Showing papers on "Undecidable problem published in 1999"

On the verification of broadcast protocols

Abstract: We analyze the model-checking problems for safety and liveness properties in parameterized broadcast protocols. We show that the procedure suggested previously for safety properties may not terminate, whereas termination is guaranteed for the procedure based on upward closed sets. We show that the model-checking problem for liveness properties is undecidable. In fact, even the problem of deciding if a broadcast protocol may exhibit an infinite behavior is undecidable.

Symbolic Verification of Communication Protocols with Infinite StateSpaces using QDDs

Abstract: We study the verification of properties of communication protocols modeled by a finite set of finite-state machines that communicate by exchanging messages via unbounded FIFO queues. It is well-known that most interesting verification problems, such as deadlock detection, are undecidable for this class of systems. However, in practice, these verification problems may very well turn out to be decidable for a subclass containing most “real” protocols. Motivated by this optimistic (and, we claim, realistic) observation, we present an algorithm that may construct a finite and exact representation of the state space of a communication protocol, even if this state space is infinite. Our algorithm performs a loop-first search in the state space of the protocol being analyzed. A loop-first search is a search technique that attempts to explore first the results of successive executions of loops in the protocol description (code). A new data structure named Queue-content Decision Diagram (QDD) is introduced for representing (possibly infinite) sets of queue-contents. Operations for manipulating QDDs during a loop-first search are presented. A loop-first search using QDDs has been implemented, and experiments on several communication protocols with infinite state spaces have been performed. For these examples, our tool completed its search, and produced a finite symbolic representation for these infinite state spaces.

Model Checking Knowledge and Time in Systems with Perfect Recall

Abstract: This paper studies model checking for the modal logic of knowledge and linear time in distributed systems with perfect recall. It is shown that this problem (1) is undecidable for a language with operators for until and common knowledge, (2) is PSPACE-complete for a language with common knowledge but without until, (3) has nonelementary upper and lower bounds for a language with until but without common knowledge. Model checking bounded knowledge depth formulae of the last of these languages is considered in greater detail, and an automata-theoretic decision procedure is developed for this problem, that yields a more precise complexity characterization.

Handbook of computability theory

Abstract: Part 1: Fundamentals of Computability Theory. Part 2: Reducibilities and Degrees. Part 3: Generalized Computability Theory. Part 4: Mathematics and Computability Theory. Part 5: Logic and Computability Theory. Part 6: Computer Science and Computability Theory.

Message Sequence Graphs and Decision Problems on Mazurkiewicz Traces

Abstract: Message sequence chEirts (MSC) are a graphical specification language widely used for designing communication protocols. Our starting point are two decision problems concerning the correctness and the consistency of a design based by MSC graphs. Both problems are shown to be undecidable, in general. Using a natural connectivity assumption from Mazurkiewicz trace theory we show both problems to be EXPSPACE-complete for locally synchronized graphs. The results are based on new complexity results for star-connected rational trace languages.

Model Checking Knowledge and Time in Systems with Perfect Recall (Extended Abstract)

Abstract: This paper studies model checking for the modal logic of knowledge and linear time in distributed systems with perfect recall. It is shown that this problem (1) is undecidable for a language with operators for until and common knowledge, (2) is PSPACE-complete for a language with common knowledge but without until, (3) has nonelementary upper and lower bounds for a language with until but without common knowledge.M odel checking bounded knowledge depth formulae of the last of these languages is considered in greater detail, and an automata-theoretic decision procedure is developed for this problem, that yields a more precise complexity characterization.

Three-Processor Tasks Are Undecidable

Abstract: We show that no algorithm exists for deciding whether a finite task for three or more processors is wait-free solvable in the asynchronous read-write shared-memory model. This impossibility result implies that there is no constructive (recursive) characterization of wait-free solvable tasks. It also applies to other shared-memory models of distributed computing, such as the comparison-based model.

Undecidability results on two-variable logics

Abstract: It is a classical result of Mortimer that $L^2$ , first-order logic with two variables, is decidable for satisfiability. We show that going beyond $L^2$ by adding any one of the following leads to an undecidable logic:– very weak forms of recursion, viz.¶(i) transitive closure operations¶(ii) (restricted) monadic fixed-point operations¶– weak access to cardinalities, through the Hartig (or equicardinality) quantifier¶– a choice construct known as Hilbert's $\epsilon$ -operator. In fact all these extensions of $L^2$ prove to be undecidable both for satisfiability, and for satisfiability in finite structures. Moreover most of them are hard for $\Sigma^1_1$ , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than first-order logic.

