Showing papers on "Undecidable problem published in 1987"

Undecidable Optimization Problems for Database Logic Programs

Abstract: Datalog is the language of logic programs without function symbols. It is used as a database query language. If it is possible to eliminate recursion from a Datalog program F’, then t’ is said to be bounded. It is shown that the problem of deciding whether a given Datalog program is bounded is undecidable, even for linear programs (i.e., programs in which each rule contains at most one occurrence of a recursive predicate). It is then shown that every semantic property of Datalog programs is undecidable if it is stable, is strongly nontrivial, and contains An earlier version of this work appeared under the same title in the Proceedings of the 2nd IEEE Symposium on Logic i~z Computer Science (Ithaca, N.Y.). IEEE, New York, 1987, pp. 106-115. Most of the research reported here was done while H. Gaifman was visiting the AI Center of SRI International whose support he wishes to acknowledge. He also wishes to thank IBM Watson Research Center and IBM Almaden Research Center for support in the summer of 1989, when the concluding work on this paper was done. The research reported here was done partly while H. Mairson was at the Computer Science Department of Stanford University and was supported by the Office of Naval Research (ONR) contract NOO014-85-C-0731 and partly while he was at the Programming Research Group of Oxford University. The research reported here was done while Y. Sagiv was visiting the Computer Science Department of Stanford University and was supported by a grant of AT & T Foundation, a grant of IBM Corporation and the National Science Foundation (NSF) grant 1ST 84-12791. Authors’ addresses: H. Gaifman and Y. Sagiv, Hebrew University, Jerusalem 91904, Israel; H. Mairson, Department of Computer Science, Brandeis University, Waltham, MA 02254; M. Y. Vardi, IBM Almaden Research Center, K53-802, 650 Harry Road, San Jose, CA 95120-6099. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage. the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 01993 ACM 0004-5411/93/0700-0683 $01.50 Journal of the Awxmt]on for Computing Machinery, VO1 40, No 3. July 1993. PP 683-713 684 H. GAIFMAN ET AL. boundedness. In particular, the property of being first-order 1s undecidable and (assuming that PTIME 1s different from LOGSPACE and from NC) the same holds for the property of being equivalent to a linear program and for the properties of being in LOGSPACE and of being in NC

The problems of mathematics

Abstract: We are living in the Golden Age of mathematics, with more research being done than ever before. Yet many people view mathematics as a static, completed subject. This book for general readers aims to open the door to the rapid modern growth of mathematics and its power and beauty. It surveys many areas of current research in non-technical terms, describing what the problems are, where they come from, how they get solved, what mathematicians are like, what you can do with the answers when you get them, and how solving them or failing to solve them changes peoples' views of mathematics and the way it is advancing. Topics include Fermat's Last Theorem, the Riemann hypothesis, the Poincare Conjecture, prime numbers, non-Euclidean geometry, infinity, the four-color problem, probability, catastrophe theory, chaos, fractals, algorithms, and undecidable propositions. A final chapter discusses the relations between mathematics and its applications. Each topic is developed within a historical framework, and a number of recent breakthroughs are presented for the first time in layman's terms.

Classifying the Computational Complexity of Problems

Abstract: One of the more significant achievements of twentieth century mathematics, especially from the viewpoints of logic and computer science, was the work of Church, Godel and Turing in the 1930's which provided a precise and robust definition of what it means for a problem to be computationally solvable, or decidable, and which showed that there are undecidable problems which arise naturally in logic and computer science. Indeed, when one is faced with a new computational problem, one of the first questions to be answered is whether the problem is decidable or undecidable. A problem is usually defined to be decidable if and only if it can be solved by some Turing machine, and the class of decidable problems defined in this way remains unchanged if “Turing machine” is replaced by any of a variety of other formal models of computation. The division of all problems into two classes, decidable or undecidable, is very coarse, and refinements have been made on both sides of the boundary. On the undecidable side, work in recursive function theory, using tools such as effective reducibility, has exposed much additional structure such as degrees of unsolvability. The main purpose of this survey article is to describe a branch of computational complexity theory which attempts to expose more structure within the decidable side of the boundary.Motivated in part by practical considerations, the additional structure is obtained by placing upper bounds on the amounts of computational resources which are needed to solve the problem. Two common measures of the computational resources used by an algorithm are time, the number of steps executed by the algorithm, and space, the amount of memory used by the algorithm.

