Showing papers on "Undecidable problem published in 2003"

On the decidability and complexity of query answering over inconsistent and incomplete databases

Abstract: In databases with integrity constraints, data may not satisfy the constraints. In this paper, we address the problem of obtaining consistent answers in such a setting, when key and inclusion dependencies are expressed on the database schema. We establish decidability and complexity results for query answering under different assumptions on data (soundness and/or completeness). In particular, after showing that the problem is in general undecidable, we identify the maximal class of inclusion dependencies under which query answering is decidable in the presence of key dependencies. Although obtained in a single database context, such results are directly applicable to data integration, where multiple information sources may provide data that are inconsistent with respect to the global view of the sources.

A generic approach to the static analysis of concurrent programs with procedures

Abstract: We present a generic aproach to the static analysis of concurrent programs with procedures. We model programs as communicating pushdown systems. It is known that typical dataflow problems for this model are undecidable, because the emptiness problem for the intersection of context-free languages, which is undecidable, can be reduced to them. In this paper we propose an algebraic framework for defining abstractions (upper approximations) of context-free languages. We consider two classes of abstractions: finite-chain abstractions, which are abstractions whose domains do not contain any infinite chains, and commutative abstractions corresponding to classes of languages that contain a word if and only if they contain all its permutations. We show how to compute such approximations by combining automata theoretic techniques with algorithms for solving systems of polynomial inequations in Kleene algebras.

Undecidable problems for probabilistic automata of fixed dimension

Abstract: We prove that several problems associated with probabilistic finite automata are undecidable for automata whose number of input letters and number of states are fixed. As a corollary of one of our results we prove that the problem of determining if the set of all products of two 47 x 47 matrices with nonnegative rational entries is bounded is undecidable.

Numerical document queries

Abstract: A query against a database behind a site like Napster may search, e.g., for all users who have downloaded more jazz titles than pop music titles. In order to express such queries, we extend classical monadic second-order logic by Presburger predicates which pose numerical restrictions on the children (content) of an element node and provide a precise automata-theoretic characterization. While the existential fragment of the resulting logic is decidable, it turns out that satisfiability of the full logic is undecidable. Decidable satisfiability and a querying algorithm even with linear data complexity can be obtained if numerical constraints are only applied to those contents of elements where ordering is irrelevant. Finally, it is sketched how these techniques can be extended also to answer questions like, e.g., whether the total price of the jazz music downloaded so far exceeds a user's budget.

Quantum algorithm for Hilbert's tenth problem

Abstract: We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain Hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the Church—Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.

A Coverage Checking Algorithm for LF

Abstract: Coverage checking is the problem of deciding whether any closed term of a given type is an instance of at least one of a given set of patterns. It can be used to verify if a function defined by pattern matching covers all possible cases. This problem has a straightforward solution for the first-order, simply-typed case, but is in general undecidable in the presence of dependent types. In this paper we present a terminating algorithm for verifying coverage of higher-order, dependently typed patterns. It either succeeds or presents a set of counterexamples with free variables, some of which may not have closed instances (a question which is undecidable). Our algorithm, together with strictness and termination checking, can be used to certify the correctness of numerous proofs of properties of deductive systems encoded in a system for reasoning about LF signatures.

Query containment and rewriting using views for regular path queries under constraints

Abstract: In this paper we consider general path constraints for semistructured databases. Our general constraints do not suffer from the limitations of the path constraints previously studied in the literature. We investigate the containment of regular path queries under general path constraints. We show that when the path constraints and queries are expressed by words, as opposed to languages, the containment problem becomes equivalent to the word rewrite problem for a corresponding semi-Thue system. Consequently, if the corresponding semi-Thue system has an undecidable word problem, the word query containment problem will be undecidable too. Also, we show that there are word constraints, where the corresponding semi-Thue system has a decidable word rewrite problem, but the general query containment under these word constraints is undecidable. In order to overcome this, we exhibit a large, practical class of word constraints with a decidable general query containment problem.Based on the query containment under constraints, we reason about constrained rewritings -using views- of regular path queries. We give a constructive characterization for computing optimal constrained rewritings using views.