On the undecidability of freeness of matrix semigroups

Abstract: We slightly improve the result of Klarner, Birget and Satterfield, showing that the freeness of finitely presented multiplicative semigroups of 3×3 matrices over ℕ is undecidable even for triangular matrices. This is achieved by proving a new variant of Post correspondence problem. We also consider the freeness problem for 2×2 matrices. On the one hand, we show that it cannot be proved undecidable using the above methods which work in higher dimensions, and, on the other hand, we give some evidence that its decidability, if so, is also a challenging problem.

Complexity of products of modal logics

Abstract: A relatively new way of combining modal logics is to consider their products. The main application of these product logics lies in the description of parallel computing processes. Axiomatics and decidability of the validity problem have been rather extensively investigated and many logics behave well in these respects. In this paper we look at the product construction from a computational complexity point of view. We show that in many cases there is a drastic increase in complexity, e.g., all products containing the finite S5×S5 products as models have an nexptime-hard satisfaction problem. Products with a functional modality however do not lead to an increase in complexity. For the products K× S5 and S5× S5, we provide a matching upper bound. Combining (modal) logics is a very active area, witness e.g., [4] and the book [1]. A rather special way of combining two modal logics is to consider their products. This approach started with [20], and has recently been developed in great detail in [5]. In temporal logic, products of two logics have been used to describe the temporal logic of intervals (cf. the “product treatment” of the system HS from [8] in [15]: Chapter 4 and the references therein). Almost all products of temporal logics are undecidable, sometimes the validities are not even recursively enumerable [8, 22]. Products of modal (and modal and temporal) logics have applications in the theory of parallel computing [18]. Here we are concerned with the general mathematical theory of products of modal logics, in particular the complexity of several natural decision problems, like the validity problem. With respect to (Hilbert style) axiomatizability a lot of general results are obtained in [5], cf e.g., Theorem 5.7. That paper also contains decidability results for a large number of cases. The general trend for these results is that they are rather hard to prove, but become a lot easier if one of the logics is S5, though even then the filtration arguments are rather involved, and lead to models whose size is in general double exponential in the length of the formula which is to be satisfied. The upper bounds we obtain from these proofs are very bad, in the general case (when none of the logics is S5), the decision-algorithm is non-elementary, and when one of the logics is S5 we only obtain a non-deterministic double exponential time upper bound for the satisfaction problem. Questions concerning computational complexity, have hitherto not been addressed, and we will make a start here. The overall trend is that these logics have a very bad complexity for the satisfaction problem: in many simple cases it is nexptime-hard. Also, even if the satisfaction problem is decidable, the problem whether for a formula φ there exists a model The author is supported by UK EPSRC grant No. GR/K54946.

Expressive number restrictions in description logics

Abstract: Number restrictions are concept constructors that are available in almost all implemented Description Logic systems. However, they are mostly available only in a rather weak form, which considerably restricts their expressive power. On the one hand, the roles that may occur in number restrictions are usually of a very restricted type, namely atomic roles or complex roles built using either intersection or inversion. In the present paper, we increase the expressive power of Description Logics by allowing for more complex roles in number restrictions. As role constructors, we consider composition of roles (which will be present in all our logics) and intersection, union, and inversion of roles in different combinations. We will present two decidability results (for the basic logic that extends ALC by number restrictions on roles with composition, and for one extension of this logic), and three undecidability results for three other extensions of the basic logic. On the other hand, with the rather weak form of number restrictions available in implemented systems, the number of role successors of an individual can only be restricted by a fixed non-negative integer. To overcome this lack of expressiveness, we allow for variables ranging over the non-negative integers in place of the fixed numbers in number restrictions. The expressive power of this constructor is increased even further by introducing explicit quantifiers for the numerical variables. The Description Logic obtained this way turns out to have an undecidable satisfiability problem. For a restricted logic we show that concept satisfiability is decidable.

