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Showing papers on "Undecidable problem published in 1985"


Journal ArticleDOI
TL;DR: Cellular automata are used to provide explicit examples of various formally undecidable and computationally intractable problems and it is suggested that such problems are common in physical models, and some other potential examples are discussed.
Abstract: Physical Processes are reviewed as computations, and the difficulty of answering questions about them is characterized in terms of the difficulty of performing the corresponding computations. Cellular automata are used to provide explicit examples of various formally undecidable and computationally intractable problems. It is suggested that such problems are common in physical models, and some other potential examples are discussed.

223 citations


Journal ArticleDOI
TL;DR: It is shown that the implication problem is undecidable for the class of functional and inclusion dependencies, even if the inclusion dependencies are restricted to be binary.
Abstract: The implication problem for a class of dependencies is the following: given a finite set of dependencies, determine if they logically imply another dependency. We show that the implication problem is undecidable for the class of functional and inclusion dependencies. This holds true even if the inclusion dependencies are restricted to be binary. It may be noted that the implication problem is known to be decidable for functional and unary inclusion dependencies and also for inclusion dependencies without functional dependencies.

182 citations


Book ChapterDOI
TL;DR: It is hoped that the paper, which contains also NP-, PSPACE-, Π01-and Π02-hardness results for various logical systems, will enhance interest in the appealing medium of domino problems as a useful set of reduction tools for exhibiting “bad behavior”.
Abstract: In recent years many diverse logical systems for reasoning about programs have been shown to posses a highly undecidable, viz Π11- complete, validity problem. All such known results are reproved in this paper in a uniform and transparent manner by reductions from recurring domino problems. These are simple variants of the classical unbounded domino (or tiling) problems introduced by Wang and the bounded versions defined by Lewis. While the former are (weakly) undecidable and the latter complete in various complexity classes, the problems in the new class are Σ11-complete. It is hoped that the paper, which contains also NP-, PSPACE-, Π01-and Π02-hardness results for various logical systems, will enhance interest in the appealing medium of domino problems as a useful set of reduction tools for exhibiting “bad behavior”.

181 citations


Proceedings ArticleDOI
01 Jun 1985
TL;DR: A linear-time algorithm that detects many recursive definitions in deductive database systems, and specifies a useful subset of recursive definitions for which the algorithm is complete, which can be used to optimize a recursive definition and improve the efficiency of the compiled evaluation algorithms.
Abstract: Some recursive definitions in deductive database systems can be replaced by equivalent nonrecursive definitions. In this paper we give a linear-time algorithm that detects many such definitions, and specify a useful subset of recursive definitions for which the algorithm is complete. It is unlikely that our algorithm can be extended significantly, as recent results by Gaifman [5] and Vardi [19] show that the general problem is undecidable. We consider two types of initialization of the recursively defined relation: arbitrary initialization, and initialization by a given nonrecursive rule. This extends earlier work by Minker and Nicolas [10], and by Ioannidis [7], and is related to bounded tableau results by Sagiv [14]. Even if there is no equivalent equivalent nonrecursive definition, a modification of our algorithm can be used to optimize a recursive definition and improve the efficiency of the compiled evaluation algorithms proposed in Henschen and Naqvi [6] and in Bancilhon et al. [3].

92 citations


Proceedings ArticleDOI
21 Oct 1985
TL;DR: It is shown here that a natural formalization of the automatic inference of omitted type information is undecidable, and the proof is directly applicable to some practical situations, and provides a partial explanation of the difficulties encountered in other cases.
Abstract: Polymorphic type systems combine the reliability and efficiency of static type-checking with the flexibility of dynamic type checking. Unfortunately, such languages tend to be unwieldy unless they accommodate omission of much of the information necessary to perform type checking. The automatic inference of omitted type information has emerged as one of the fundamental new implementation problems of these languages. We show here that a natural formalization of the problem is undecidable. The proof is directly applicable to some practical situations, and provides a partial explanation of the difficulties encountered in other cases.

41 citations


Journal ArticleDOI
TL;DR: It is proved that the class of NTS grammars has an undecidable inclusion problem and it is shown that one can decide whether a given c.f. grammar is NTS or not.