On undecidability of equicontinuity classification for cellular automata

Abstract: Equicontinuity classification is a popular classification of cellular automata based on their dynamical behavior. In this paper we prove that most of its classes are undecidable.

A Generic Approach to the Static Analysis of Concurrent Programs with Procedures

Abstract: We present a generic aproach to the static analysis of concurrent programs with procedures. We model programs as communicating pushdown systems. It is known that typical dataflow problems for this model are undecidable, because the emptiness problem for the intersection of context-free languages, which is undecidable, can be reduced to them. In this paper we propose an algebraic framework for defining abstractions (upper approximations) of context-free languages. We consider two classes of abstractions: finite-chain abstractions, which are abstractions whose domains do not contain any infinite chains, and commutative abstractions corresponding to classes of languages that contain a word if and only if they contain all its permutations. We show how to compute such approximations by combining automata theoretic techniques with algorithms for solving systems of polynomial inequations in Kleene algebras.

A coverage checking algorithm for IF

Abstract: Coverage checking is the problem of deciding whether any closed term of a given type is an instance of at least one of a given set of patterns It can be used to verify if a function defined by pattern matching covers all possible cases This problem has a straightforward solution for the first-order, simply-typed case, but is in general undecidable in the presence of dependent types In this paper we present a terminating algorithm for verifying coverage of higher-order, dependently typed patterns It either succeeds or presents a set of counterexamples with free variables, some of which may not have closed instances (a question which is undecidable) Our algorithm, together with strictness and termination checking, can be used to certify the correctness of numerous proofs of properties of deductive systems encoded in a system for reasoning about LF signatures

On the extension problem for partial permutations

Abstract: A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl*V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators *, **, ?, =, = n , and P do not preserve decidability. In addition, several joins, including A V U, are shown to be undecidable.

New results on the computability and complexity of points--to analysis

Abstract: Given a program and two variables p and q, the goal of points-to analysis is to check if p can point to q in some execution of the program. This well-studied problem plays a crucial role in compiler optimization. The problem is known to be undecidable when dynamic memory is allowed. But the result is known only when variables are allowed to be structures. We extend the result to show that, the problem remains undecidable, even when only scalar variables are allowed. Our second result deals with a version of points-to analysis called flow-insensitive analysis, where one ignores the control flow of the program and assumes that the statements can be executed in any order. The problem is known to be NP-Hard, even when dynamic memory is not allowed and variables are scalar. We show that when the variables are further restricted to have well-defined data types, the problem is in P. The corresponding flow-sensitive version, even with further restrictions, is known to be PSPACE-Complete. Thus, our result gives some theoretical evidence that flow-insensitive analysis is easier than flow-sensitive analysis. Moreover, while most variations of the points-to analysis are known to be computationally hard, our result gives a rare instance of a non-trivial points-to problem solvable in polynomial time.

Computability over Topological Structures

Abstract: Computable analysis is the Turing machine based theory of computability on the real numbers and other topological spaces. Similarly as Ersov’s concept of numberings can be used to deal with discrete structures, Kreitz and Weihrauch’s concept of representations can be used to handle structures of continuum cardinality. In this context the choice of representations is very sensitively related to the underlying notion of approximation, hence to topology. In this paper we summarize some basic ideas of the representation based approach to computable analysis and we introduce an abstract and purely set theoretic characterization of this theory which can be considered as a generalization of the classical concept of µ-recursive functions. Together with this characterization we introduce the notion of a perfect topological structure. In particular, these structures are effectively categorical, i.e. they characterize their own computability theory. Important examples of perfect structures are provided by metric spaces and additional attention is paid to their effective subsets.