Representability is not decidable for finite relation algebras

Abstract: We prove that there is no algorithm that decides whether a finite relation algebra is representable. Representability of a finite relation algebra A is determined by playing a certain two player game G(A) over 'atomic A-networks'. It can be shown that the second player in this game has a winning strategy if and only if A is representable. Let be a finite set of square tiles, where each edge of each tile has a colour. Suppose includes a special tile whose four edges are all the same colour, a colour not used by any other tile. The tiling problem we use is this: is it the case that for each tile T 2 there is a tiling of the plane Z ◊ Z using only tiles from in which edge colours of adjacent tiles match and with T placed at (0,0)? It is not hard to show that this problem is undecidable. From an instance of this tiling problem we construct a finite relation algebra RA( ) and show that the second player has a winning strategy in G(RA( )) if and only if is a yes- instance. This reduces the tiling problem to the representation problem and proves the latter's undecidability.

Interaction between path and type constraints

Abstract: XML [7], which is emerging as an important standard for data exchange on the World Wide Web, highlights the importance of the semistructured data. Although the XML standard itself does not require any schema or type system, a number of proposals [6, 15, 18] have been developed that roughly correspond to data definition languages. These allow one to constrain the structure of XML data by imposing a schema on it. These and other proposals also advocate the need for integrity constraints, another form of constraints that should, for example, be capable of expressing inclusion constraints and inverse relationships. The latter have recently been studied as path constraints in the context of semistructured data [4, 11]. It is likely that future XML proposals will involve both forms of constraint, and it is therefore appropriate to understand the interaction between them. This paper investigates that interaction. In particular it studies constraint implication problems, which are important both in understanding the semantics of type/constraint systems and in query optimization. A number of results on path constraint implication are established in the presences and absences of type systems. These results demonstrate that adding a type systems may in some cases simplify reasoning about path constraints and in other cases make it harder. For example, it is shown that there is a path constraint implication problem that is decidable in PTIME in the untyped context, but that becomes undecidable when a type system is added. On the other hand, there is an implication problem that is undecidable in the untyped context, but becomes not only decidable in cubic time but also finitely axiomatizable when a type system is imposed. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-98-16. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/106

Decidable integration graphs

Abstract: Integration graphsare a computational model developed in the attempt to identify simple hybrid systems with decidable analysis problems. We start with the class ofconstant 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 astimers. All other continuous variables are calledintegrators. 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. To restrict the model even further, we impose the requirement that no test that refers to integrators may appear within a loop in the graph. This restricted class of CSHS is calledintegration graphs. The main results of the paper are that the reachability problem of integration graphs is decidable for two special cases: the case of a single timer and the case of a single test involving integrators. The expressive power of the integration-graphs formalism is demonstrated by showing that some typical problems studied within the context of the calculus of durations and timed statecharts can be formulated as reachability problems for restricted integration graphs, and a high fraction of these fall into the subclasses of a single timer or a single test involving integrators.

The Emptiness Problem for Intersection Types

Abstract: We study the intersection type assignment system as defined by Barendregt, Coppo and Dezani. For the four essential variants of the system (with and without a universal type and with and without subtyping) we show that the emptiness (inhabitation) problem is recursively unsolvable. That is, there is no effective algorithm to decide if there is a closed term of a given type. It follows that provability in the logic of “strong conjunction” of Mints and Lopez-Escobar is also undecidable.

The Complexity of Decision Procedures in Relevance Logic II

Abstract: Relevance logic is distinguished among non-classical logics by the richness of its mathematical as well as philosophical structure. This richness is nowhere more evident than in the difficulty of the decision problem for the main relevant propositional logics. These logics are in general undecidable [22]; however, there are important decidable subsystems. The decision procedures for these subsystems convey the impression of great complexity. The present paper gives a mathematically precise form to this impression by showing that the decision problem for R →, the implication fragment of R, is exponential space hard. This means that any Turing machine which solves this decision problem must use an exponential amount of space (relative to the input size) on infinitely many inputs.