40 citations


Book ChapterDOI
01 Dec 1985
TL;DR: It is shown that unification in the equational theory defined by the one-sided distributivity law x × (y+z)=x×y+x×z is decidable and that unification is undecidable if the laws of associativity x+y)+z and unit element 1×x=x× 1=x are added.
Abstract: We show that unification in the equational theory defined by the one-sided distributivity law x × (y+z)=x×y+x×z is decidable and that unification is undecidable if the laws of associativity x+(y+z)=(x+y)+z and unit element 1×x=x× 1=x are added. Unification under one-sided distributivity with unit element is shown to be as hard as Markov's problem, whereas unification under two-sided distributivity, with or without unit element, is NP-hard. A quadratic time unification algorithm for one-sided distributivity, which may prove interesting since available universal unification procedures fail to provide a decision procedure for this theory, is outlined. The study of these problems is motivated by possible applications in circuit synthesis and by the need for gaining insight in the problem of combining theories with overlapping sets of operator symbols.

38 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that there exist undecidable Diophantine equations for which neither the existence nor the non-existence of integer solutions may be proved in a given axiomatization of arithmetic.
Abstract: Summary In 1979 Jan-Erik Roos published a fascinating paper [22] making explicit connections between the Poincare-Betti series of loop spaces or local rings and the Hilbert series of finitely presented graded Hopf algebras. This paper develops a way to model Diophantine equations within the category of graded Hopf algebras. Combining this with Roos' work, we obtain loop spaces and local rings whose series reflect the solution sets of arbitrary Diophantine equations. By Matiyasevic's negative solution to Hilbert's tenth problem ([15], [16]), there is no algorithm for deciding in general whether or not a given Diophantine equation has solutions. For us, one consequence is that no algorithm exists to decide whether a given finitely presented graded algebra is "generic" (see [3]). As to rings, there is no finite procedure for evaluating whether an arbitrary ring's Poincare series equals a given series, even though by [10] the sequence of coefficients is recursive. Likewise, given two finite simply-connected CW complexes, there is no guaranteed method to tell, in general, whether their loop spaces have the same rational homotopy type. At the same time, Matiyasevic showed that there exist undecidable Diophantine equations, i.e., polynomial equations for which neither the existence nor the non-existence of integer solutions may be proved in a given axiomatization of arithmetic. We obtain "undecidable local rings," "undecidable spaces," and "undecidable topological maps." These concepts will be made precise in Section 4. A few other consequences of the link between Diophantine equations and Hilbert series may be listed. First, a graded algebra (resp. loop space or local ring) exists whose Hilbert (resp. Poincare) series radius of convergence may be proved to be a transcendental number. Also, the Hilbert series can represent a transcendental function which solves no algebraic differential equation. Lastly, we prove a stability theorem for "Diophantine solution sets of bounded complexity."

31 citations


Book ChapterDOI
09 Sep 1985
TL;DR: This work constructs a special class of finite state tree transducers that are code in recognizable tree languages and proves decidability of confluence for ground term rewriting systems.
Abstract: It is well known that confluence (which is equivalent to Church-Rosser property) is undecidable for arbitrary term rewriting systems. We prove here decidability of confluence for ground term rewriting systems. To obtain this result, we construct a special class of finite state tree transducers that we code in recognizable tree languages. Our work illustrates how tree language theory is useful in term rewriting systems study and we give easily some other results in the ground case (as decidability of uniform termination).

28 citations


Journal ArticleDOI
TL;DR: The self-embedding property is shown to be undecidable and partially decidable, and it follows that the nonself-embedded property is not partially decided, even for globally finite term rewriting systems.

26 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the question of whether a given congruence class is finite is undecidable (in fact, Π 2 -complete) and that the problem is not even semidecidable in the sense that it requires space at least exponential in the square root of the input length.

Journal ArticleDOI
TL;DR: This survey states some old and new solved and unsolved problems on quantum mechanics, some of which turned out to be undecidable.

Journal ArticleDOI
TL;DR: It is shown that the satisfiability problem even for-free regular terms is undecidable, and similar techniques are used to show that a very natural extension of the Process Logic of Harel, Kozen and Parikh is Undecidable.
Abstract: Regular terms with the Kleene operations $ \cup $, ; and $ * $ can be thought of as operators on languages, generating other languages. An equation $\tau _1 = \tau _2 $ between two such terms is said to be satisfiable just in case languages exist which make this equation true. We show that the satisfiability problem even for $ * $-free regular terms is undecidable. Similar techniques are used to show that a very natural extension of the Process Logic of Harel, Kozen and Parikh is undecidable.

