Showing papers on "Undecidable problem published in 2001"

On XML integrity constraints in the presence of DTDs

Abstract: The paper 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.

Synthesizing distributed systems

Abstract: In system synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. When the system is distributed, the goal is to construct the system's underlying processes. Results on multi-player games imply that the synthesis problem for linear specifications is undecidable for general architectures, and is nonelementary decidable for hierarchical architectures, where the processes are linearly ordered and information among them flows in one direction. In this paper, we present a significant extension of this result. We handle both linear and branching specifications, and we show that a sufficient condition for decidability of the synthesis problem is a linear or cyclic order among the processes, in which information flows in either one or both directions. We also allow the processes to have internal hidden variables, and we consider communications with and without delay. Many practical applications fall into this class.

Abstract Interpretation Based Formal Methods and Future Challenges

Abstract: In order to contribute to the solution of the software reliability problem, tools have been designed to analyze statically the run-time behavior of programs. Because the correctness problem is undecidable, some form of approximation is needed. The purpose of abstract interpretation is to formalize this idea of approximation. We illustrate informally the application of abstraction to the semantics of programming languages as well as to static program analysis. The main point is that in order to reason or compute about a complex system, some information must be lost, that is the observation of executions must be either partial or at a high level of abstraction. A few challenges for static program analysis by abstract interpretation are finally briefly discussed. The electronic version of this paper includes a comparison with other formal methods: typing, model-checking and deductive methods.

A uniform method for proving lower bounds on the computational complexity of logical theories

Abstract: A new method for obtaining lower bounds on the computational complexity of logical theories is presented. It extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable. New inseparability results related to the well known inseparability result of Trakhtenbrot and Vaught are the foundation of the method. Their use yields hereditary lower bounds (i.e., bounds which apply uniformly to all subtheories of a theory). By means of interpretations lower bounds can be transferred from one theory to another. Complicated machine codings are replaced by much simpler definability considerations, viz., the kinds of binary relations definable with short formulas on large finite sets. Numerous examples are given, including new proofs of essentially all previously known lower bounds for theories, and lower bounds for various theories of finite trees, which turn out to be particularly useful.

XML with data values: typechecking revisited

Abstract: We investigate the type checking problem for XML queries: statically verifying that every answer to a query conforms to a given output DTD, for inputs satisfying a given input DTD. This problem had been studied by a subset of the authors in a simplified framework that captured the structure of XML documents but ignored data values. We revisit here the type checking problem in the more realistic case when data values are present in documents and tested by queries. In this extended framework, type checking quickly becomes undecidable. However, it remains decidable for large classes of queries and DTDs of practical interest. The main contribution of the present paper is to trace a fairly tight boundary of decidability for type checking with data values. The complexity of type checking in the decidable cases is also considered.

On the products of linear modal logics

Abstract: We study two‐dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4.3, S4.3, GL.3, Grz.3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems posed by Gabbay and Shehtman. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatization for the square K4.3 × K4.3 of the minimal liner logic using non‐structural Gabbay‐type inference rules.

The Decidability of Model Checking Mobile Ambients

Abstract: The ambient calculus is a formalism for describing the mobility of both software and hardware The ambient logic is a modal logic designed to specify properties of distributed and mobile computations programmed in the ambient calculus In this paper we investigate the border between decidable and undecidable cases of model checking mobile ambients for some fragments of the ambient calculus and the ambient logic Recently, Cardelli and Gordon presented a model-checking algorithm for a fragment of the calculus (without name restriction and without replication) against a fragment of the logic (without composition adjunct) and asked the question, whether this algorithm could be extended to include either replication in the calculus or composition adjunct in the logic Here we answer this question negatively: it is not possible to extend the algorithm, because each of these extensions leads to undecidability of the problem On the other hand, we extend the algorithm to the calculus with name restriction and logic with new constructs for reasoning about restricted names

Undecidable problems of decentralized observation and control

Abstract: We introduce a notion of decentralized observability for discrete-event systems, which we call joint observability. We prove that checking joint observability of a regular language w.r.t. one observer is decidable, whereas for two (or more) observers the problem becomes undecidable. Based on this result, we show that a related decentralized control problem is also undecidable. We finally provide an extensive study relating our work to existing work in the literature.

Definition and application of a five-parameter characterization of one-dimensional cellular automata rule space

Abstract: Cellular automata (CA) are important as prototypical, spatially extended, discrete dynamical systems. Because the problem of forecasting dynamic behavior of CA is undecidable, various parameter-based approximations have been developed to address the problem. Out of the analysis of the most important parameters available to this end we proposed some guidelines that should be followed when defining a parameter of that kind. Based upon the guidelines, new parameters were proposed and a set of five parameters was selected; two of them were drawn from the literature and three are new ones, defined here. This article presents all of them and makes their qualities evident. Then, two results are described, related to the use of the parameter set in the Elementary Rule Space: a phase transition diagram, and some general heuristics for forecasting the dynamics of one-dimensional CA. Finally, as an example of the application of the selected parameters in high cardinality spaces, results are presented from experiments involving the evolution of radius-3 CA in the Density Classification Task, and radius-2 CA in the Synchronization Task.

