Showing papers on "Undecidable problem published in 1989"

On the complexity of space bounded interactive proofs

Abstract: Two results on interactive proof systems with two-way probabilistic finite-state verifiers are proved. The first is a lower bound on the power of such proof systems if they are not required to halt with high probability on rejected inputs: it is shown that they can accept any recursively enumerable language. The second is an upper bound on the power of interactive proof systems that halt with high probability on all inputs. The proof method for the lower bound also shows that the emptiness problem for one-way probabilistic finite-state machines is undecidable. In the proof of the upper bound some results of independent interest on the rate of convergence of time-varying Markov chains to their halting states are obtained. >

Safety of datalog queries over infinite databases

Abstract: A query is safe with respect to a set of constraints if for every database that satisfies the constraints the query is guaranteed to yield a finite set of answers. We study here the safety problem for Datalog programs with respect to finiteness constraints. We show that safety can be viewed as a combination of two properties: weak safety, which guarantees the finiteness of intermediate answers, and termination, which guarantees the finiteness of the evaluation. We prove that while weak safety is decidable, termination is not. We then consider monadic programs, i.e., programs in which all intensional predicates are monadic, and show that safety is decidable in polynomial time for monadic programs. While we do not settle the safety problem, we show that a closely related problem, the decision problem for safety with respect to functional dependencies, is undecidable even for monadic programs.

Decidability and expressiveness for first-order logics of probability

Abstract: Decidability and expressiveness issues for two first-order logics of probability are considered. In one the probability is on possible worlds, whereas in the other it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. It is shown that, when the probability is on the domain, if the language contains only unary predicates, then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is Pi /sub 1//sup 2/ as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, Pi /sub infinity //sup 1/. Thus, the logic cannot be axiomatized in either case. When the probability is on the set of possible worlds, the validity problem is Pi /sub 1//sup 2/ complete with as little as one unary predicate in the language, even without equality. With equality, Pi /sub infinity //sup 1/ hardness with only a constant symbol is obtained. In many applications it suffices to restrict attention to domains of a bounded size; it is shown that the logics are decidable in this case. >

Formal analysis of real-time equational rule-based systems

Abstract: A study is made of the real-time performance of a class of rule-based programs written in the language EQL. Response time is defined in terms of the computation paths of a program leading to fixed points; investigated is the complexity of the problem of analyzing these programs to meet response-time requirements. It is shown that the response-time analysis problem is in general undecidable and is PSPACE-hard in the case where all the variables have finite domains. A general analysis strategy which seems to be quite effective in practical cases is proposed. This strategy aims at avoiding the combinatorial state-space explosion problem inherent in brute-force approaches. Based on this strategy, a suite of analysis tools has been implemented to verify that the variables in an EQL program always converge to stable values in bounded time. The tools have been successfully applied to real-life programs. >

The undecidability of the DA-unification problem

Abstract: We show that the D A -uniflcation problem is undecidable. That is, given two binary function symbols ⊕ and ⊗, variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following D A -axioms hold: Two terms are D A -unifiable (i.e. an equation is solvable in D A ) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory D A . This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained.

On the strength of the interpretation method

Abstract: In spite of the fact that true arithmetic reduces to the monadic second-order theory of the real line, Peano arithmetic cannot be interpreted in the monadic second-order theory of the real line. ?0. Introduction. The decision problem for the monadic second-order theory of the real line was posed by Grzegorczyk in 1951 (Gr), and was proved undecidable in 1976 by Shelah (Sh). Shelah reduced the first-order theory of true arithmetic to the monadic second-order theory of the real line. In other words, he constructed an algorithm that, given a sentence p in the first-order language of arithmetic, produces a sentence p' in the monadic second-order language of order such that p is true in the standard model of arithmetic if and only if p' is true in the real line. Shelah's proof was analyzed, strengthened and generalized in several papers by the present authors (see (Gu2)). In particular, the following somewhat strange results were proved in (GS): Let V be a model of ZFC, B the Boolean algebra of regular open subsets of the real line in V, and VB the corresponding Boolean-valued model of ZFC. The full second-order VB-theory of arithmetic reduces to the monadic second-order V- theory of the real line. If V satisfies the continuum hypothesis then the full second- order VB-theory of the real line reduces to the monadic second-order V-theory of the real line. In spite of these strong reducibility results, we prove in this paper that Peano arithmetic is not interpretable in the monadic second-order theory of the real line. Actually, we prove a stronger result which is also more convenient to prove. To formulate the stronger result we need a couple of definitions. DEFINITION 0.1. We define a first-order theory with equality which will be called the weak set theory. The signature of the weak set theory consists of one binary predicate symbol P, and the axioms of the weak set theory are as follows: (a) Vx3yVz(P(z, y) +-* z = x). (b) VxVy3uVz(P(z, u) +-+ (P(z, x) or P(z, y))).