Journal ArticleDOI
TL;DR: This work addresses the problem of whether the communication of a network of two communicating finite state machines is deadlock-free and bounded, and shows that the problem is undecidable if the two machines exchange two types of messages, and one of the channels is known to be bounded.
Abstract: Consider a network of two communicating finite state machines which exchange messages over two one-directional, unbounded channels, and assume that each machine receives the messages from its input channel based on some fixed (partial) priority relation. We address the problem of whether the communication of such a network is deadlock-free and bounded. We show that the problem is undecidable if the two machines exchange two types of messages. The problem is also undecidable if the two machines exchange three types of messages, and one of the channels is known to be bounded. However, if the two machines exchange two (or less) types of messages, and one channel is known to be bounded, then the problem becomes decidable. The problem is also decidable if one machine sends one type of message and the second machine sends two (or less) types of messages; the problem becomes undecidable if the second machine sends three types of messages. The problem is also decidable if the message priority relation is empty. W...

Proceedings Article
18 Aug 1985
TL;DR: The method uses the language, inference rules and proofs of non-monotonic logics, but ignores theoremhood, and defines states of the reasoning process, and focuses on current proof as the criterion for belief.
Abstract: Artificial Intelligence needs a formal theory of the process of non-monotonic reasoning Ideally, such a theory would decide, for every proposition and state of the process, whether the program should believe the proposition in that state, or remain agnostic Without non-monotonic inference rules, nonmonotonic inferences cannot be explained in the same relational, rule-based fashion as other inferences But with such rules, theoremhood is often formally undecidable and thus a useless criterion for our purpose So how could any system be a "non-monotonic logic programming language"? Our method uses the language, inference rules and proofs of non-monotonic logics, but ignores theoremhood Instead, it defines states of the reasoning process, and focuses on current proof as the criterion for belief It defines "admissible beliefs" and "valid proof" for given states, and we prove in [5] that a belief is currently admissible iff it is currently proven The primitive nonmonotonic condition is "currently unproven" The theory, Logical Process Theory, can accept a range of non-monotonic logics It was inspired by Doyle's RMS [3] and is similar to his more recent theory in [4| A model implementation, WATSON, exists and has been used to write a small diagnostic reasoner, which reasons non-monotonic ally using violation of expectations and an abstraction hierarchy


Journal ArticleDOI
TL;DR: It is proved that the first order metatheory of the classical propositional logic is undecidable and this result answers negatively a question of J. K. van Benthem as to whether the interpolation theorem in some sense completes the metatheories of the calculus.
Abstract: ?0. Introduction. In this paper we investigate the first order metatheory of the classical propositional logic. In the first section we prove that the first order metatheory of the classical propositional logic is undecidable. Thus as a mathematical object even the simplest of logics is, from a logical standpoint, quite complex. In fact it is of the same complexity as true first order number theory. This result answers negatively a question of J. F. A. K. van Benthem (see [van Benthem and Doets 1983]) as to whether the interpolation theorem in some sense completes the metatheory of the calculus. Let us begin by motivating the question that we answer. In [van Benthem and Doets 1983] it is claimed that a Jolklore prejudice has it that interpolation was the final elementary property of first order logic to be discovered. Even though other properties of the propositional calculus have been discovered since Craig's orginal paper [Craig 1957] (see for example [Reznikoff 1965]) there is a lot of evidence for the fundamental nature of the property. In abstract model theory for example one finds that very few logics have the interpolation property. There are two well-known open problems in this area. These are 1. Is there a logic satisfying the full compactness theorem as well as the interpolation theorem that is not equivalent to first order logic even for finite models? 2. Is there a logic stronger than L(Q), the logic with the quantifier there exist uncountably many, that is countably compact and has the interpolation property? The first question is due to Friedman [Friedman 1975]; the basic question of whether or not compactness and interpolation characterize first order logic remains open. Mundici [Mundici 1981] has shown that if such a logic exists and has a weak Lowenheim-Skolem property then it will be the same as first order logic on countable models of finite type. The second and weaker question is attributed to Feferman [Makowsky, Shelah and Stavi 1976]. Here again there are only partial answers. Shelah recently claims to have shown that a positive answer to Feferman's question is consistent with ZFC. Perhaps it is to these sorts of results (or the lack of them) that folklore prejudice should look towards for its vindication. Further discussion of these topics can be found in [Barwise and Feferman], particularly in Kaufmann's contribution.

