Showing papers on "Undecidable problem published in 1991"

Automata on infinite objects

Abstract: Publisher Summary This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.

Benefits of Relaxing Punctuality

Abstract: : The most natural, compositional way of modeling real time systems uses a dense domain for time. The satisfiability of real time constraints that are capable of expressing punctual it in this model is, however, known to be undecidable. The authors introduce a temporal language that can constrain the time difference between events only with finite (yet arbitrary) precision and show the resulting logic to be EXPACE-complete. This result allows the authors to develop an algorithm for the verification of timing properties of real time systems with a dense semantics.

The benefits of relaxing punctuality

Abstract: The most natural, compositional, way of modeling real-time systems uses a dense domain for time. The satistiability of timing constraints that are capable of expressing punctuality in this model, however, is known to be undecidable. We introduce a temporal language that can constrain the time difference between events only with finite, yet arbitrary, precision and show the resulting logic to be EXPSPACE-complete. This result allows us to develop an algorithm for the verification of timing properties of real-time systems with a dense semantics. Categories and Subject Descriptors: C.3 (Special-Purpose and Application-Based Systems)--real- time systems; D.2.1 (Software Engineeriogk Requirements/Specifications-languages; F.3.1 (Lugics and Meanings of ~rugmms): Specifying and Verifying and Reasoning about Programs-lo& of programs; mechanical verification; specification techniques; F.4.3 (Mathematical Logic and Formal Languages): Formal Languages-classes f decision problems

Generalized shifts: unpredictability and undecidability in dynamical systems

Abstract: A class of shift-like dynamical systems is presented that displays a wide variety of behaviours. Three examples are presented along with some general definitions and results. A correspondence with Turing machines allows one to discuss issues of predictability and complexity. These systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos, and many simple questions about their dynamics are undecidable. The author discusses the complexity of various sets they generate, including periodic points, basins of attraction, and time series. Finally, he shows that they can be embedded in smooth maps in R2, or smooth flows in R3.

Sufficient-completeness, ground-reducibility and their complexity

Abstract: The sufficient-completeness property of equational algebraic specifications has been found useful in providing guidelines for designing abstract data type specifications as well as in proving inductive properties using the induction-less-induction method. The sufficient-completeness property is known to be undecidable in general. In an earlier paper, it was shown to be decidable for constructor-preserving, complete (canonical) term rewriting systems, even when there are relations among constructor symbols. In this paper, the complexity of the sufficient-completeness property is analyzed for different classes of term rewriting systems. A number of results about the complexity of the sufficient-completeness property for complete (canonical) term rewriting systems are proved: (i) The problem is co-NP-complete for term rewriting systems with free constructors (i.e., no relations among constructors are allowed), (ii) the problem remains co-NP-complete for term rewriting systems with unary and nullary constructors, even when there are relations among constructors, (iii) the problem is provably in “almost” exponential time for left-linear term rewriting systems with relations among constructors, and (iv) for left-linear complete constructor-preserving rewriting systems, the problem can be decided in steps exponential innlogn wheren is the size of the rewriting system. No better lower-bound for the complexity of the sufficient-completeness property for complete (canonical) term rewriting system with nonlinear left-hand sides is known. An algorithm for left-linear complete constructor-preserving rewriting systems is also discussed. Finally, the sufficient-completeness property is shown to be undecidable for non-linear complete term rewriting systems with associative functions. These complexity results also apply to the ground-reducibility property (also called inductive-reducibility) which is known to be directly related to the sufficient-completeness property.

Classical physics and Penrose's thesis

Abstract: We expose and discussPenrose's thesis: “Nature produces harnessable noncomputable processes, but none at the classical level.” We then suggest a partial counterexample to it, based on aGedanken experiment about an undecidable family of integrable Hamiltonian systems that could lead to a sort of idealized solution to the Halting problem for Turing machines.