Detecting Equalities of Variables: Combining Efficiency with Precision

Abstract: Detecting whether different variables have the same value at a program point is generally undecidable. Though the subclass of equalities, whose validity holds independently from the interpretation of operators (Herbrand-equivalences), is decidable, the technique which is most widely implemented in compilers, value numbering, is restricted to basic blocks. Basically, there are two groups of algorithms aiming at globalizations of value numbering: first, a group of algorithms based on the algorithm of Kildall, which uses data flow analysis to gather information on value equalities. These algorithms are complete in detecting Herbrand-equivalences, however, expensive in terms of computational complexity. Second, a group of algorithms influenced by the algorithm of Alpern, Wegman and Zadeck. They do not fully interpret the control flow, which allows them to be particularly efficient, however, at the price of being significantly less precise than their Kildall-like counterparts. In this article we discuss how to combine the best features of both groups by aiming at a fair balance between computational complexity and precision. We propose an algorithm, which extends the one of Alpern, Wegman and Zadeck. The new algorithm is polynomial and, in practice, expected to be almost as efficient as the original one. Moreover, for acyclic control flow it is as precise as Kildall's one, i. e. it detects all Herbrand-equivalences.

Multi-Dimensional Description Logics

Abstract: In this paper, we construct a new concept description language intended for representing dynamic and intensional knowledge The most important feature distinguishing this language from its predecessors in the literature is that it allows applications of modal operators to all kinds of syntactic terms: concepts, roles and formulas Moreover, the language may contain both local (ie, state-dependent) and global (ie, state-independent) concepts, roles and objects All this provides us with the most complete and natural means for reflecting the dynamic and intensional behaviour of application domains We construct a satisfiability checking (mosaic-type) algorithm for this language (based on ALC) in (i) arbitrary multimodal frames, (ii) frames with universal accessibility relations (for knowledge) and (iii) frames with transitive, symmetrical and euclidean relations (for beliefs) On the other hand, it is shown that the satisfaction problem becomes undecidable if the underlying frames are arbitrary linear orders or the language contains the common knowledge operator for n ≥ 2 agents

Achilles, Turtle, and Undecidable Boundedness Problems for Small DATALOG Programs

Abstract: DATALOG is the language of logic programs without function symbols. It is considered to be the paradigmatic database query language. If it is possible to eliminate recursion from a DATALOG program then it is bounded. Since bounded programs can be executed in parallel constant time, the possibility of automatized boundedness detecting is believed to be an important issue and has been studied in many papers. Boundedness was proved to be undecidable for different kinds of semantical assumptions and syntactical restrictions. Many different proof techniques were used. In this paper we propose a uniform proof method based on the discovery of, as we call it, the Achilles--Turtle machine, and make strong improvements on most of the known undecidability results. In particular we solve the famous open problem of Kanellakis showing that uniform boundedness is undecidable for single rule programs (called also sirups). This paper is the full version of [J. Marcinkowski, Proc. 13th STACS, Lecture Notes in Computer Science 1046, pp. 427--438], and [J. Marcinkowski, 11th IEEE Symposium on Logic in Computer Science, pp. 13--24].

Chapter 6 - The Recursively Enumerable Degrees

Abstract: Decision problems are the motivating force in the search for a formal definition of algorithm that constituted the beginnings of recursion (computability) theory. In most settings, the notion of a recursively enumerable (r.e.) set is found: the theorems of a axiomatized theory, the solvable Diophantine equations, and the true equations among words in a finitely presented group. Typically, such decision problems amount to deciding if a particular r.e. set is computable. All these sets are simply non-computable. Another view sees them as more complicated or harder to compute than the recursive sets. This is the view that leads to the notion of relative computability (reducibility) introduced by Turing. The equivalence classes under this notion of relative computability are first called the “degrees of recursive unsolvability.” The starting point for the investigation of this fundamental notion of relative computability is the r.e. degrees. The classic results of logic, such as Godel's incompleteness theorem, Church's proof of the undecidability of predicate logic, and Turing's unsolvability of the Halting problem, each proved that there is a nonrecursive r.e. degree. All the natural examples, however, of nonrecursive r.e. sets supplied by standard theories that could be proven undecidable or from other natural definitions of noncomputable r.e. sets, turned out to have the same complexity.

Parametric Temporal Logic for Model Measuring

Abstract: We extend the standard model checking paradigm of linear temporal logic, LTL, to a "model measuring" paradigm where one can obtain more quantitative information beyond a "Yes/No" answer. For this purpose, we define a parametric temporal logic, PLTL, which allows statements such as "a request p is followed in at most x steps by a response q", where x is a free variable. We show how one can, given a formula ϕ(x1,..., xk) of PLTL and a system model K, not only determine whether there exists a valuation of x1,..., xk under which the system K satisfies the property ϕ, but if so find valuations which satisfy various optimality criteria. In particular, we present algorithms for finding valuations which minimize (or maximize) the maximum (or minimum) of all parameters. These algorithms exhibit the same PSPACE complexity as LTL model checking.We show that our choice of syntax for PLTL lies at the threshold of decidability for parametric temporal logics, in that several natural extensions have undecidable "model measuring" problems.