What Will Be Eventually True of Polynomial Hybrid Automata

Abstract: Hybrid automata have been introduced in both control engineering and computer science as a formal model for the dynamics of hybrid discrete-continuous systems. While computability issues concerning safety properties have been extensively studied, liveness properties have remained largely uninvestigated. In this article, we investigate decidability of state recurrence and of progress properties.First, we show that state recurrence and progress are in general undecidable for polynomial hybrid automata. Then, we demonstrate that they are closely related for hybrid automata subject to a simple model of noise, even though these automata are infinite-state systems. Based on this, we augment a semi-decision procedure for recurrence with a semidecision method for length-boundedness of paths in such a way that we obtain an automatic verification method for progress properties of linear and polynomial hybrid automata that may only fail on pathological, practically uninteresting cases. These cases are such that satisfaction of the desired progress property crucially depends on the complete absence of noise, a situation unlikely to occur in real hybrid systems.

Inadequacy of computable loop invariants

Abstract: Hoare logic is a widely recommended verification tool. There is, however, a problem of finding easily checkable loop invariants; it is known that decidable assertions do not suffice to verify while programs, even when the pre- and postconditions are decidable. We show here a stronger result: decidable invariants do not suffice to verify single-loop programs. We also show that this problem arises even in extremely simple contexts. Let N be the structure consisting of the set of natural numbers together with the functions S(x)=x+1,D(x)=2(x)=***x/2***. There is a single-loop program *** using only three variables x,y,z such that the asserted program x=y=z=0 *** false is partially correct on N but any loop invariant I(x,y,z) for this asserted program is undecidable.

Verification of hybrid systems: formalization and proof rules in PVS

Abstract: Combining discrete state-machines with continuous behavior, hybrid systems are a well-established mathematical model for discrete systems acting in a continuous environment. As a priori infinite state systems, their computational properties are undecidable in the general model and the main line of research concentrates on model checking of finite abstractions of restricted subclasses of the general model. In our work, we use deductive methods, falling back upon the general-purpose theorem prover PVS. To do so we extend the classical approach for the verification of state-based programs by developing an inductive proof method to deal with the parallel composition of hybrid systems. It covers shared variable communication, label-synchronization, and especially the common continuous activities in the parallel composition of hybrid automata. Besides hybrid systems and their parallel composition, we formalized their operational step semantics and a number of proof-rules within PVS, for one of which we give also a rigorous completeness proof. Moreover the theory is applied to the verification of a number of examples.

On the Complexity of Constant Propagation

Abstract: Constant propagation (CP) is one of the most widely used optimizations in practice (cf. [9]). Intuitively, it addresses the problem of statically detecting whether an expression always evaluates to a unique constant at run-time. Unfortunately, as proved by different authors [4, 16], CP is in general undecidable even if the interpretation of branches is completely ignored. On the other hand, it is certainly decidable in more restricted settings, like on loop-free programs (cf. [7]). In this paper, we explore the complexity of CP for a three-dimensional taxonomy. We present an almost complete complexity classification, leaving only two upper bounds open.

The first-order theory of ordering constraints over feature trees

Abstract: The system FT< of ordering constraints over feature trees has been introduced as an extension of the system FT of equality constraints over feature trees. We investigate the first-order theory of FT< and its fragments in detail, both over finite trees and over possibly infinite trees. We prove that the first-order theory of FT< is undecidable, in contrast to the first-order theory of FT which is well-known to be decidable. We show that the entailment problem of FT< with existential quantification is PSPACE-complete. So far, this problem has been shown decidable, coNP-hard in case of finite trees, PSPACE-hard in case of arbitrary trees, and cubic time when restricted to quantifier-free entailment judgments. To show PSPACE-completeness, we show that the entailment problem of FT< with existential quantification is equivalent to the inclusion problem of non-deterministic finite automata. Available at http://www.ps.uni-saarland.de/Publications/documents/FTSubTheory_98.pdf

Malnormality is undecidable in hyperbolic groups

Abstract: In answer to a question of Myasnikov, we show that there exist hyperbolic groups for which there is no algorithm to decide which finitely generated subgroups are malnormal or quasiconvex.

Liveness Verification of Reversal-Bounded Multicounter Machines with a Free Counter

Abstract: We investigate the Presburger liveness problems for nondeterministic reversal-bounded multicounter machines with a free counter (NCMFs). We show the following: - The ∃-Presburger-i.o. problem and the ∃-Presburger-eventual problem are both decidable. So are their duals, the ∀-Presburger-almost-always problem and the ∀-Presburger-always problem. - The ∀-Presburger-i.o. problem and the ∀-Presburger-eventual problem are both undecidable. So are their duals, the ∃-Presburger-almost-always problem and the ∃-Presburger-always problem. These results can be used to formulate a weak form of Presburger linear temporal logic and developits model-checking theories for NCMFs. They can also be combined with [12] to study the same set of liveness problems on an extended form of discrete timed automata containing, besides clocks, a number of reversal-bounded counters and a free counter.

