Showing papers on "Undecidable problem published in 1993"

Model-Checking in Dense Real-Time

Abstract: Model-checking is a method of verifying concurrent systems in which a state-transition graph model of the system behavior is compared with a temporal logic formula. This paper extends model-checking for the branching-time logic CTL to the analysis of real-time systems, whose correctness depends on the magnitudes of the timing delays. For specifications, we extend the syntax of CTL to allow quantitative temporal operators such as ?? <5, meaning "possibly within 5 time units." The formulas of the resulting logic, Timed CTL (TCTL), are interpreted over continuous computation trees, trees in which paths are maps from the set of nonnegative reals to system states. To model finite-state systems we introduce timed graphs-state-transition graphs annotated with timing constraints. As our main result, we develop an algorithm for model-checking, for determining the truth of a TCTL-formula with respect to a timed graph. We argue that choosing a dense domain instead of a discrete domain to model time does not significantly blow up the complexity of the model-checking problem. On the negative side, we show that the denseness of the underlying time domain makes the validity problem for TCTL ?11-hard. The question of deciding whether there exists a timed graph satisfying a TCTL-formula is also undecidable.

Real-Time Logics

Abstract: The theory of the natural numbers with linear order and monadic predicates underlies propositional linear temporal logic. To study temporal logics that are suitable for reasoning about real-time systems, we combine this classical theory of infinite state sequences with a theory of discrete time, via a monotonic function that maps every state to its time. The resulting theory of timed state sequences is shown to be decidable, albeit nonelementary, and its expressive power is characterized by ?-regular sets. Several more expressive variants are proved to be highly undecidable. This framework allows us to classify a wide variety of real-time logics according to their complexity and expressiveness. Indeed, it follows that most formalisms proposed in the literature cannot be decided. We are, however, able to identify two elementary real-time temporal logics as expressively complete fragments of the theory of timed state sequences, and we present tableau-based decision procedures for checking validity. Consequently, these two formalisms are well-suited for the specification and verification of real-time systems.

Parametric real-time reasoning

Abstract: Traditional approaches to the algorithmic veri cation of real-time systems are limited to checking program correctness with respect to concrete timing properties (e.g., \message delivery within 10 milliseconds"). We address the more realistic and more ambitious problem of deriving symbolic constraints on the timing properties required of real-time systems (e.g., \message delivery within the time it takes to execute two assignment statements"). To model this problem, we introduce parametric timed automata | nite-state machines whose transitions are constrained with parametric timing requirements. The emptiness question for parametric timed automata is central to the veri cation problem. On the negative side, we show that in general this question is undecidable. On the positive side, we provide algorithms for checking the emptiness of restricted classes of parametric timed automata. The practical relevance of these classes is illustrated with several veri cation examples. There remains a gap between the automata classes for which we know that emptiness is decidable and undecidable, respectively, and this gap is related to various hard and open problems of logic and automata theory.

What can be computed locally

Abstract: The purpose of this paper is a study of computation that can be done locally in a distributed network, where "locally" means within time (or distance) independent of the size of the network. Locally checkable labeling (LCL) problems are considered, where the legality of a labeling can be checked locally (e.g., coloring). The results include the following: There are nontrivial LCL problems that have local algorithms. There is a variant of the dining philosophers problem that can be solved locally. Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. It is undecidable, in general, whether a given LCL has a local algorithm. However, it is decidable whether a given LCL has an algorithm that operates in a given time $t$. Any LCL problem that has a local algorithm has one that is order-invariant (the algorithm depends only on the order of the processor IDs).

Integration Graphs: A Class of Decidable Hybrid Systems

Abstract: Integration Graphs are a computational model developed in the attempt to identify simple Hybrid Systems with decidable analysis problems. We start with the class of constant slope hybrid systems (cshs), in which the right hand side of all differential equations is an integer constant. We refer to continuous variables whose right hand side constants are always 1 as timers. All other continuous variables are called integrators. The first result shown in the paper is that simple questions such as reachability of a given state are undecidable for even this simple class of systems.

Simultaneous Stabilizability of Three Linear Systems Is Rationally Undecidable

Abstract: We show that the simultaneous stabilizability of three linear systems, that is the question of knowing whether three linear systems are simultaneously stabilizable, is rationally undecidable. By this we mean that it is not possible to find necessary and sufficient conditions for simultaneous stabilization of the three systems in terms of expressions involving the coefficients of the three systems and combinations of arithmetical operations (additions, subtractions, multiplications, and divisions), logical operations (''and'' and ''or''), and sign test operations (equal to, greater than, greater than or equal to,...).

