Showing papers on "Undecidable problem published in 1990"

Model-checking for real-time systems

Abstract: This research extends CTL model-checking to the analysis of real-time systems, whose correctness depends on the magnitudes of the timing delays. For specifications, the syntax of CTL is extended to allow quantitative temporal operators. The formulas of the resulting logic, TCTL, are interpretation over continuous computation trees, trees in which paths are maps from the set of nonnegative reals to system states. To model finite-state systems the notion of timed graphs is introduced-state-transition graphs extended with a mechanism that allows the expression of constant bounds on the delays between the state transition. As the main result, an algorithm is developed for model checking, that is, for determining the truth of a TCTL formula with respect to a timed graph. It is argued that choosing a dense domain, instead of a discrete domain, to model time does not blow up the complexity of the model-checking problem. On the negative side, it is shown that the denseness of the underlying time domain makes TCTL II/sub 1//sup 1/-hard. The question of deciding whether a given TCTL formula is implementable by a timed graph is also undecidable. >

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.

Distributed reactive systems are hard to synthesize

Abstract: The problem of synthesizing a finite-state distributed reactive system is considered. Given a distributed architecture A, which comprises several processors P/sub 1/, . . ., P/sub k/ and their interconnection scheme, and a propositional temporal specification phi , a solution to the synthesis problem consists of finite-state programs Pi /sub 1/, . . ., Pi /sub k/ (one for each processor), whose joint (synchronous) behavior maintains phi against all possible inputs from the environment. Such a solution is referred to as the realization of the specification phi over the architecture A. Specifically, it is shown that the problem of realizing a given propositional specification over a given architecture is undecidable, and it is nonelementarily decidable for the very restricted class of hierarchical architectures. An extensive characterization of architecture classes for which the realizability problem is elementarily decidable and of classes for which it is undecidable is given. >

Unpredictability and undecidability in dynamical systems

Abstract: We show that motion with as few as three degrees of freedom (for instance, a particle moving in a three-dimensional potential) can be equivalent to a Turing machine, and so be capable of universal computation. Such systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos: Even if the initial conditions are known exactly, virtually any question about their long-term dynamics is undecidable.

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

On the predictability of coupled automata: an allegory about chaos

Abstract: The authors show a sharp dichotomy between systems of identical automata with symmetric global control whose behavior is easy to predict and those whose behavior is hard to predict. The division pertains to whether the global control rule is invariant with respect to permutations of the states of the automaton. It is also shown that testing whether the global control rule has this invariance property is an undecidable problem. It is argued that there is a natural analog between complexity in the present model and chaos in dynamical systems. >

Polynomial time query processing in temporal deductive databases

Abstract: We study conditions guaranteeing polynomial time computability of queries in temporal deductive databases. We show that if for a given set of temporal rules, the period of its least models is bounded from the above by a polynomial in the database size, then also the time to process yes-no queries (as well as to compute finite representations of all query answers) can be polynomially bounded. We present a bottom-up query processing algorithm BT that is guaranteed to terminate in polynomial time if the periods are polynomially bounded. Polynomial periodicity is our most general criterion, however it can not be directly applied. Therefore, we exhibit two weaker criteria, defining inflationary and I-periodic sets of temporal rules. We show that it can be decided whether a set of temporal rules is inflationary. I-periodicity is undecidable (as we show), but it can be closely approximated by a syntactic notion of multi-separability.

Method schemas

Abstract: The concept of method schemas is proposed as a simple model for object-oriented programming with features such as classes with methods and inheritance, method name overloading, and late binding. An important issue is to check whether a given method schema can possibly lead to inconsistencies in some interpretations. The consistency problem for method schemas is studied. The problem is shown to be undecidable in general. Decidability is obtained for monadic and/or recursion-free method schemas. The effect of covariance is considered. The issues of incremental consistency checking and of a sound algorithm for the general case are briefly discussed.

