Showing papers on "Undecidable problem published in 1998"

On the decidability of query containment under constraints

Abstract: Query containment under constraints is the problem of checking whether for every database satisfying a given set of constraints, the result of one query is a subset of the result of another query. Recent research points out that this is a central problem in several database applications, and we address it within a setting where constraints are specified in the form of special inclusion dependencies over complex expressions, built by using intersection and difference of relations, special forms of quantification, regular expressions over binary relations, and cardinality constraints. These types of constraints capture a great variety of data models, including the relational, the entity-relational, and the object-oriented model. We study the problem of checking whether q is contained in q′ with respect to the constraints specified in a schema S, where q and q′ are nonrecursive Datalog programs whose atoms are complex expressions. We present the following results on query containment. For the case where q does not contain regular expressions, we provide a method for deciding query containment, and analyze its computational complexity. We do the same for the case where neither S nor q, q′ contain number restrictions. To the best of our knowledge, this yields the first decidability result on containment of conjunctive queries with regular expressions. Finally, we prove that the problem is undecidable for the case where we admit inequalities in q′.

Symbolic Methods for Exploring Infinite State Spaces

Abstract: In this thesis, we introduce a general method for computing the set of reachable states of an infinite-state system. The basic idea, inspired by well-known statespace exploration methods for finite-state systems, is to propagate reachability from the initial state of the system in order to determine exactly which are the reachable states. Of course, the problem being in general undecidable, our goal is not to obtain an algorithm which is guaranteed to produce results, but one that often produces results on practically relevant cases. Our approach is based on the concept ofmeta-transition, which is a mathematical object that can be associated to the model, and whose purpose is to make it possible to compute in a finite amount of time an infinite set of reachable states. Different methods for creating meta-transitions are studied. We also study the properties that can be verified by state-space exploration, in particular linear-time temporal properties. The state-space exploration technique that we introduce relies on a symbolic representation system for the sets of data values manipulated during exploration. This representation system has to satisfy a number of conditions. We give a generic way of obtaining a suitable representation system, which consists of encoding each data value as a string of symbols over some finite alphabet, and to represent a set of values by a finite-state automaton accepting the language of the encodings of the values in the set. Finally, we particularize the general representation technique to two important domains: unbounded FIFO buffers, and unbounded integer variables. For each of those domains, we give detailed algorithms for performing the required operations on represented sets of values.

Complexity of Two-Dimensional Patterns

Abstract: In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of “regular language” or “local rule” that are equivalent in d=1 lead to distinct classes in d≥2. We explore the closure properties and computational complexity of these classes, including undecidability and L, NL, and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d≥2 has a periodic point of a given period, and that certain “local lattice languages” are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t −d unless it maps every initial condition to a single homogeneous state.

Termination of graph rewriting is undecidable

Abstract: It is shown that it is undecidable in general whether a graph rewriting system (in the “double pushout approach”) is terminating. The proof is by a reduction of the Post Correspondence Problem. It is also argued that there is no straightforward reduction of the halting problem for Turing machines or of the termination problem for string rewriting systems to the present problem.

Decidability and undecidability results for the termination problem of active database rules

Abstract: Active database systems enhance the functionality of traditional databases through the use of active rules or `triggers'. One of the principal questions for such systems is that of termination - is it possible for the rules to recursively activate one another indefinitely, given an initial triggering event. In this paper, we study the decidability of the termination problem, our aim being to delimit the boundary between the decidable and the undecidable. We present two families of rule languages, the one literal languages where each update is permitted to have just one atom in its body, and the unary languages where only unary relations may be updated, but higher arity relations may be accessed through views. Within each of these, we identify members close to the boundary of (un)decidability. Our context is similar to the while query language and the dynamics gives an interesting contrast to Datalog with negation; our results shed insights on the power of triggers as well as comparison of the termination problem to boundedness and query containment.

Second-order unification and type inference for Church-style polymorphism

Abstract: We prove that the second-order unification problem is undecidable even if the functional free variables may only be applied to ground terms. Despite this strong restriction our proof uses elementary techniques, and does not rely on the undecidability of the tenth Hilbert's problem which itself has a very difficult proof. We apply this result to obtain a surprising result of the undecidability of type inference for the Church-style system F - polymorphic λ-calculus.

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 Unique-normal-form property of ground systems for the first time. Next we prove undecidability of these problems for a fixed string rewriting system using our reductions. Finally, we prove partial decidability results for Confluence of 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 undecidable, and we show a linear equational theory with decidable word problem but undecidable linear equational matching.