On the first-order expressibility of recursive queries

Abstract: A Datalog program is bounded iff it is equivalent to a recursion-free Datalog program. We show that, for some classes of Datalog programs, expressibility in first-order query languages coincides with boundedness. Our results imply that testing first-order expressibility is undecidable for binary programs, decidable for monadic programs, and complete for S02.

Combining Matching Algorithms: The Rectangular Case

Abstract: The problem of combining matching algorithms for equational theories with disjoint signatures is studied. It is shown that the combined matching problem is in general undecidable but that it becomes decidable if all theories are regular. For the case of regular theories an efficient combination algorithm is developed. As part of that development we present a simple algorithm for solving the word problem in the combination of arbitrary equational theories with disjoint signatures.

Word Problems over TRaces Which are Solvable in Linear Time

Abstract: Finite complete replacement systems over free partially commutative monoids have a decidable word problem, and if these systems are weight-reducing, then the word problem is solvable in square time. It is unknown whether this time bound is optimal; but the restriction to semi-Thue systems or vector replacement systems yields linear time. In direction to close this gap we first give a global condition for trace replacement systems which insures a linear time bound, too; and we prove that this condition is decidable. Then we give a stronger but local condition which leads to the notion of cones and blockes which are certain traces. Trivial examples of cones and blocks are words and vectors respectively. We show that under a mild additional assumption we may decide the confluence of a noetherian trace replacement system if all left-hand sides are cones or blocks. (In general confluence is undecidable even for length-reducing systems.) Altogether we obtain a sufficient and computable condition such that the word problem of a trace replacement system is solvable in linear time.

Analysis of communicating processes for non-progress

Abstract: The problem of testing two processes (specified as finite-state machines) communicating asynchronously with each other using send and receive commands over a set of message types is considered for two forms of nonprogress: deadlock and unspecified reception. Since the nonprogress problem is undecidable, a dataflow approach is used to obtain sufficient conditions under which the two processes are free of deadlock and unspecified reception. The approximation analysis is based on weakening the receive operation. Polynomial time algorithms are presented to perform the analysis. This problem arises in the context of dataflow analysis of the processes that communicate by message passing and in the context of showing correctness of protocol specifications. Diagrams are provided for some networks that can be certified to be free of unspecified receptions using the algorithms. The problem of testing for deadlock in more than two processes still remains open. >

Computational problems in equational theorem proving

Abstract: We study computational aspects of equational reasoning related to the complexity of determining whether a term rewriting system is terminating, and to automated theorem proving in first-order logic. Detection of homeomorphic embedding can be used as an aid in detecting potential nontermination of rewriting. We show that, given a pair of ground terms, the problem of determining the existence of homeomorphic embedding of one term into another is NP-complete when functions may be associative and commutative, solving an open problem posed in (46). We present polynomial algorithms for a number of restricted cases. We show that determining whether there exists a matching that results in a homeomorphic embedding of one term into another is NP-complete. We demonstrate a polynomial algorithm for detecting whether, given two terms s and t, s is semi-unifiable with t. This procedure is also useful for detecting nontermination of rewriting. We show that it is undecidable whether the Knuth-Bendix completion procedure generates a crossed pair of rules. This resolves an open question posed in (23), where the applicability of such a test to detection of divergence is discussed. Our proof technique generalizes: we provide a simple proof that the universal matching problem is undecidable for regular canonical theories, a result first proved in (21). We also prove that the universal unification problem is undecidable for permutative canonical theories, and discuss other generalizations. We move to an examination of the use of equational methods for reasoning in propositional calculus, which can be expressed in terms of computing the Grobner Basis of a set of polynomials over a Boolean polynomial ring. We show that exponential time and space are necessary and sufficient for computing such a basis. We show that the exponential lower bound holds even when the original basis corresponds to a set of propositional Horn clauses. We show that even under very limiting restrictions on the input form of a set of first-order polynomials it remains undecidable whether the set is consistent. We examine the effect of limiting inference to idempotent inference and reduction. We show that it is undecidable whether this procedure will terminate, even when the system is guaranteed to be noetherian. We discuss experimental results based on two strategies based upon these restrictions.

Homogeneous models and almost decidability

Abstract: Countable homogeneous models are ‘simple’ objects from a model theoretic point of view. From a recursion theoretic point of view they can be complex. For instance the elementary theory of such a model might be undecidable, or the set of complete types might be recursively complex. Unfortunately even if neither of these conditions holds, such a model still can be undecidable. This paper investigates countable homogeneous models with respect to a weaker notion of decidability called almost decidable . It is shown that for theories that have only countably many type spectra, any countable homogeneous model of such a theory that has a Σ 2 type spectrum is almost decidable.

