Showing papers on "Undecidable problem published in 1994"
TL;DR: Alur et al. as discussed by the authors proposed timed automata to model the behavior of real-time systems over time, and showed that the universality problem and the language inclusion problem are solvable only for the deterministic automata: both problems are undecidable (II i-hard) in the non-deterministic case and PSPACE-complete in deterministic case.
7,096 citations
TL;DR: This work introduces a temporal logic for the specification of real-time systems that employs a novel quantifier construct for referencing time: the freeze quantifier binds a variable to the time of the local temporal context.
Abstract: We introduce a temporal logic for the specification of real-time systems. Our logic, TPTL, employs a novel quantifier construct for referencing time: the freeze quantifier binds a variable to the time of the local temporal context.TPTL is both a natural language for specification and a suitable formalism for verification. We present a tableau-based decision procedure and a model-checking algorithm for TPTL. Several generalizations of TPTL are shown to be highly undecidable.
665 citations
TL;DR: It is shown that the reversibility problem is undecidable in case of two-dimensional CA, and it is proved that the corresponding surjectivity problem-the problem of deciding if the global function is surjective-is Undecidable for two- dimensional CA.
249 citations
TL;DR: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.
Abstract: We show that the equality problem for rational series with multiplicities in the tropical semiring is undecidable.
212 citations
TL;DR: A reduction from the halting problem for two-counter Turing machines is used to show that the subtyping and typing relations of F?
Abstract: F? is a typed ?-calculus with subtyping and bounded second-order polymorphism. First introduced by Cardelli and Wegner, it has been widely studied as a core calculus for type systems with subtyping. We use a reduction from the halting problem for two-counter Turing machines to show that the subtyping and typing relations of F? are undecidable.
117 citations
11 Apr 1994
TL;DR: The decidability of the model checking problem for several μ-calculi and Petri nets is analysed and it is found that a very simple subset of the modal μ-Calculus is undecidable.
Abstract: The decidability of the model checking problem for several μ-calculi and Petri nets is analysed. The linear time μ-calculus without atomic sentences is decidable; if simple atomic sentences are added, it becomes undecidable. A very simple subset of the modal μ-calculus is undecidable.
94 citations
IBM1
TL;DR: It is shown that, although the two logics capture quite different intuitions about probability, there is a precise sense in which they are equi-expressive, in which the logic cannot be axiomatized in either case.
Abstract: We consider decidability and expressiveness issues for two first-order logics of probability. In one, the probability is on possible worlds, while 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. We show 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 ?21 complete, 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, ?1∞ hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is ?21 complete with as little as one unary predicate in the language, even without equality. With equality, we get ?1∞ hardness with only a constant symbol. We then turn our attention to an analysis of what causes this overwhelming complexity. For example, we show that if we require rational probabilities then we drop from ?21 to ?11. In many contexts it suffices to restrict attention to domains of bounded size; fortunately, the logics are decidable in this case. Finally, we show that, although the two logics capture quite different intuitions about probability, there is a precise sense in which they are equi-expressive.
83 citations
TL;DR: All other equivalences in the linear/branching time hierarchy are examined and it is shown that none of them are decidable for normed BPA processes.
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.
82 citations
11 Jul 1994
TL;DR: This paper considers the verification of a particular class of infinite-state systems, namely systems consisting of finite-state processes that communicate via unbounded lossy FIFO channels and shows that the following problems are undecidable.
Abstract: We consider the verification of a particular class of infinite-state systems, namely systems consisting of finite-state processes that communicate via unbounded lossy FIFO channels. This class is able to model e.g. link protocols such as the Alternating Bit Protocol and HDLC. In an earlier paper, we showed that several interesting verification problems are decidable for this class of systems, namely (1) the reachability problem: is a set of states reachable from some other state of the system, (2) safety property over traces formulated as regular sets of allowed finite traces, and (3) eventuality properties: do all computations of a system eventually reach a given set of states. In this paper, we show that the following problems are undecidable, namely
The model checking problem in propositional temporal logics such as Propositional Linear Time Logic (PTL) and Computation Tree Logic (CTL).