Subtree-elimination algorithms in deductive databases

Abstract: A deductive database consists of a set of stored facts, and a set of logical rules (typically, recursive Horn clauses) that are used to manipulate these facts. A number of optimizations in such databases involve the transformation of sets of logical rules (programs) to simpler, more efficiently evaluable programs. We consider a class of optimizations in which the transformation is a simple syntactic restriction on the form of the original program, and in which the correctness of the transformation indicates the existence of a normal form for the proof trees generated by the program. For example, the existence of basis-linearizability in a nonlinear program indicates that the program is inherently linear, and permits the use of special-purpose query evaluators for linear recursions. The canonical example of a basis-linearizable program is the program that computes the transitive closure of a binary relation; the corresponding normal form for the proof trees is that of right-linearity. Similarly, if a program is sequencable, then it is conducive to a pipelined evaluation. In addition, the existence of k-boundedness in a program permits the elimination of recursion overhead in evaluating the program. We investigate the complexity of detecting such optimization opportunities, and provide correct (but not always complete) algorithms for this purpose. Each of the problems that are mentioned above may be described in terms of the subtree-elimination problem, which we define and analyze. We relate the detection of basis-linearizability, sequencability and 1-boundedness to the complexity classes ${\cal NC}$, ${\cal P}$ and ${\cal NP}$, and show that the first two of these problems are, in general, undecidable. The techniques used in our analysis provide a complete description of the complexity of deciding the equivalence of conjunctive queries (single-rule, nonrecursive programs), and tight undecidability results for the detection of program equivalence.

Decision problems for term rewriting systems and recognizable tree languages

Abstract: We study the connections between recognizable tree languages and rewrite systems. We investigate some decision problems. Particularly, let us consider the property (P): a rewrite system S is such that, for every recognizable tree language F, the set of S-normal forms of terms in F is recognizable too. We prove that the property (P) is undecidable. We prove that the existential fragment of the theory of ground term algebras modulo a congruence \(\mathop \leftrightarrow \limits^* E\) generated by a set E of equations such that there exists a finite, noetherian, confluent rewrite system S satisfying (P) with \(\mathop \leftrightarrow \limits^* S = \mathop \leftrightarrow \limits^* E\) is undecidable. Nevertheless, we develop a decision procedure for the validity of linear formulas in a fiagment of such a theory.

On the complexity of ML typeability with overloading

Abstract: We examine the complexity of type checking in an ML-style type system that permits functions to be overloaded with different types. In particular, we consider the extension of the ML type system proposed by Wadler and Blott in the appendix of [WB89], with global overloading only, that is, where the only overloading is that which exists in an initial type assumption set; no local overloading via over and inst expressions is allowed. It is shown that under a correct notion of well-typed terms, the problem of determining whether a term is well typed with respect to an assumption set in this system is undecidable. We then investigate limiting recursion in assumption sets, the source of the undecidability. Barring mutual recursion is considered, but this proves too weak, for the problem remains undecidable. Then we consider a limited form of recursion called parametric recursion. We show that although the problem becomes decidable under parametric recursion, it appears harder than conventional ML typability, which is complete for DEXPTIME [Mai90].

Linear bounded automata and rewrite systems: influence of initial configurations on decision properties

Abstract: We prove that termination is undecidable for non-length-increasing string rewriting systems, using linear-bounded automata. On the other hand, we prove the undecidability of confluence for terminating rewriting systems when terms begin by a fixed symbol. These two results illustrate that sometimes restriction of problem to recognizable domains modify decidability properties, sometimes it does not. (We only consider finite terms).

Hamiltonian paths in infinite graphs

Abstract: A tight connection is exhibited between infinite paths in recursive trees and Hamiltonian paths in recursive graphs. A corollary is that determining Hamiltonicity in recursive graphs is highly undecidable, viz, Σ 1 1 -complete. This is shown to hold even for highly recursive graphs with degree bounded by 3. Hamiltonicity is thus an example of an interesting graph problem that is outside the arithmetic hierarchy in the infinite case.

Hamiltonian paths in infinite graphs

Abstract: A tight connection is exhibited between infinite paths in recursive trees and Hamiltonian paths in recursive graphs. A corollary is that determining Hamiltonicity in recursive graphs is highly undecidable, viz, Σ 1 1 -complete. This is shown to hold even for highly recursive graphs with degree bounded by 3. Hamiltonicity is thus an example of an interesting graph problem that is outside the arithmetic hierarchy in the infinite case.