Journal ArticleDOI
TL;DR: The equivalence problem for multiplicity sets of regular languages is shown to be undecidable and the proposed solution to this problem is simple and elegant.

Journal ArticleDOI
TL;DR: This paper presents two results, both of which strengthen the general conclusion that repeat is a stronger construct than loop, in several respects, and shows that QDL, the first-order version of dynamic logic, is more expressive with repeat than with loop.

Journal ArticleDOI
TL;DR: Some assertions concerning decidability and undecidability of Luzin's problems on constituents are proved as mentioned in this paper, however, these assertions do not hold for the case of as mentioned in this paper.
Abstract: Some assertions concerning decidability and undecidability of Luzin's problems on constituents are proved.Bibliography: 34 titles.

Book ChapterDOI
TL;DR: The Paris-Harrington Theorem is a variant of the Finite Ramsey Theorem, which, though provable in ordinary set theory (and so a theorem of ordinary mathematical practice), is not provable for formal number theory as discussed by the authors.
Abstract: Publisher Summary This chapter discusses the varieties of arboreal experience. The Paris–Harrington Theorem is a variant of the Finite Ramsey Theorem, which, though provable in ordinary set theory (and so a theorem of ordinary mathematical practice), is not provable in formal number theory. This independence being of a character different from that of Godel's undecidable sentence or that associated with independence results in set theory, the result has been much touted by logicians seeking social acceptance by the rest of the mathematical world as something closely akin to the coming of the millennium. This is unfortunate as the result really is remarkable philosophically because of this new character of independence and mathematically for reasons requiring a new paragraph. The difference between the usual Finite Ramsey Theorem and the Paris–Harrington Theorem is surprisingly consequential, concerning computationally hairy combinatorial problems with unfeasible bounds; but the former's unfeasible bounds are readily intelligible, while the latter's are not. The relation between Kruskal's Theorem and ordinals is fairly close. In fact, Kruskal's Theorem is most memorably stated, if not in terms of ordinals and well-ordering, at least in terms of well-quasi-ordering.


Book ChapterDOI
23 Sep 1985
TL;DR: It is shown, that the set of most general DS-unifiers is recursively enumerable and that such a set may be infinite and that it is undecidable, whether two terms are DS- unifiable.
Abstract: The many-sorted first order calculus ∑RP* is extended to a many-sorted calculus, which allows declarations, i.e. a term t of sort S can be declared to be of some lesser sort S’. The heart of such a calculus is the unification algorithm for terms, which respects the declarations. In this paper it is shown, that the set of most general DS-unifiers is recursively enumerable and that such a set may be infinite. Furthermore, it is shown, that it is undecidable, whether two terms are DS-unifiable.

Book ChapterDOI
01 Dec 1985
TL;DR: In this paper, networks of communicating finite state machines (CFSM's), that explicitly allow zero testing (i.e. empty channel detection), are considered and it is shown that the boundedness problem is decidable for the class of FIFO networks consisting of two such CFSM, where one of the two machines is allowed to send only a single type of message to the other.
Abstract: In this paper, we consider networks of communicating finite state machines (CFSM's), that explicitly allow zero testing (i.e. empty channel detection). In our main result, we show that the boundedness problem is decidable for the class of FIFO networks consisting of two such CFSM's, where one of the two machines is allowed to send only a single type of message to the other. This result, we feel, is somewhat surprising since the zero testing capability is precisely the required extension needed in order to render the problem undecidable for the related class of vector addition systems with states (VASS's) of dimension two. Note that both have the ability to store two nonnegative integers which can be conditionally tested for zero. The reason for the disparity appears to be that such a class of extended VASS's would be capable of more synchronized behavior (since the actions of the two counters can be controlled by a single finite state control). The rest of the paper examines other classes of networks which allow empty channel detection. These results seem to indicate that our main result can not be extended.