The problem of deciding eventuality properties with fair channels: do all computations eventually reach a given set of states if the unreliable channels are fair in the sense that they deliver infinitely many messages if infinitely many messages are transmitted. This problem can model the question of whether a link protocol, such as HDLC, will eventually reliably transfer messages across a medium that is not permanently broken.
80 citations
Book•
26 Aug 1994TL;DR: A mechanical proof of the Church-Rosser theorem is given and derived inference rules are explained, showing the representability of metatheory and the undecidable sentence.
Abstract: 1. Introduction 2. The statement of the incompleteness theorem 3. Derived inference rules 4. The representability of metatheory 5. The undecidable sentence 6. A mechanical proof of the Church-Rosser theorem 7. Conclusions.
67 citations
01 Jul 1994
TL;DR: A new notion of λ-term reduction is developed and used to prove that the problem of typability at rank 2 is reducible to the problemof acyclic semi-unification, which is an undecidable problem at every rank k≥3.
Abstract: We examine the problem of type inference for a family of polymorphic type systems containing the power of Core-ML. This family comprises the levels of the stratification of the second-order l-calculus (system F) by “rank” of types. We show that typability is an undecidable problem at every rank k≥3. While it was already known that typability is decidable at rank 2, no direct and easy-to-implement algorithm was available. We develop a new notion of l-term reduction and use it to prove that the problem of typability at rank 2 is reducible to the problem of acyclic semi-unification. We also describe a simple procedure for solving acyclic semi-unification. Issues related to principle types are discussed.
11 Jul 1994
TL;DR: The general model checking problem for CTL* formulae over RA is shown undecidable, but the decidability result is obtained for another substantial subset of the logic, called A-CTL*+, which includes all ”linear time” formULae.
Abstract: This paper investigates automated model checking possibilities for CTL* formulae over infinite transition systems represented by relational automata (RA). The general model checking problem for CTL* formulae over RA is shown undecidable, the undecidability being observed already on the class of Restricted CTL formulae. The decidability result, however, is obtained for another substantial subset of the logic, called A-CTL*+, which includes all ”linear time” formulae.
01 Jan 1994
TL;DR: In this paper, it was shown that semantic query optimization can be completely done in recursive rules provided that order constraints and negated EDB subgoals appear only in the recursive rules, but not in the it's.
Abstract: Semantic query optimization refers to the process of using integrity constraints (ic ‘s) in order to optimize the evaluation of queries. The process is well understood in the case of unions of select-project-join queries (i. e., nonrecursive datalog). For arbitrary datalog programs, however, the issue has largely remained an unsolved problem. This paper studies this problem and shows when semantic query optimization can be completely done in recursive rules provided that order constraints and negated EDB subgoals appear only in the recursive rules, but not in the it’s. If either order constraints or negated EDB subgoals are introduced in it’s, then the problem of semantic query optimization becomes undecidable. Since semantic query optimization is closely related to the containment problem of a datalog program in a union of conjunctive queries, our results also imply new decidability and undecidability results for that problem when order constraints and negated EDB subgoals are used.
01 Jan 1994
TL;DR: In this article, it was shown that Hilbert's Tenth problem over a field of rational functions with constant field is undecidable over a finite field of characteristic other than 2 and that the same holds for characteristic 2.
Abstract: In Hilbert's Tenth problem for fields of rational functions over finite fields (Invent. Math. 103 (1991)) Pheidas showed that Hilbert's Tenth problem over a field of rational functions with constant field a finite field of characteristic other than 2 is undecidable. We show that the same holds for characteristic 2
TL;DR: It is shown that ray-tracing problems in some three-dimensional simple optical systems (purely geometrical optics) are undecidable.