Type reconstruction in the presence of polymorphic recursion

Abstract: We study the problem of type-checking functional programs in three extensions of ML. One distinguishing feature of these extensions is that they allow recursive definitions to be polymorphically typed. Although the motivation for these extensions comes from pragmatic considerations of programming language design, we show that the typability problem for each one of these extensions is polynomial-time equivalent to the Semi-Unification Problem and, therefore, undecidable

On the expressivity of feature logics with negation, functional uncertainty, and sort equations

Abstract: Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.

The undecidability of the semi-unification problem

Abstract: The Semi-Unification Problem (SUP) is a natural generalization of both first-order unification and matching. The problem arises in various branches of computer science and logic. Although several special cases of SUP are known to be decidable, the problem in general has been open for several years. We show that SUP in general is undecidable, by reducing what we call the "boundedness problem" of Turing machines to SUP. The undecidability of this boundedness problem is established by a technique developed in the mid-1960s to prove related results about Turing machines

Order-sorted feature theory unification

Abstract: Order-sorted feature (OSF) terms provide an adequate representation for objects as flexible records They are sorted, attributed, possibly nested, structures, ordered thanks to a subsort ordering Sort definitions offer the functionality of classes imposing structural constraints on objects These constraints involve variable sorting and equations among feature paths, including self-reference Formally, sort definitions may be seen as axioms forming an OSF theory OSF theory unification is the process of normalizing an OSF term, using sort-unfolding to enforce structural constraints imposed on sorts by their definitions It allows objects to inherit, and thus abide by, constraints from their classes A formal system is thus obtained that logically models record objects with recursive class definitions accommodating multiple inheritance We show that OSF theory unification is undecidable in general However, we propose a set of confluent normalization rules which is complete for detecting inconsistency of an object with respect to an OSF theory These rules translate into an efficient algorithm using structure-sharing and lazy constraint-checking Furthermore, a subset consisting of all rules but one is confluent and terminating This yields a practical complete normalization strategy, as well as an effective compilation scheme

Inclusion is Undecidable for Pattern Languages

Abstract: The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It has been open ever since. In this paper, we settle this open problem and show that inclusion is undecidable for (both erasing and nonerasing) pattern languages. In addition, we show that a special case of the inclusion problem, i.e., the inclusion problem for terminal-free erasing pattern languages, is decidable.

Undecidable Problems in Fractal Geometry.

Abstract: In this paper , a relationship between the classical theory of computat ion and fract al geometry is est ablish ed. It erated Function Syst ems are used as tool s to define fractals. It is shown t hat two questions about Iterat ed Function Systems are und ecidable: to t est if t he attractor of a given It erat ed Fun ction Syst em and a given line segment int ersect and to test if a given It erated Funct ion Syst em is totally disconnected. T he proofs are very simple and are obtained by reducing the Post Correspondence P roblem and by interp ret ing strings as numb ers and conca tenation operat ions as compositions of affine t ransformations. These results show that for every Turing machine there exists a fract al set which can be viewed, in a certain sense, as geomet rically encoding the complement of the language accepted by t he machine. One can build a fractal-based geometrical model of computation which is computationally universal.

The halting problem of one binary Horn clause is undecidable

Abstract: This paper proposes a codification of the halting problem of any Turing machine in the form of only one right-linear binary Horn clause as follows: p(t) ← p(tt). where t (resp. tt) is any (resp. linear) term. Recursivity is well-known to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes.

On the unification problem for Cartesian closed categories

Abstract: An axiomatization of the isomorphisms that hold in all Cartesian closed categories (CCCs), discovered independently by S.V. Soloviev (1983) and by K.B. Bruce and G. Longo (1985), leads to seven equalities. It is shown that the unification problem for this theory is undecidable, thus setting an open question. It is also shown that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible. CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by M. Rittri (1990, 1991). It also has potential applications to the problem of polymorphic higher-order unification, which in turn is relevant to theorem proving, logic programming, and type reconstruction in higher-order languages. >

Halting Problem of One Binary Horn Clause is Undecidable

Abstract: This paper proposes a codification of the halting problem of any Turing machine in the form of only one right-linear binary Horn clause as follows: p(t) ← p(tt). where t (resp. tt) is any (resp. linear) term. Recursivity is well-known to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes.