A decidable class of bounded recursions

Abstract: Detecting bounded recursions is a powerful optimization technique for recursions database query languages as bounded recursions can be replaced by equivalent nonrecursive definitions. The problem is of theoretical interest because by varying the class of recursions considered one can generate instances that vary from linearly decidable to NP-hard to undecidable. In this paper we review and clarify the existing definitions of boundedness. We then specify a sample criterion that guarantees that the condition in Vaughton [7] is necessary and sufficient for boundedness. The programs satisfying this criterion subsume and extend previously known decidable classes of bounded linear recursions.

Safety of recursive Horn clauses with infinite relations

Abstract: A database query is said to be safe if its result consists of a finite set of tuples If a query is expressed using a set of pure Horn Clauses, the problem of determining whether it is safe is in general undecidable In this paper, we show that the problem is decidable when terms involving function symbols (including arithmetic) are represented as distinct occurrences of uninterpreted infinite predicates over which certain finiteness dependencies hold. We present a sufficient condition for safety when some monotonicity constraints also hold.

Bounds on the propagation of selection into logic programs

Abstract: We consider the problem of propagating selections (i.e., bindings of variables) into logic programs. In particular, we study the class of binary chain programs and define selection propagation as the task of finding an equivalent program containing only unary derived predicates. We associate a context free grammar L(H) with every binary chain program H. We show that, given H propagating a selection involving some constant is possible iff L(H) is regular, and therefore undecidable. We also show that propagating a selection of the form p(X,X) is possible iff L(H) is finite, and therefore decidable. We demonstrate the connection of these two cases, respectively, with the weak monadic second order theory of one successor and with monadic generalized spectra. We further clarify the analogy between chain programs and languages from the point of view of program equivalence and selection propagation heuristics.

Decidability of the purely existential fragment of the theory of term algebras

Abstract: This paper is concerned with the question of the decidability and the complexity of the decision problem for certain fragments of the theory of free term algebras. The existential fragment of the theory of term algebras is shown to be decidable through the presentation of a nondeterministic algorithm, which, given a quantifier-free formula P, constructs a solution for P if it has one and indicates failure if there are no solutions. It is shown that the decision problem is in NP by proving that, if a quantifier-free formula P has a solution, then there is one that can be represented as a dag in space at most cubic in the length of P. The decision problem is shown to be complete for NP by reducing 3-SAT to that problem. Thus it is established that the existential fragment of the theory of pure list structures in the language of NIL, CONS, CAR, CDR, =, ≤ (subexpression) is NP-complete. It is further shown that even a slightly more expressive fragment of the theory of term algebras, the one that allows bounded universal quantifiers, is undecidable.

Dominical categories: Recursion theory without elements

Abstract: Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem. Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developed without reference to elements . By Godel's generalized incompleteness theorem for consistent arithmetical system T we mean any statement of the following sort: (1) if every recursive set is definable in T , then T is essentially undecidable [41]; or (2) if all recursive functions are definable in T , then T is essentially undecidable [41]; or (3) if every recursive set is definable in T , then T 0 and R 0 (the sets of Godel numbers of the theorems and refutables of T ) are recursively inseparable [39]; or (4) if all re sets are representable in T , then T 0 is creative [28], [39]; or (5) if T is a Rosser theory (i.e., all disjoint re sets are strongly separable in T ), then T 0 and R 0 are effectively inseparable [39].