The computability and complexity of optical beam tracing

Abstract: The ray-tracing problem is considered for optical systems consisting of a set of refractive or reflective surfaces. It is assumed that the position and the tangent of the incident angle of the initial light ray are rational. The computability and complexity of the ray-tracing problems are investigated for various optical models. The results show that, depending on the optical model, ray tracing is sometimes undecidable, sometimes PSPACE-hard, and sometimes in PSPACE. >

On subsumption and semiunification in feature algebras

Abstract: A generalization of term subsumption, or matching, to a class of mathematical structures called feature algebras is discussed It is shown how these generalize both first-order terms and the feature structures used in computational linguistics The notion of subsumption generalizes to a natural notion of homomorphism between elements of these algebras, and the authors characterize the notion, showing how it corresponds to a mapping which preserves partial information In the setting of feature algebras, unification corresponds naturally to solving constraints involving equalities between strings of unary functions symbols, and semiunification also allows inequalities representing subsumption constraints Their generalization allows the authors to show that the semiunification problem for finite feature algebras is undecidable This implies that the corresponding problem for rational trees (cyclic terms) is also undecidable Thus a partial solution to the decidability of this problem, which has been open for several years, is produced >

Undecidability principle and the uncertainty principle even for classical systems.

Abstract: It is shown that for any physical system there is an infinite number of measurements which are related to one of the undecidable problems. The undecidability of the physical systems indicates that there is an infinite number of incomputable correlation functions for each physical system. These results show that there is an inherent and irreducible limitation on the knowledge of the nature of physical systems. This uncertainty principle is caused by the undecidability principle.

A Unified Approach to the Deadlock Detection Problem in Networks of Communicating Finite State Machines

Abstract: We consider the deadlock detection problem (DDP) of networks of communicating finite state machines (NCFSMs). The DDP problem is known to be undecidable for NCFSMs. In this paper, we provide a characterization of those subclasses of networks for which the deadlock problem is decidable. We also provide a proof technique based on our characterization and illustrate our technique on an example.

Recursively indefinite databases

Abstract: We define recursively indefinite databases, a new type of logical database in which indefinite information arises from partial knowledge of the fixpoint of a datalog program. Although, in general, query answering is undecidable, there exists a broad class of queries for which it is decidable, a result we establish by making connections with the theory of hypergraph edge replacement graph grammars. We analyze the complexity of query answering for this class of queries under various constraints and demonstrate a class of databases which generalizes disjunctive databases, but without increasing data complexity.

Some Results on Equational Unification

Abstract: A clear distinction is made between the (elementary) unification problem where there is only one pair of terms to be unified, and the problem where many such pairs have to be simultaneously unified — it is shown that there exists a finite, depth-reducing, and confluent term-rewriting system R such that the (single) E-unification problem mod R is decidable, while the simultaneous E-unification problem is undecidable It is also shown that E-unification is undecidable for variable-permuting theories, thus settling an open problem The corresponding E-matching problem is shown to be PSPACE-complete

IDLOG: extending the expressive power of deductive database languages

Abstract: The expressive power of pure deductive database languages, such as DATALOG and stratified DATALOGS, is limited in a sense that some useful queries such as functions involving aggregation are not definable in these languages. Our concern in this paper is to provide a uniform logic framework for deductive databases with greater expressive power. It has been shown that with a linear ordering on the domain of the database, the expressive power of some database languages can be enhanced so that some functions involving aggregation can be defined. Yet, a direct implementation of the linear ordering in deductive database languages may seem unintuitive, and may not be very efficient to use in practice. We propose a logic for deductive databases which employs the notion of “identifying each tuple in a relation”. Through the use of these tuple-identifications, different linear orderings are defined as a result. This intuitively explains the reason why our logic has greater expressive power. The proposed logic language is non-deterministu in nature. However, non-determinism is not the real reason for the enhanced expressive power. A deterministic subset of the programs in this language is computational complete in the sense that it defines all the computable deterministic queries. Although the problem of deciding whether a program is in this subset is in general undecidable, we do provide a rather general sufficient test for identifying such programs. Also discussed in this paper is an extended notion of queries which allows both the input and the output of a query to contain interpreted constants of an infinite domain. We show that extended queries involving aggregation can also be defined in the language.