Taking it to the limit: on infinite variants of NP-complete problems

Abstract: Infinite, recursive versions of NP optimization problems are defined. For example, MAX CLIQUE becomes the question of whether a recursive graph contains an infinite clique. The work was motivated by trying to understand what makes some NP problems highly undecidable in the infinite case, while others remain on low levels of the arithmetical hierarchy. Two results are proved; one enables using knowledge about the infinite case to yield implications to the finite case, and the other enables implications in the other direction. Taken together, the two results provide a method for proving (finitary) problems to be outside the syntactic class MAX NP, hence outside MAX SNP too. The technique is illustrated with many examples. >

The Gödel Incompleteness Theorem and Decidability over a Ring

Abstract: Here we give an exposition of Godel’s result in an algebraic setting and also a formulation (and essentially an answer) to Penrose’s problem. The notions of computability and decidability over a ring R underly our point of view. Godel’s Theorem follows from the Main Theorem: There is a definable undecidable set ovis Z. By way of contrast, Tarski’s Theorem asserts that every definable set over the reals or any real closed field R is decidable over R. We show a converse to this result: Any sufficiently infinite ordered field with this latter property is necessarily real closed.

On weakly confluent monadic string-rewriting systems

Abstract: It is investigated as to how far the various decidability results for finite, monadic, and confluent string-rewriting systems can be carried over to the class of finite monadic string-rewriting systems that are only weakly confluent. Here a monadic string-rewriting system R on some alphabet Σ is called weakly confluent if it is confluent on all the congruence classes [a]R, with a ∈ Σ ⋓ {e}. After establishing that the property of weak confluence is tractable for finite monadic string-rewriting systems, we prove that many decision problems that are tractable for finite, monadic, and confluent systems are, in fact, undecidable for finite monadic systems that are only weakly confluent. An example is the word problem. On the other hand, for finite, monadic, and weakly confluent systems that present groups, the validation problem for linear sentences is decidable. Many decision problems, among them the word problem and the generalized word problem, can be expressed through linear sentences and, hence, they all are decidable in this setting. The paper closes with a specialized completion procedure for finite, monadic string-rewriting systems presenting groups. Given a system of this form, the completion procedure tries to construct an equivalent system of the same form that, in addition, is weakly confluent. The correctness and completeness of this procedure are shown, and some detailed examples are presented. This procedure, together with the decidability results mentioned before, presents an elegant and uniform way to perform computations in context-free groups effectively.

On propositional quantifiers in provability logic.

Abstract: The first order theory of the Diagonalizable Algebra of Peano Arith- metic (DA(PA)) represents a natural fragment of provability logic with proposi- tional quantifiers. We prove that the first order theory of the O-generated subalgebra of DA(PA) is decidable but not elementary recursive; the same theory, enriched by a single free variable ranging over DA(PA), is already undecidable. This gives a negative answer to the question of the decidability of provability logics for recur- sive progressions of theories with quantifiers ranging over their ordinal notations. We also show that the first order theory of the free diagonalizable algebra on n independent generators is undecidable iff n Φ 0.

Decidability and Undecidability of Theories with a Predicate for the Primes

Abstract: It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semenov on decidability of monadic theories, and a proof of Semenov's result is presented.

On Arnold's Hilbert Symposium Problems

Abstract: We prove that stability is undecidable for dynamical systems whose right-hand side is explicitly written in the language of elementary analysis.

Word Problem for Thue Systems with a Few Relations

Abstract: The history of investigations on the word problem for Thue systems is presented with the emphasis on undecidable systems with a few relations. The best known result, a Thue system with only three relations and undecidable word problem, is presented with details. Bibl. 43 items.

The Word Problem for Thue Rewriting Systems

Abstract: This paper is divided into two parts that can be read independently. Part I is devoted to examples of undecidable Thue rewriting systems. Part II presents the results obtained so far on the conjecture that one-relation Thue systems are decidable.

Calculi for Bags and their Complexity

Abstract: In this paper, we propose calculi to express queries over bags (i.e. multisets), and study their complexity. We show that the calculus for bags is undecidable in general. Nevertheless, simple syntactic restrictions on the calculus result in computable languages. We provide here two restricted calculi with bounded complexity, and show that the restrictions are minimal. Indeed, any looser restriction leads to non computable queries.