Decidability questions for fairness in Petri nets

Abstract: For a Petri net and a group of transitions E a procedure is given to decide if there is an infinite firing sequence which is fdp w. r. t. E. An analysis of the complexity shows that even the problem if there is an arbitrary infinite firing sequence in a bounded net is hard for DSPACE(exp). If (strong) fairness is required the problem becomes undecidable, this is even shown for the question if there is an infinite firing sequence which is fair w. r. t. one transition t.

Complexity of place/transition nets

Abstract: This is a survey of results about the complexity of decision problems for various questions about Petri nets that arise in the analysis of systems. The border between undecidable and decidable problems is discussed first and then problems are presented by decreasing complexity. As a consequence of the results presented it will follow that one has to concentrate on very restricted classes of systems in order to get practically relevant algorithms that work well for all cases, since even seemingly simple classes of Petri nets have simple problems with a provable high lower bound for the complexity of their sol’ution.

Detecting looping simplifications

Abstract: A generalization of tree matching and unification algorithms is presented. Given the equation s=t, this algorithm can often quickly determine that the rewrite rule s→t leads to an infinite sequence of “simplifications”. The rule t→s can be tested in the same way. Rules leading to infinite simplifications should not be included in a rewrite system. In general, the problem of deciding whether a set of rewrite rules leads to infinite simplifications is undecidable. The algorithm that is used for this problem is a cross between a unification algorithm for terms with overlapping variables and a matching algorithm. In the simplest case it attempts to find α, σ M and σ U such that σ M σ U =σ U t/a. In other words, is there a substitution σ U such that in the rule σ U s→σ U t the left side matches a subpart of the right side. The same basic algorithm can be used to test more complex cases of looping involving the interaction of several rules, but it is limited to those cases where each application of a rule occurs inside of the previous rule application. Experiments suggest that the simplest form of the algorithm is about 80 percent effective in eliminating bad orientations of rules. The algorithm never rules out a good orientation of a rule, and so it is most useful when one wants to consider all possible rule orientations.

A note on a canonical theory with undecidable unification and matching problem

Abstract: A natural example of a canonical theory with undecidable unification and matching problem is presented.

Unsolvable Decision Problems for Prolog Programs

Abstract: The paper presents a general method by which various natural decision problems for programs in PROLOG and extensions of PROLOG can easily be shown to be recursively unsolvable. A particularly interesting application of this method gives an affirmative answer to Flannagan's [1985] conjecture that the floundering property for queries with respect to MU-PROLOG programs is undecidable.

The undecidability of the unification and matching problem for canonical theories

Abstract: The problem whether there exists a unifying substitution for two terms is considered in the class of theories which can be embedded into canonical term rewriting systems. The problem is shown to be undecidable, even if we restrict the substitutions to matching ones. This implies that the class of admissible canonical theories is a proper subset of the class of canonical theories.

Some direct interconnections between formal language theory and discrete-time linear system theory

Abstract: In an attempt to bridge the gap between theoretical computer science and, system theory, three problems from developmental system theory are discussed in parallel with the corresponding problems for discrete-time linear systems. The question of existence of algorithmic solutions to each of the problems is investigated. Based on elementary linear algebra, it is shown that each of the three problems is decidable for discrete-time linear systems while the equivalent problems for developmental systems are either undecidable or the existence of algorithmic solutions is an open problem. Developmental systems are studied in both theoretical biology and formal language theory so the discussion presented in this paper provides some fundamental links between these two subjects and linear systems theory.

Solutions for the distributed termination problem

Abstract: Considering a network of communicating finite state machines which exchange messages over Channels, we discuss the distributed termination problem in the more general context of asynchronous environment. We first whow that this problem is undecidable. Then we discuss the possibility of superimposing algorithms that detect termination. At least we examine the case of faulty processors.