It is undecidable whether the Knuth-Bendix completion procedure generates a crossed pair

Abstract: We reduce an instance of Turing machine acceptance to the problem of detecting whether the Knuth-Bendix completion procedure generates a crossed pair of rules. This resolves an open question posed in [5]. Our proof technique generalizes; using similar reductions, we can show that a number of other questions related to whether the Knuth-Bendix completion procedure generates certain types of rules are all undecidable. We suggest that the techniques illustrated herein may be useful in answering a number of related questions about the Knuth-Bendix completion procedure, and discuss several examples; in particular, we demonstrate how our construction provides a simple proof that the universal matching problem is undecidable for regular canonical theories, a result first proved in [4], and prove that the universal unification problem is undecidable for permutative canonical theories.

Time-Bounded Reasoning in First Order Knowledge Base Systems

Abstract: In first order predicate logic, it is undecidable whether a formula is deducible from a set of axioms. In order to realize a practical knowledge base system in the framework of the first order logic, we must overcome this problem.

About Connections Between Syntactical and Computational Complexity

Abstract: We prove in this paper three kinds of results: 1/ It is undecidable whether a term rewriting system preserves recognizability. 2/ ground term rewriting systems with recognizable control are as powerfull as general rewriting systems. 3/ Influence on decidability and complexity of termination of syntactical restrictions on term rewriting systems reduced to one or two rules.

Integration of Functions Defined with Rewriting Rules in Datalog.

Abstract: This work is devoted to the integration of functions in Datalog. Functions are defined with a rewrite relation. We define a fixpoint semantics for Horn programs containing both relations and rewriting rules. The principal contribution is the bounded semantics. We study the two following problems: the computability of the fixpoint in presence of an infinite universe and the completeness of the semantics restricted to interpretations containing only bounded depth terms. We prove that both problems are undecidable in general, but decidable subcases are presented.

A practical approach to semantic configuration management

Abstract: A configuration management (CM) tool is supposed to build a consistent software system following incremental changes to the system. The notion of consistency usually is purely syntactic, having to do with the sorts of properties analyzed by compilers. Semantic consistency traditionally has been studied in the field of formal methods and has been considered an impractical goal for CM.Although the semantic CM problem is undecidable, it is possible to obtain a structural approximation of the semantic effects of a change in a finite number of steps. Our approximation technique is formalized in logic and is based on information-theoretic properties of programs. The method in its present form applies to many but not all software systems, and it is programming-language independent. To the best of our knowledge, the semantic CM problem has not been formalized previously in nonsemantic terms, and we believe that our simplified formulation offers the potential for considerably more powerful debugging and configuration management tools.

On the equivalence problem for deterministic multitape automata and transducers

Abstract: We discuss the technique for testing the equivalence of two deterministic automata by constructing a language that matches the computations of two equivalent automata on the same input word. Specifically, we propose to use HDTOL languages that are powerful enough to match computations of many equivalent deterministic multitape automata, and at the same time, have nice decidable properties. Using this new technique of HDTOL matching, we show that the inclusion problem between an arbitrary deterministic multitape automaton and a simple one is decidable in both directions. Further, we show that the computations of two simple automata can be HDTOL matched on their common domain. This implies that the equivalence problem for transducers based on simple automata is decidable. The latter result is the best possible since the problem is undecidable even for transducers based on automata with parallel loops.

QM Axiom Representations with Imaginary & Transfinite Numbers and Exponentials

Abstract: A presentation is made showing how imaginary numbers, exponentials, and transfinite ordinals can be given logical meanings that are applicable to the definitions for the axioms of Quantum Mechanics (QM). This is based on a proposed logical definition for axioms which includes an axiom statement and its negation as parts of an undecidable statement which is forced to the tautological truth value: true. The logical algebraic expression for this is shown to be isomorphic to the algebraic expression defining the imaginary numbers ± i (V-l). This supports a progressive and Hegelian view of theory development. This means that thesis and antithesis axioms in the QM theory structure which should be carried along at present could later on be replaced by a synthesis to a deeper theory prompted by subsequently discovered new experimental facts and concepts. This process :ould repeat at a later time since the synthesis theory axioms would then be considered as a lew set of thesis statements from which their paired antithesis axiom statements would be derived. The present epistemological methods of QM, therefore, are considered to be a good way of temporarily leapfrogging defects in our conceptual and experimental knowledge until a deeper determinate theory is found. These considerations bring logical meaning to exponential forms like the Psi and wave functions. This is derives from the set theoretic meaning for simple forms like 2 which is blown to be the set of all subsets of the (discrete) set, A. The equal symbol in equations which are axioms, and all its other symbols, can be mapped to a transfinite ordinal, [maginary exponential forms (like e*″) can be shown to stand for the (continuous) set of all subsets or the set of all experimental situations (which thus includes arbitrary sets of experimental situations) which are based on the axiom, 0, a transfinite ordinal.