theory is decidable for both structural and non-structural subtyping, and the decidability results are shown by reduction to a decision problem on tree automata.
Abstract: We investigate the first-order of subtyping constraints. We show that the first-order theory of non-structural subtyping is undecidable, and we show that in the case where all constructors are either unary or nullary, the first-order theory is decidable for both structural and non-structural subtyping. The decidability results are shown by reduction to a decision problem on tree automata. This work is a step towards resolving long-standing open problems of the decidability of entailment for non-structural subtyping.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the problem of computing a fluid limit of a queueing system or of a constrained homogeneous random walk in d + is undecidable.
Abstract: We investigate stability of scheduling policies in queueing systems. To this day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we introduce a certaingeneralized priority policy and prove that the stability of this policy is algorithmically undecidable. We also prove that stability of a homogeneous random walk in d + is undecidable. Finally, we show that the problem of computing a fluid limit of a queueing system or of a constrained homogeneous random walk is undecidable. To the best of our knowledge these are the first undecidability results in the area of stability of queueing systems and random walks in d +. We conjecture that stability of common policies like First-In-First-Out and priority policy is also an undecidable problem.

Journal Article
TL;DR: In this paper, the authors consider the problem of verifyability of hybrid systems and show that for most non-trivial subclasses of hybrid automata, reachability is undecidable.
Abstract: Hybrid systems combining discrete and continuous dynamics arise as mathematical models of various artificial and natural systems, and as an approximation to complex continuous systems. A very important problem in the analysis of the behavior of hybrid systems is reachability. It is well-known that for most non-trivial subclasses of hybrid systems this and all interesting verification problems are undecidable. Most of the proved decidability results rely on stringent hypothesis that lead to the existence of a finite and computable partition of the state space into classes of states which are equivalent with respect to reachability. This is the case for classes of rectangular automata [4] and hybrid automata with linear vector fields [9]. Most implemented computational procedures resort to (forward or backward) propagation of constraints, typically (unions of convex) polyhedra or ellipsoids [1, 6, 8]. In general, these techniques provide semi-decision procedures, that is, if the given final set of states is reachable, they will terminate, otherwise they may fail to. Maybe the major drawback of set-propagation, reach-set approximation procedures is that they pay little attention to the geometric properties of the specific (class of) systems under analysis. An interesting and still decidable class of hybrid system are the (2-dimensional) polygonal differential inclusions (or SPDI for short).

Book ChapterDOI
27 Jul 2002
TL;DR: An interesting and still decidable class of hybrid system are the (2-dimensional) polygonal differential inclusions (or SPDI for short), which pay little attention to the geometric properties of the specific systems under analysis.
Abstract: Hybrid systems combining discrete and continuous dynamics arise as mathematical models of various artificial and natural systems, and as an approximation to complex continuous systems. A very important problem in the analysis of the behavior of hybrid systems is reachability. It is well-known that for most non-trivial subclasses of hybrid systems this and all interesting verification problems are undecidable. Most of the proved decidability results rely on stringent hypothesis that lead to the existence of a finite and computable partition of the state space into classes of states which are equivalent with respect to reachability. This is the case for classes of rectangular automata [4] and hybrid automata with linear vector fields [9]. Most implemented computational procedures resort to (forward or backward) propagation of constraints, typically (unions of convex) polyhedra or ellipsoids [1, 6, 8]. In general, these techniques provide semi-decision procedures, that is, if the given final set of states is reachable, they will terminate, otherwise they may fail to. Maybe the major drawback of set-propagation, reach-set approximation procedures is that they pay little attention to the geometric properties of the specific (class of) systems under analysis. An interesting and still decidable class of hybrid system are the (2-dimensional) polygonal differential inclusions (or SPDI for short).

Book ChapterDOI
08 Apr 2002
TL;DR: A logic is defined, in some sense a maximal fragment of temporal logic with a decidable model-checking problem for the class of ground tree rewriting graphs, which are generated by ground tree (or term) rewriting systems.
Abstract: We consider infinite graphs that are generated by ground tree (or term) rewriting systems. the vertices of these graphs are trees. thus, with a finite tree automaton one can represent a regular set of vertices. It is shown that for a regular set T of vertices the set of vertices from where one can reach (respectively, infinitely often reach) the set T is again regular. Furthermore it is shown that the problems, given a tree t and a regular set T, whether all paths starting in t eventually (respectively, infinitely often) reach T, are undecidable. We then define a logic which is in some sense a maximal fragment of temporal logic with a decidable model-checking problem for the class of ground tree rewriting graphs.