Undecidable equivalences for basic process algebra

Abstract: A recent theorem shows that strong bisimilarity is decidable for the class of normed BPA processes, which correspond to a class of context-free grammars generating the ϵ-free context-free languages. Huynh and Tian (Technical Report UTDCS-31-90, University of Texas at Dallas, 1990) have shown that readiness and failure equivalence are undecidable for BPA processes. In this paper we examine all other equivalences in the linear/branching time hierarchy and show that none of them are decidable for normed BPA processes.

Decidability, Behavioural Equivalences and Infinite Transition Graphs

Abstract: This thesis studies behavioural equivalences on labelled infinite transition graphs and the role that they can play in the context of modal logics and notions from language theory. A natural class of such infinite graphs is that corresponding to the SnS -definable tree languages first studied by Rabin. We show that a modal mu-calculus with label set f0; : : : ; n 1g can define these tree languages up to an observational equivalence. Another natural class of infinite transition graphs is that of normed BPA processes, which correspond to the graphs of leftmost derivations in context-free grammars without useless productions. A remarkable result is that strong bisimulation is decidable for these graphs. After an outline of the existing proofs due to Baeten et al. and Caucal we present a much simpler proof using a tableau system closely related to the branching algorithms employed in language theory following Korenjak and Hopcroft. We then present a result due to Colin Stirling, giving a weakly sound and complete sequent-based equational theory for bisimulation equivalence for normed BPA processes from the tableau system. Moreover, we show how to extract a fundamental relation (as in the work of Caucal) from a successful tableau. We then introduce silent actions and consider a class of normed BPA processes with the restriction that processes cannot terminate silently, showing that the decidability result for strong bisimilarity can be extended to van Glabbeek’s branching bisimulation equivalence for this class of processes. We complete the picture by establishing that all other known behavioural equivalences and a number of preorders are undecidable for normed BPA processes.

Fast left-linear semi-unification

Abstract: Semi-unification is a generalization of both unification and matching with applications in proof theory, term rewriting systems, polymorphic type inference, and natural language processing. It is the problem of solving a set of term inequalities M1 ≤ N1, ..., M k <- N k , where ≤ is interpreted as the subsumption preordering on (first-order) terms. Whereas the general problem has recently been shown to be undecidable, several special cases are decidable.

An undecidable problem for countable markov chains

Abstract: A nondeterministic defense system is considered for a linearly ordered sequences of nodes of a countable Markov chain with an environment. We prove undecidability of the problem of existence of a sequence of signals from attacking subsystems that render the central nodes completely defenseless.

Undecidable Properties of Syntactic Theories

Abstract: Since we are looking for unification algorithms for a large enough class of equational theories, we are interested in syntactic theories because they have a nice decomposition property which provides a very simple unification procedure. A presentation is said resolvent if any equational theorem can be proved using at most one equality step at the top position. A theory which has a finite and resolvent presentation is called syntactic. In this paper we give decidability results about open problems in syntactic theories: unifiability in syntactic theories is not decidable, resolventness of a presentation and syntacticness of a theory are even not semidecidable. Therefore we claim that the condition of syntacticness is too weak to get unification algorithms directly.

Computability theory : concepts and applications

Abstract: Turing machines turing machines as recognisers universality undecidability alternative models post's correspondence problem recursive function theory formal models of arithmetic Godel's incompleteness theorem computer science and computability theory.

Proof schemata in Hilbert-type axiomatic theories

Abstract: A derivation schema in an axiomatic theory is defined as a finite sequence of analysis of applications of rules and axioms. A derivation by a schema U is any derivation whose list of analyses of applications of rules and axioms is precisely U. A derivation schema is admissible if a corresponding derivation can be constructed. Let G be a Hilbert-type axiomatic theory. The following problems are considered: a) to decide whether a given derivation scheme is admissible in G; b) to decide whether a formula is derivable by a given derivation schema in G. In the usual formulations of the predicate calculus without equality, the first problem is shown to be decidable, the second undecidable.

Infinitary queries and their asymptotic probabilities I: Properties definable in transitive closure logic

Abstract: We present new general method for proving that for certain classes of finite structures the limit law fails for properties expressible in transitive closure logic. In all such cases also all associated asymptotic problems are undecidable.