Design Strategies for Rewrite Rules

Abstract: Term rewriting systems employed for the specification of abstract data types and for very high level programming languages based on rewrite rules are often required to have properties, such as confluence, termination, and sufficient-completeness, which are undecidable. Rather than attempting, as it is usually done, to check these properties in a system a posteriori, i.e. after the rules have been designed, we propose two strategies for addressing this problem a priori, i.e. during the design phase of rules. We propose the concepts of under- and over-specification, determine sufficient and/or necessary conditions to avoid them, show how to obtain these conditions in a constructive way, and relate them to a number of desirable properties which have appeared in the literature. Our approach is based on the completeness and parsimony properties of sets of tuples of terms and on a recursive mechanism which extends primitive recursion from natural numbers to abstract data types. We prove a number of results, illustrate their application to the design of rewriting systems by means of examples, and discuss the power and the limitations of our approach.

Combinatorial rewriting on traces

Abstract: There are two main problems in working with replacement systems over free partially commutative monoids: For finite noetherian systems confluence is undecidable, in general, and the known algorithm to compute irreducible normal forms need time square in the derivation length instead of linear. We first give a decidable and sufficient condition for finite noetherian systems such that confluence becomes decidable. This condition is weaker than the known ones before. Then we give a decidable and sufficient condition such that irreducible normal forms are computable in time linear to the derivation length. Furthermore, we prove that the first condition is implied by the second. We also present a new uniform algorithm for computing normal forms using Zielonka's theory of asynchronous automata.

Hard problems for simple logic programs

Abstract: A number of optimizations have been proposed for Datalog programs involving a single intensional predicate (“single-IDB programs”). Examples include the detection of commutativity and separability ([Naug88],[RSUV89], [Ioan89a]) in linear logic programs, and the detection of ZYT-linearizability ([ZYT88], [RSUV89], [Sara89], [Sara90]) in nonlinear programs. We show that the natural generalizations of the commutativity and ZYT-linearizability problems (respectively, the sequencability and base-case linearizability problems) are undecidable. Our constructions involve the simulation of context-free grammars using single-IDB programs that have a bounded number of initialisation rules. The constructions may be used to show that containment (or equivalence) is undecidable for such programs, even if the programs are linear, or if each program contains a single recursive rule. These results tighten those of [Shmu87] and [Abit89].