Simulation Preorder on Simple Process Algebras

Abstract: We consider the problem of simulation preorder/equivalence between infinite-state processes and finite-state ones. We prove that simulation preorder (in both directions) and simulation equivalence are intractable between all major classes of infinite-state systems and finite-state ones. This result is obtained by showing that the problem whether a BPA (or BPP) process simulates a finitestate 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. The decidability border for the mentioned problem is also established. Simulation preorder (in both directions) and simulation equivalence are decidable in EXPTIME between pushdown processes and finite-state ones. On the other hand, simulation preorder is undecidable between PA and finite-state processes in both directions. The obtained results also hold for those PA and finite-state processes which are deterministic and normed, and thus immediately extend to trace preorder. Regularity (finiteness) w.r.t. simulation and trace equivalence is also shown to be undecidable for PA. Finally, we describe a way how to utilize decidability of bisimulation problems to solve certain instances of undecidable simulation problems.We apply this method to BPP processes.

Overview of complexity and decidability results for three classes of elementary nonlinear systems

Abstract: It has become increasingly apparent this last decade that many problems in systems and control are NP-hard and, in some cases, undecidable. The inherent complexity of some of the most elementary problems in systems and control points to the necessity of using alternative approximate techniques to deal with problems that are unsolvable or intractable when exact solutions are sought.

A finite relation algebra with undecidable network satisfaction problem

Abstract: We deene a nite relation algebra and show that the network satisfaction problem is undecidable for this algebra 1. be a relation algebra (see JT52] for the original ax-iomatisation or Mad91] for an introduction to relation algebra). An atom a of A is a minimal, non-zero element. At(A) denotes the set of all atoms of A. A is atomic if for all non-zero a 2 A there exists 2 At(A) with a. In the following we assume A is atomic.

POINTLIKE SETS, HYPERDECIDABILITY AND THE IDENTITY PROBLEM FOR FINITE SEMIGROUPS Dedicated to the Memory of Marcel-Paul Schiitzenberger

Abstract: In this paper, we give a relationship between the identity problem and the problem of deciding whether certain subsets of nilpotent semigroups are pointlike. We then use this to give an example of a pseudovariety which has a decidable membership problem, but for which one cannot decide pointlike sets. Then, by modifying the equations, we show that no graph is fundamentally hyperdecidable by constructing, for each graph, a labeling over a nilpotent semigroup for which we cannot decide inevitability with respect to the pseudovariety defined by these equations. 1. Pointlikes and the Identity Problem An identity in a set of variables X is a formal equality u = v in the free semigroup X+. A semigroup S is said to satisfy the identity, written S \= u = v, if, for every homomorphism S \= u = v. The identity problem for finite semigroups is then, given a finite set of identities E, can one decide for any identity u = v whether E \= u = v in all finite semigroups or, in other words, is u = v a consequence of E for finite semigroups. A well known result of Albert, Baldinger, and the first author [1] says the answer is no. In the same paper, an example of a finite set of identities E such that Com V PV(S) has undecidable membership, where PV(E) denotes the pseudovariety of semigroups satisfying E, is given. The close tie between the decidability of the membership problem for joins and the decidability of pointlike sets, explored in the second author's paper [11], led the first author to conjecture that pointlikes are not decidable for such PV(E). Recall a subset A of a finite semigroup S is called pointlike for a pseudovariety V if under all relational morphisms into V, A relates to a point. Many important pseudovarieties have been proven to have decidable pointlikes, or even stronger properties, see Sec. 2 below. The second author then made explicit the following close connection between the decidability of pointlikes and the decidability of the identity problem.

On s-regular preffix-rewriting systems and automatic structures

Abstract: Underlying the notion of an automatic structure is that of a synchronously regular (s-regular for short) set of pairs of strings. Accordingly we consider s-regular prefix-rewriting systems showing that even for fairly restricted systems of this form confluence is undecidable in general. Then a close correspondence is established between the existence of an automatic structure that yields a prefix-closed set of unique representatives for a finitely generated monoid and the existence of an s-regular canonical prefix-rewriting system presenting that monoid.