Journal ArticleDOI
TL;DR: This paper presents simple loop programming languages which are, computationally, strictly more powerful, i.e. which can compute more than the class of Presburger functions.
Abstract: This paper is concerned with the semantics (or computational power) of very simple loop programs over different sets of primitive instructions. Recently, a complete and consistent Hoare axiomatics for the class of {x←0, x←y, x←x+1, x←x∸1, do x...end} programs which contain no nested loops, was given, where the allowable assertions were those formulas in the logic of Presburger arithmetic. The class of functions computable by such programs is exactly the class of Presburger functions. Thus, the resulting class of correctness formulas has a decidable validity problem. In this paper, we present simple loop programming languages which are, computationally, strictly more powerful, i.e. which can compute more than the class of Presburger functions. Furthermore, using a logical assertion language that is also more powerful than the logic of Presburger arithmetic, we present a class of correctness formulas over such programs that also has a decidable validity problem. In related work, we examine the expressive power of loop programs over different sets of primitive instructions. In particular, we show that an {x←0, x←y, x←x+1, do x ... end, if x=0 then y←z}-program which contains no nested loops can be transformed into an equivalent {x←0, x←y, x←x+1, do x ... end}-program (also without nested loops) in exponential time and space. This translation was earlier claimed, in the literature, to be doable in polynomial time, but then this was subsequently shown to imply that PSPACE=PTIME. Consequently, the question of translatability was left unanswered. Also, we show that the class of functions computable by {x←0, x←y, x←x+1, x←x∸1, do x ... end, if x=0 then x←c}-programs is exactly the class of Presburger functions. When the conditional instruction is changed to “if x=0 then x←y+1”, then the class of computable functions is significantly enlarged, enough so, in fact, as to render many decision problems (e.g. equivalence) undecidable.

Book
01 Jan 1985
TL;DR: This paper presents a new transformational approach to Partial Correctness Proof Calculi for Algol 68-like Programs with Finite Modes and Simple Side Effects and describesal Complexity for Classes of Ianov-Schemes.
Abstract: Preface (M Karpinksi and J Van Leeuwen) Input-Driven Languages are Recognized in log n Space (B v Braunmuehl and R Verbeek) Constructive Mathematics as a Programming Logic I: Some Principles of Theory (RL Constable) Space and Reversal Complexity of Probabilistic One-Way Turing Machines (R Freivalds) Recurring Dominoes: Making the Highly Undecidable Highly Understandable (D Harel) A New Transformational Approach to Partial Correctness Proof Calculi for Algol 68-like Programs with Finite Modes and Simple Side Effects (H Langmaack) Effective Determination of the Zeros of p-ADIC Exponential Functions (A MacIntyre) The Logic of Games and Its Applications (R Parikh) A Fast Parallel Construction of Disjoint Paths in Networks (E Shamir and E Upfal) Descriptional Complexity for Classes of Ianov-Schemes (P Trum and D Wotschke)

Journal ArticleDOI
Michio Oyamaguchi1
TL;DR: This paper considers the problem of deciding, for a given specification of an abstract data type, whether it is a data type extension, and shows that the problem is undecidable, but there exists a large class of specifications which are data type extensions.

Journal ArticleDOI
TL;DR: The following problem is shown to be undecidable: Given a reduced finite Thue systemS over an alphabet Σ with |Σ| = 2, isS equivalent to a finite Church-Rosser system?
Abstract: The following problem is shown to be undecidable: Given a reduced finite Thue systemS over an alphabet Σ with |Σ| = 2, isS equivalent to a finite Church-Rosser system?


Journal ArticleDOI
Aldo Ursini1
TL;DR: It is proved that the only decidable variety of diagonalizable algebras is that defined by ‘τ0=1’, and any variety containing an algebra in which τ0≠1 is hereditarily undecidable.
Abstract: We make use of a Theorem of Burris-McKenzie to prove that the only decidable variety of diagonalizable algebras is that defined by ‘τ0=1’. Any variety containing an algebra in which τ0≠1 is hereditarily undecidable. Moreover, any variety of intuitionistic diagonalizable algebras is undecidable.