01 Jan 2002
TL;DR: It is shown that in the context of modal logic, inflationary fixed points are far more expressive than least fixed points, and already relatively simple logics such as the transitive closure logic lead to undecidable query languages on constraint databases.
Abstract: Fixed-point logics are logics with an explicit operator for forming fixed points of definable mappings. They are particularly well suited for modelling recursion in logical languages and consequently they have found applications in various areas of theoretical computer science such as database theory, finite model theory, and computer-aided verification. The topic of this thesis is the study of fixed-point logics with respect to their expressive power. Of particular interest are logics based on inflationary fixed points and their comparison to least fixed-point logics. The first part focuses on fixed-point extensions of first-order logic. In the main result we show that inflationary and least fixed-point logic – the extensions of first-order logic by least and inflationary fixed points – have the same expressive power on all structures, i.e. LFP = IFP. In the second part of this thesis, we study fixed-point extensions of modal logic. Such logics are widely used in the field of computer-aided verification. Again, the least fixed-point extension of modal logic, the modal μ-calculus, is of particular interest and is among the best studied logics in this area. The main contribution of the second part is the introduction and study of the corresponding inflationary fixed-point logic. Contrary to the case of first-order logic mentioned above, where least and inflationary fixed points lead to equivalent logics, it is shown that in the context of modal logic, inflationary fixed points are far more expressive than least fixed points. On the other hand, they are algorithmically far more complex. Besides the two main results, we study a variety of different fixed-point logics and develop methods to compare their expressive power. Finally, in the third part, we study fixed-point logics as query languages for constraint databases. It is shown that already relatively simple logics such as the transitive closure logic lead to undecidable query languages on constraint databases. Therefore we consider suitable restrictions of fixedpoint logics to obtain tractable query languages, i.e. languages with polynomial time evaluation. A detailed overview of the results presented in this thesis can be found in the second part of the introduction.

Journal ArticleDOI
TL;DR: The reconstruction of the reals without the axiom of choice is the principle focus of this paper and the calculus of set-valued functions including the generalized integral and its application to quantum gravity is introduced.

Journal ArticleDOI
TL;DR: In this article, it was shown that the subtyping problem induced by Mitchell's containment relation for second-order polymorphic types is undecidable for the polymorphic lambda calculus extended by an appropriate subsumption rule.
Abstract: We prove that the subtyping problem induced by Mitchell's containment relation for second-order polymorphic types is undecidable. It follows that type-checking is undecidable for the polymorphic lambda-calculus extended by an appropriate subsumption rule.

Journal Article
TL;DR: In this paper, the authors present precise theorems that should help the reader understand to which extent statements like introducing dependent types in a programming language implies that type checking is undecidable, are justified.
Abstract: Functional programming languages often feature mechanisms that involve complex computations at the level of types. These mechanisms can be analyzed uniformly in the framework of dependent types, in which types may depend on values. The purpose of this chapter is to give some background for such an analysis. We present here precise theorems, that should hopefully help the reader to understand to which extent statements like introducing dependent types in a programming language implies that type checking is undecidable, are justified.

Journal ArticleDOI
TL;DR: For a hierarchy of properties of term rewriting systems related to termination the authors prove relative undecidability: for implications X → Y in the hierarchy the property X is undecidable for term rewrite systems satisfying Y.
Abstract: For a hierarchy of properties of term rewriting systems related to termination we prove relative undecidability: For implications X → Y in the hierarchy the property X is undecidable for term rewriting systems satisfying Y. For most implications we obtain this result for term rewriting systems consisting of a single rewrite rule.

Journal ArticleDOI
TL;DR: Grid rules cover all rules which terminate by a total division order and total division orders are shown to be irrelevant for the termination problem of one-rule string rewriting.
Abstract: Termination of string rewriting is known undecidable. Termination of string rewriting with only one rule is neither known decidable nor known undecidable. This paper presents a decision procedure for rules $u\rightarrow v$ such that some letter b from u occurs as often or less often in v. We call such rules "grid" rules. By far most rules are grid rules. Grid rules cover all rules which terminate by a total division order. Thus total division orders are shown to be irrelevant for the termination problem of one-rule string rewriting.