Symmetry in distributed systems

Abstract: A distributed computing system can be considered to be symmetric because of its topology or because of its behavior. Unfortunately, these different definitions can categorize the same system differently. The choice of definition becomes important when one is trying to prove that certain problems, such as the Dining Philosophers problem, cannot be solved on symmetric distributed computing systems. Since the behavioral definitions are based on the possible patterns of computation, it is much easier to use them for these proofs. However, behavioral definitions frequently are undecidable; topological definitions admit straightforward decision procedures. This thesis presents a new definition for symmetry, called similarity, that, while based on behavior of processes, is decidable given the initial configuration of the system. The decision procedure for similarity depends partly on the model of computation being used, but a way to discover these decision procedures is given and is used to find decision procedures for a wide range of models of distributed computation. Distributed versions of these decision procedures form the basis of solutions to the problem of selecting a process as the leader. The thesis also shows how to use similarity to compare the relative power of different models of computation, including models with shared variables with and without locking, models using synchronous and asynchronous message-passing, models making different assumptions about fairness, and models based on probabilistic techniques.

Generalizations and Strengthenings of Gödel’s Incompleteness Theorem

Abstract: In 1931 in the journal Monatshefte fur Mathematik und Physik a short paper (a bit more than 20 pages) of an Austrian mathematician and logician Kurt Godel was published — paper which has turned out to be one of the greatest and most important papers in mathematical logic and foundations of mathematics. Its title was “Uber formal unentscheidbare Satze der ‘Principia Mathematica’ und verwandter Systeme. I”. In it Godel proved that arithmetic of natural numbers and all systems containing it are essentially incomplete provided they are consistent. It means that there are sentences which are undecidable in them, i.e. sentences φ such that neither φ, nor ¬φ are theorems. What’s more, we know which sentence of the pair φ, ¬φ is true in the basic model of the theory, i.e. in the model to the description of which the theory was formulated. This incompleteness is essential, i.e. it cannot be removed by adding the undecidable sentences as a new axioms because new undecidable sentences will appear (undecidable in the new, richer theory). This theorem (so called 1st Godel theorem) indicates the cognitive limitations of the deductive method.

The use of 3-D sensing techniques for on-line collision-free path planning

Abstract: The state of the art in collision prevention for manipulators with revolute joints, showing that it is a particularly computationally hard problem, is discussed. Based on the analogy with other hard or undecidable problems such as theorem proving, an extensible multi-resolution architecture for path planning, based on a collection of weak methods is proposed. Finally, the role that sensors can play for an on-line use of sensor data is examined.

Some Results about Confluence on a Given Congruence Class

Abstract: It is undecidable in general whether or not a term-rewriting system is confluent on a given congruence class. This result is shown to hold even when the term-rewriting systems under consideration contain unary function symbols only, and all their rules are length-reducing. On the other hand, for certain subclasses of these systems confluence on a given congruence class is decidable.

Observations on the disjointness problem for rational subsets of free partially commutative monoids

Abstract: Let I be a partially commutative alphabet of size three. Let M denote the free partially commutative monoid generated by I. The disjointness problem for rational subsets of M is: for two given rational (described by regular expressions) subsets, X, Y of M decide if XnY=0. In this paper we show that the problem is decidable for every commutativity region over the alphabet I. It is known (see (3)) that the problem is undecidable in the case of the four letters alphabet. Hence we give a sharp bound on the number of letters for which the problem is decidable. A similar situation occurs for the unique decipherability problem with partially commutative alphabets. It was shown in (4) that this problem is decidable for alphabets of size three and that it is undecidable for alphabets of size four. We show that the unique decipherability problem with partially commutative alphabet I is a special case of the disjointness problem of rational subsets of the monoid generated by I. This and our algorithm for the disjointness problem give alternative and much simpler proof of the decidability of the unique decipherability problem with partially commutative alphabets of size three. Let I={a,b,c}. It was proved in (5) using multicounter machines that if a commutes with c and b, and b does not commute with c then the disjointness problem is decidable. We give here a simpler proof for this case and prove the decidability for all other possible commutativity relations for three letters alphabet.