Real-time model-checking: Parameters everywhere

Abstract: In this paper, we study the model-checking and parameter synthesis problems of the logic TCTL over discrete-timed automata where parameters are allowed both in the model and in the property. We show that the model-checking problem of TCTL extended with parameters is undecidable over discrete-timed automata with only one parametric clock. The undecidability result needs equality in the logic. When equality is not allowed, we show that the model-checking and the parameter synthesis problems become decidable.

Universality and language inclusion for open and closed timed automata

Abstract: The algorithmic analysis of timed automata is fundamentally limited by the undecidability of the universality problem. For this reason and others, there has been considerable interest in restricted classes of timed automata. In this paper we study the universality problem for two prominent such subclasses: open and closed timed automata. This problem is described as open in [6,8] in the case of open timed automata. We show here that the problem is undecidable for open timed automata over strongly monotonic time (no two events are allowed to occur at the same time), and decidable over weakly monotonic time. For closed timed automata, we show that the problem is undecidable regardless of the monotonicity assumptions on time. As a corollary, we settle the various language inclusion problems over these classes of timed automata.

Typechecking XML views of relational databases

Abstract: Motivated by the need to export relational databases as XML data in the context of the Web, we investigate the typechecking problem for transformations of relational data into tree data (XML). The problem consists of statically verifying that the output of every transformation belongs to a given output tree language (specified for XML by a DTD), for input databases satisfying given integrity constraints. The typechecking problem is parameterized by the class of formulas defining the transformation, the class of output tree languages, and the class of integrity constraints. While undecidable in its most general formulation, the typechecking problem has many special cases of practical interest that turn out to be decidable. The main contribution of this article is to trace a fairly tight boundary of decidability for typechecking in this framework. In the decidable cases we examine the complexity, and show lower and upper bounds. We also exhibit a practically appealing restriction for which typechecking is in PTIME.

On program equivalence in languages with ground-type references

Abstract: Using game semantics we prove that program equivalence is undecidable in finitary Idealized Algol with active expressions as well as in its call-by-value counterpart. It is also shown that strategies corresponding to Idealized Algol terms of respectively second, third and higher orders define exactly regular, context-free and recursively enumerable languages.

The Undecidability of Grisin's Set Theory

Abstract: We investigate a contractionless naive set theory, due to Grisin [11] We prove that the theory is undecidable

An E-unification algorithm for analyzing protocols that use modular exponentiation

Abstract: Modular multiplication and exponentiation are common operations in modern cryptography. Unification problems with respect to some equational theories that these operations satisfy are investigated. Two different but related equational theories are analyzed. A unification algorithm is given for one of the theories which relies on solving syzygies over multivariate integral polynomials with noncommuting indeterminates. For the other theory, in which the distributivity property of exponentiation over multiplication is assumed, the unifiability problem is shown to be undecidable by adapting a construction developed by one of the authors to reduce Hilbert's 10th problem to the solvability problem for linear equations over semi-rings. A new algorithm for computing strong Grobner bases of right ideals over the polynomial ring Z is proposed; unlike earlier algorithms proposed by Baader as well as by Madlener and Reinert which work only for right admissible term orderings with the boundedness property, this algorithm works for any right admissible term ordering. The algorithms for some of these unification problems are expected to be integrated into Naval Research Lab.'s Protocol Analyzer (NPA), a tool developed by Catherine Meadows, which has been successfully used to analyze cryptographic protocols, particularly emerging standards such as the Internet Engineering Task Force's (IETF) Internet Key Exchange [11] and Group Domain of Interpretation [12] protocols. Techniques from several different fields - particularly symbolic computation (ideal theory and Groebner basis algorithms) and unification theory - are thus used to address problems arising in state-based cryptographic protocol analysis.