Computation, logic, philosophy : a collection of essays

Abstract: One. Broad Issues.- 1. On Formalization.- 1.1 Systematization [1955(53)].- 1.2 Communication.- 1.3 Clarity and consolidation.- 1.4 Rigour.- 1.5 Approximation to intuition.- 1.6 Application to philosophy.- 1.7 Too many digits.- 1.8 Ideal language.- 1.9 How artificial language?.- 1.10 The paradoxes.- 2. The Concept of Computability [(1953)].- 2.1 Formalizing intuitive concepts.- 2.2 The intuitive concept of computability.- 2.3 Computation by theoretical machines.- 2.4 General recursive functions.- 2.5 Constructive proofs.- 2.6 Effective methods.- 2.7 Speed functions.- 2.8 Transfinite recursions.- 2.9 The indeterminate domain of computable functions.- 3. Process and Existence in Mathematics [1961(60)].- 4. Logic, Computation and Philosophy [1971(66)].- 4.1 Logic and logical positivism.- 4.2 What is mathematics?.- 4.3 Logic and computation.- 4.4 Relatively undecidable propositions and absolutely unsolvable problems.- 4.5 Foundations of set theory.- 4.6 What is mathematics? (continued).- Two. Automated Theorem Proving(ATP).- 5. Computer Theorem Proving and Artificial Intelligence [1984(82)].- Appendix: Citation for Haowang as Winner of "Milestone" Award in Automatic Theorem-Proving.- 6. Proving Theorems by Pattern Recognition, I [1960(59)].- 6.1 Introduction.- 6.2 A program that does 9 chapters of Principia in 9 minutes.- 6.3 The E1A case solved with sequential tables.- 6.4 General remarks.- 7. Observations on ATP.- 7.1 Mechanical mathematics and inferential analysis [1963(61)].- 7.2 The mechanization of mechanical arguments [1963(62)a].- 7.3 Formalization and automatic theorem-proving [1965(64)].- 8. Some Data for ATP.- 8.1 On axioms of conditional set existence [1967(66)].- 8.2 Natural hulls and set existence [1967(66)a].- 8.3 A theorem on definitions of the Zermelo-Neumann ordinals [1967(66)b].- 9. Proving Theorems by Pattern Recognition, II [1961(60)a].- 9.1 A survey of the decision problem.- 9.2 The Skolem, case.- 9.3 The A2E satisfiability case.- 9.4 The A1E1A1 satisfiability case.- 9.5 A proof procedure for the predicate calculus.- 9.6 Remarks on mathematical disciplines.- Three. Decidability and Complexity.- 10. Games, Logic and Computers [1965a].- Appendix: Notes on a Class of Tiling Problems [1975(60)].- 11. Dominoes and the AEA Case of the Decision Problem [1963(62)].- 12. Towards Feasible Solutions of the Tautology Problem (with B.Dunhan) [1976(74)].- 12.1 Computational complexity and Boolean validity.- 12.2 A brief overview with some general observations.- 12.3 Some basic properties of Boolean validity.- 12.4 Some calculations and classifications.- 12.5 Hard examples and negative results.- 12.6 A feasible decision procedure for biconditional expressions.- 12.7 Two partial methods and an indication of two generic methods.- 13. Ranked Matching and Hospital Interns (with D.A.Martin) [(1977)].- 13.1 Preliminary.- 13.2 Deletion of useless names: Operations I and II.- 13.3 The canonical form T1 of T.- 13.4 The student and hospital optimal assignments.- 13.5 Mixed assignments and a characterization of all stable assignments.- 13.6 The marriage problem.- Four. Topics from Theory to Practice.- 14. Logical Fragments Relevant to Computer Science.- 14.1 Logic of many-sorted theories [1952(50)].- 14.2 Ackermann's consistency proof [1962(53)].- 14.3 Partial systems of number theory [1962(55)].- 14.4 The calculus of partial predicates and its extension to set theory [1961(61)].- 14.5 Model theory [1974(71)].- 15. Computers and Mathematical Activity.- 15.1 Remarks on machines, sets and the decision problem [1964(63)].- 15.2 Logic and computers [1965].- 15.3 Remarks on mathematics and computers [1970(67)].- 15.4 On the long-range prospects of automatic theorem-proving [1970(68)a].- 16. On Information Processing of the Chinese Language [1979].- The List of the Publications of the Author.

Integration of functions in logic database systems

Abstract: We extend Datalog, a logic programming language for rule-based systems, by respectively integrating types, negation and functions. This extention of Datalog is called MilAnt. Furthermore, MilAnt consistency is defined as a stronger form of consistency for functions. It is known that consistency for functions is undecidable. We prove that MilAnt consistency is decidable and an algorithm is given to detect the MilAnt consistency of a MilAnt program. To this end, we use a mixture of dependencies that are local to a rule and dependencies that are global for the whole program.

Deciding properties of nonregular programs

Abstract: The problem of deciding the validity of formulas in extensions of propositional dynamic logic (PDL) is considered. The extensions are obtained by adding programs defined by nonregular languages. In the past, a number of very simple languages were shown to render this problem highly undecidable, whereas other very similar-looking languages were shown to retain decidability. Understanding this rather strange phenomenon and generalizing the isolated extensions have remained elusive. The authors provide decision procedures for two wide classes of extensions, thus shedding light on the general problem. The proofs are novel, in that they explicitly consider the machines that accept the languages, in this case special classes of PDAs and stack automata. It is shown that the emptiness problem for stack automata on infinite trees is decidable, a result of independent interest, and the result is combined with the construction of certain tree models for the corresponding formulas. >

The undecidability of the second order predicate unification problem

Abstract: We prove that the second order predicate unification problem is undecidable by reducing the second order term unification problem to it.