The Π₃-theory of the computably enumerable Turing degrees is undecidable

Abstract: We show the undecidability of the Π3-theory of the partial order of enumerable Turing degrees. 0. Introduction. Recursively enumerable (henceforth called enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups. Putting the enumerable sets into context with each other in various ways yields structures whose study has for long been a mainstay of computability theory. If the sets are related in the most elementary way, namely by inclusion, one obtains a distributive lattice E with very complex algebraic properties. Another way to compare sets is to look at the information content. Turing reducibility is a very general, but the most widely accepted concept of relative computability: a setX of natural numbers is Turing-reducible to Y iff the answer to “n ∈ X?” can be determined by a Turing machine computation which can use answers to oracle questions “y ∈ Y ?” during the computation. (For more restricted notions of relative computability one would for instance place a priori 1991 Mathematics Subject Classification. 03D25,03D35.

Termination Analysis for Functional Programs

Abstract: Proving termination is a central problem in software development and formal methods for termination analysis are essential for program verification. However, since the halting problem is undecidable and totality of functions is not even semi-decidable, there is no procedure to prove or disprove the termination of all algorithms.

The first-order theory of linear one-step rewriting is undecidable

Abstract: The theory of one-step rewriting for a given rewrite system R and signature σ is the first-order theory of the following structure: its universe consists of all σ-ground terms, and its only predicate is the relation “x rewrites to y in one step by R”. The structure contains no function symbols and no equality. We show that there is no algorithm deciding the ∃ ∗ ∀ ∗ - fragment of this theory for an arbitrary finite, linear and non-erasing term-rewriting system. With the same technique we prove that the theory of encompassment plus one-step rewriting by the rule f(x) → g(x) and the modal theory of one-step rewriting are undecidable.

Set Constraints in Some Equational Theories

Abstract: Set constraints are relations between sets of ground terms over a given alphabet. They give a natural formalism for many problems in program analysis, type inference, order-sorted unification, and constraint logic programming. In this paper we start studies of set constraints in the environment given by equational specifications. We show that in the case of associativity (i.e., in free monoids) as well as in the case of associativity and commutativity (i.e., in commutative monoids) the problem of consistency of systems of set constraints is undecidable; in linear nonerasing shallow theories the consistency of systems of positive set constraints is NEXPTIME-complete and in linear shallow theories the problem for positive and negative set constraints is decidable.

Decision problems concerning thinness and slenderness of formal languages

Abstract: A language L is called thin if for almost all n there is at most one word of length n in L. A language L is called slender if there is a positive integer k such that for any n there are at most k words of length n in L. The notions of Parikh thin and Parikh slender languages are defined similarly by counting the words with the same Parikh vectors instead of the words of the same length. In this paper we discuss decision problems concerning these four properties. It is shown that all four properties are decidable for bounded semilinear languages but undecidable for DT0L languages. As a consequence all these problems are decidable for context-free languages.

Controllers for discrete event systems via morphisms

Abstract: We study the problem of synthesising controllers for discrete event systems. Traditionally this problem is tackled in a linear time setting. Moreover, the desired subset of the computations of the uncontrolled system (often called a plant) is specified by automata theoretic means. Here we formulate the problem in a branching time framework. We use a class of labelled transition systems to model both the plant and the specification. We deploy behaviour preserving morphisms to capture the role of a controller; the controlled behaviour of the plant should be related via a behaviour preserving morphism to the specification at the level of unfoldings. One must go over to unfoldings in order to let the controller use memory of the past to carry out its function. 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 the size of the finite controller, if one exists can be bounded by a polynomial in the sizes of the plant and the specification. Such a controller can also be effectively constructed. We then prove that in a natural concurrent setting, the problem of checking for the existence of a (finite) controller is undecidable.

Deciding Global Partial-Order Properties

Abstract: Model checking of asynchronous systems is traditionally based on the interleaving model, where an execution is modeled by a total order between events. Recently, the use of partial order semantics that allows independent events of concurrent processes to be unordered is becoming popular. Temporal logics that are interpreted over partial orders allow specifications relating global snapshots, and permit reduction algorithms to generate only one representative linearization of every possible partial-order execution during state-space search. This paper considers the satisfiability and the model checking problems for temporal logics interpreted over partially ordered sets of global configurations. For such logics, only undecidability results have been proved previously. In this paper, we present an Expspace decision procedure for a fragment that contains an eventuality operator and its dual. We also sharpen previous undecidability results, which used global predicates over configurations. We show that although our logic allows only local propositions (over events), it becomes undecidable when adding some natural until operator.

Rigid Reachability

Abstract: We show that rigid reachability, the non-symmetric form of rigid E-unification, is undecidable already in the case of a single constraint. From this we infer the undecidability of a new rather restricted kind of second-order unification. We also show that certain decidable subclasses of the problem which are ρ-complete in the equational case become EXPTIME-complete when symmetry is absent. By applying automata-theoretic methods, simultaneous monadic rigid reachability with ground rules is shown to be in EXPTIME.