Automatic Verification of Recursive Procedures with One Integer Parameter

Abstract: Context-free processes (BPA) have been used for dataflow analysis in recursive procedures with applications in optimizing compilers (Proceedings of FOSSaCS'99, Lecture Notes in Computer Science, Vol. 1578, Springer, Berlin, 1999, pp. 14-30). We introduce a more refined model called BPA(Z) that can model not only recursive dependencies, but also the passing of an integer parameter to a subroutine. Moreover, this parameter can be tested against conditions expressible in Presburger arithmetic. This new and more expressive model can still be analyzed automatically. We define Z-input 1-CM, a new class of 1-counter machines (cm) that take integer numbers as input, to describe sets of configurations of BPA(Z). We show that the Post* (the set of successors) of a set of BPA(Z)-configurations described by a Z-input 1-CM can be effectively constructed. The Pre* (set of predecessors) of a regular set can be effectively constructed as well. However, the Pre* of a set described by a Z-input 1-CM cannot be represented by a Z-input 1-CM, in general, and has an undecidable membership problem. Then we develop a new temporal logic based on reversal-bounded counter machines (i.e. machines which use counters such that the change between increasing and decreasing mode of each counter is bounded (J. Assoc. Comput. Mach. 25 (1978) 116) that can be used to describe properties of BPA(Z) and show that the model-checking problem is decidable.

Undecidability of domino games and hhp-bisimilarity

Abstract: History preserving bisimilarity (hp-bisimilarity) and hereditary history preserving bisimilarity (hhp-bisimilarity) are behavioural equivalences taking into account causal relationships between events of concurrent systems. Their prominent feature is that they are preserved under action refinement, an operation important for the top-down design of concurrent systems. It is shown that, in contrast to hp-bisimilarity, checking hhp-bisimilarity for finite labelled asynchronous transition systems is undecidable, by a reduction from the halting problem of 2-counter machines. To make the proof more transparent a novel intermediate problem of checking domino bisimilarity for origin constrained tiling systems is introduced and shown undecidable. It is also shown that the unlabelled domino bisimilarity problem is undecidable, which implies undecidability of hhp-bisimilarity for unlabelled finite asynchronous systems. Moreover, it is argued that the undecidability of hhp-bisimilarity holds for finite elementary net systems and 1-safe Petri nets.

Two-way equational tree automata for AC-like theories: decidability and closure properties

Abstract: We study two-way tree automata modulo equational theories. We deal with the theories of Abelian groups (ACUM), idempotent commutative monoids (ACUI), and the theory of exclusive-or (ACUX), as well as some variants including the theory of commutative monoids (ACU). We show that the one-way automata for all these theories are closed under union and intersection, and emptiness is decidable. For two-way automata the situation is more complex. In all these theories except ACUI, we show that two-way automata can be effectively reduced to one-way automata, provided some care is taken in the definition of the so-called push clauses. (The ACUI case is open.) In particular, the two-way automata modulo these theories are closed under union and intersection, and emptiness is decidable. We also note that alternating variants have undecidable emptiness problem for most theories, contrarily to the non-equational case where alternation is essentially harmless.

Transcending the Limits of Turing Computability

Abstract: Hypercomputation or super-Turing computation is a ``computation'' that transcends the limit imposed by Turing's model of computability. The field still faces some basic questions, technical (can we mathematically and/or physically build a hypercomputer?), cognitive (can hypercomputers realize the AI dream?), philosophical (is thinking more than computing?). The aim of this paper is to address the question: can we mathematically build a hypercomputer? We will discuss the solutions of the Infinite Merchant Problem, a decision problem equivalent to the Halting Problem, based on results obtained in \cite{Coins,acp}. The accent will be on the new computational technique and results rather than formal proofs.

Undecidability of Weak Bisimilarity for PA-Processes

Abstract: We prove that the problem whether two PA-processes are weakly bisimilar is undecidable. We combine several proof techniques to provide a reduction from Post's correspondence problem to our problem: existential quantification technique, masking technique and deadlock elimination technique.