Algorithms and Reductions for Rewriting Problems

Abstract: In this paper we initiate a study of polynomial-time reductions for some basic decision problems of rewrite systems. We then give a polynomial-time algorithm for the unique-normal-form property of ground systems for the first time. Next we prove undecidability of several problems for a fixed string rewriting system using our reductions. Finally, we prove the decidability of confluence for commutative semi-thue systems. The Confluence and Unique-normal-form property are shown Expspace-hard for commutative semi-thue systems. We also show that there is a family of string rewrite systems for which the word problem is trivially decidable but confluence is undecidable, and we show a linear equational theory with decidable word problem but undecidable linear equational matching problem.

A theory of mathematical correctness and mathematical truth

Abstract: A theory of objective mathematical correctness is developed. The theory is consistent with both mathematical realism and mathematical anti-realism, and versions of realism and anti-realism are developed that dovetail with the theory of correctness. It is argued that these are the best versions of realism and anti-realism and that the theory of correctness behind them is true. Along the way, it is shown that, contrary to the traditional wisdom, the question of whether undecidable sentences like the continuum hypothesis have objectively determinate truth values is independent of the question of whether mathematical realism is true.

From Finite State Communication Protocols to High-Level Message Sequence Charts

Abstract: The ITU standard for MSCs provides a useful framework for visualizing communication protocols. HMSCs can describe a collection of MSC scenarios in early stages of system design. They extend finite state systems by partial order semantics and asynchronous, unbounded message exchange. Usually we ask whether an HMSC can be implemented, for instance by a finite state protocol. This question has been shown to be undecidable [5]. Motivated by the paradigm of reverse engineering we study in this paper the converse translation, specifically the question whether a finite state communication protocol can be transformed into an equivalent HMSC. This kind of translation is needed when e.g. different forms of specification (HMSC, finite automata, temporal logic) must be integrated into a single one, for instance into an HMSC. We show in this paper that translating finite state automata into HMSCs is feasible under certain natural assumptions. Specifically, we show that we can test in polynomial time whether a finite state protocol given by a Buchi automaton is equivalent to an HMSC, provided that the automaton satisfies the diamond property (the precise bound is NLOGSPACE-complete). The diamond property is a natural property induced by concurrency. Under the weaker assumption of bounded Buchi automata we show that the test is co-NP-complete. Finally, without any buffer restriction the problem is shown to be undecidable.

Toward general analysis of recursive probability models

Abstract: There is increasing interest within the research community in the design and use of recursive probability models. There remains concern about computational complexity costs and the fact that computing exact solutions can be intractable for many nonrecursive models. Although inference is undecidable in the general case for recursive problems, several research groups are actively developing computational techniques for recursive stochastic languages. We have developed an extension to the traditional λ calculus as a framework for families of Turing complete stochastic languages. We have also developed a class of exact inference algorithms based on the traditional reductions of the λ calculus. We further propose that using the deBruijn notation (a λ-calculus notation with nameless dummies) supports effective caching in such systems, as the reuse of partial solutions is an essential component of efficient computation. Finally, our extension to the λ-calculus offers a foundation and general theory for the construction of recursive stochastic modeling languages as well as promise for effective caching and efficient approximation algorithms for inference.

Decision Questions Concerning Semilinearity, Morphisms, and Commutation of Languages

Abstract: Let e be a class of automata (in a precise sense to be defined) and ec the class obtained by augmenting each automaton in e with finitely many reversal-bounded counters. We first show that if the languages defined by e are effectively semilinear, then so are the languages defined by ec, and, hence, their emptiness problem is decidable. This result is then used to show the decidability of various problems concerning morphisms and commutation of languages. We also prove a surprising undecidability result: given a fixed two element code K, it is undecidable whether a given context-free language L commutes with K, i.e., LK = KL.

Applying an update method to a set of receivers

Abstract: In the context of object databases, we study the application of an update method to a collection of receivers rather than to a single one. The obvious strategy of applying the update to the receivers one after the other, in some arbitrary order, brings up the problem of order independence. On a very general level, we investigate how update behavior can be analyzed in terms of certain schema annotations, called colorings. We are able to characterize those colorings that always describe order-independedent updates. We also consider a more specific model of update methods implemented in the relational algebra. Order-independence of such algebraic methods is undecidable in general, but decidable if the expressions used are positive. Finally, we consider an alternative parallel strategy for set-oriented applications of algebraic update methods and compare and relate it to the sequential strategy.

The Equivalence Problem for Computational Models: Decidable and Undecidable Cases

Abstract: This paper presents a survey of fundamental concepts and main results in studying the equivalence problem for computer programs. We introduce some of the most-used models of computer programs, give a brief overview of the attempts to refine the boarder between decidable and undecidable cases of the equivalence problem for these models, and discuss the techniques for proving the decidability of the equivalence problem.