On Unification Problems in Restricted Second-Order Languages

Abstract: We review known results and improve known boundaries between the decidable and the undecidable cases of second-order unification with various restrictions on second-order variables As a key tool we prove an undecidability result that provides a partial solution to an open problem about simultaneous rigid E-unification

The Relation Between Second-Order Unification and Simultaneous Rigid E-Unification

Abstract: Simultaneous rigid E-unification, or SREU for short, is a fundamental problem that arises in global methods of automated theorem proving in classical logic with equality. In order to do proof search in intuitionistic logic with equality one has to handle SREU as well. Furthermore, restricted forms of SREU are strongly related to word equations and finite tree automata. It was recently shown that second-order unification has a very natural reduction to simultaneous rigid E-unification, which constituted probably the most transparent undecidability proof of SREU. Here we show that there is also a natural encoding of SREU in second-order unification. It follows that the problems are logspace equivalent. So second-order unification plays the same fundamental role as SREU in automated reasoning in logic with equality. We exploit this connection and use finite tree automata techniques to present a very elementary undecidability proof of second-order unification, by reduction from the halting problem for Turing machines. It follows from that proof that second-order unification is undecidable for all nonmonadic second-order term languages having at least two second-order variables with sufficiently high arities.

The first-order theory of ordering constraints over feature trees

Abstract: The system FT/sub /spl les// 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/sub /spl les// and its fragments, both over finite trees and over possibly infinite trees. We prove that the first-order theory of FT/sub /spl les// is undecidable, in contrast to the first-order theory of FT which is well-known to be decidable. We determine the complexity of the entailment problem of FT/sub /spl les// with existential quantification to be PSPACE-complete, by proving its equivalence to the inclusion problem of non-deterministic finite automata. Our reduction from the entailment problem to the inclusion problem is based on a new algorithm that, given an existential formula of FT/sub /spl les//, computes a finite automaton which accepts all its logic consequences.

Potential divisibility in finite semigroups is undecidable

Abstract: We prove that there is no algorithm to decide, given a finite semigroup S and two elements a, b∈S, whether there exists a bigger finite semigroup T>S where a divides b and b divides a. This solves a thirty years old problem by John Rhodes.

Physical significance of experimental Mueller matrices

Abstract: The recent result obtained by Givens Kostinski [J. Mod. Opt.40, 471 (1993)] successfully solves the old and important problem in polarization optics of characterizing a given 4×4 matrix as a Mueller matrix from a mathematical point of view. For practical purposes, however, a further elaboration on this result is needed, namely, the problem of characterizing a matrix whose elements have been empirically obtained after the measurement of a given number of independent quantities that are affected by errors. We solve this problem by first obtaining an alternative form of the Givens–Kostinski theorem that allows us to figure out an algorithm for calculating the error propagation. It turns out that the experimental matrix can be finally regarded as physically meaningful or not, or even undecidable, depending on such errors. As a tool for potential users, a routine (in both fortran and idl languages) that carries out all the numerical calculations is available via ftp at a specified address.

Kripke, Belnap, Urquhart and Relevant decidability & complexity

Abstract: The first philosophically motivated sentential logics to be proved undecidable were relevant logics like R and E But we deal here with important decidable fragments thereof, like \(R_{\longrightarrow}\) Their decidability rests on S Kripke’s gentzenizations, together with his central combinatorial lemma Kripke’s lemma has a long history and was reinvented several times It turns out equivalent to and a consequence of Dickson’s lemma in number theory, with antecedents in Hilbert’s basis theorem This lemma has been used in several forms and in various fields For example, Dickson’s lemma guarantees termination of Buchberger’s algorithm that computes the Grobner bases of polynomial ideals In logic, Kripke’s lemma is used in decision proofs of some substructural logics with contraction Our preferred form here of Dickson-Kripke is the Infinite Division Principle (IDP) We present our proof of IDP and its use in proving the finite model property for \(R_{\longrightarrow}\)

Controllers for discrete event systems via morphisms

Abstract: We study the problem of synthesising controllers for discrete event systems. Traditionally this problem is tackled in a linear time setting. Moreover, the desired subset of the computations of the uncontrolled system (often called a plant) is specified by automata theoretic means. Here we formulate the problem in a branching time framework. We use a class of labelled transition systems to model both the plant and the specification. We deploy behaviour preserving morphisms to capture the role of a controller; the controlled behaviour of the plant should be related via a behaviour preserving morphism to the specification at the level of unfoldings. One must go over to unfoldings in order to let the controller use memory of the past to carry out its function. 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 the size of the finite controller, if one exists can be bounded by a polynomial in the sizes of the plant and the specification. Such a controller can also be effectively constructed. We then prove that in a natural concurrent setting, the problem of checking for the existence of a (finite) controller is undecidable.

The Guarded Fragment of Conceptual Graphs

Abstract: Conceptual graphs (CGs) are an expressive and intuitive formalism, which plays an important role in the area of knowledge representation. Due to their expressiveness, most interesting problems for CGs are inherently undecidable. We identify the syntactically defined guarded fragment of CGs, for which both subsumption and validity is decidable in deterministic exponential time.