Abstract: The ray-tracing problem is, given an optical system and the position and direction of an initial light ray, to decide if the light ray reaches some given final position. For many years ray tracing has been used for designing and analyzing optical systems. Ray tracing is now used extensively in computer graphics to render scenes with complex curved objects under global illumination.
We show that ray-tracing problems in some three-dimensional simple optical systems (purely geometrical optics) are undecidable. These systems may consist of either reflective objects that are represented by rational quadratic equations, or refractive objects that are represented by rational linear equations. Some problems in more restricted models are shown to be PSPACE-hard or sometimes in PSPACE.
Book•
01 Jan 1994
TL;DR: In this article, Davies introduced a general theory of elementary propositions and proved the existence of a definable well-ordering of the continuum in the context of propositional calculus, which is a necessary condition for definability for transfinite von Neumann-Godel set theory sets.
Abstract: Introduction, Martin Davies. [1] The generalized gamma functions. [51 Introduction to a general theory of elementary propositions. [14] Generalized differentiation. [17] Finite combinatory processes, Formulation I. [18] Polyadic groups. [19] The Two-Valued Iterative Systems of Mathematical Logic. [20] Absolutely unsolvable problems and relatively undecidable propositions account of an anticipation. [21] Formal reductions of the general combinatorial decision problem. [22] Recursively enumerable sets of positive integers and their decision problems. [23] A variant of a recursively unsolvable problem. [24] Note on a conjecture of Skolem. [26] Recursive unsolvability of a problem of Thue. [27] Conjuntos recurrentemente numerables de enteros positives y sus problemas de decision. [34] (with S. C. Kleene) The upper semi-lattice of degrees of recursive unsolvability. ABSTRACTS: [2] Discussion of problem 433. [3] Introduction to a general theory of elementary propositions. [4] Determination of all closed systems of truth tables. [6] On a simple class of deductive systems. [7] Visual intuition in Lobachevsky space. [8] Visual intuition in spherical and elliptic space: Einstein's finite universe. [9] A non-Weierstrassian method of analytic prolongation. [10] A new method for generalizing ex in the complex domain. [11] A simple geometric proof of the equality of the Brochardt angles of a triangle. [12] Theory of generalized differentiation. [13] The mth derivative of a function of a function calculus of mth derivatives. [15] Polyadic groups (preliminary report). [16] Finite combinatory processes. Formulation. [25] Recursive unsolvability of a problem of Thue. [28] Degrees of recursive unsolvability (preliminary report). [291 [with Samuel Linial (who later changed his name to Samuel Gulden)] Recursive unsolvability of the deducibility, Tarski's completeness, and independence of axioms problems of propositional calculus. [30] Note on a relation recursion calculus. [31] Solvability, definability, provability history of an error. [32] A necessary condition for definability for transfinite von Neumann-Godel set theory sets, with an application to the problem of the existence of a definable well-ordering of the continuum (preliminary report). [33] (with S. C. Kleene) The upper semi-lattice of degrees of recursive unsolvability. [List Permissions]
24 May 1994
TL;DR: The alignment logic as discussed by the authors is a modal extension of relational calculus that allows the database programmer to reason about relations over strings of arbitrary length, and it can be expressed using a syntax reminiscent of regular expressions.
Abstract: In order to enable the database programmer to reason about relations over strings of arbitrary length we introduce alignment logic, a modal extension of relational calculus. In addition to relations, a state in the model consists of a two-dimensional array where the strings are aligned on top of each other. The basic modality in the language (a transpose, or “slide”) allows for a rearrangement of the alignment, and more complex formulas can be formed using a syntax reminiscent of regular expressions, in addition to the usual connectives and quantifiers. It turns out that the computational counterpart of the string-based portion of the logic is the class of multitape two-way finite state automata, which are devices particularly well suited for the implementation of string matching. A computational counterpart of the full logic is obtained from relational algebra by extending the selection operator into filters based on these multitape machines. Safety of formulas in alignment logic implies that new strings generated from old ones have to be of bounded length. While an undecidable property in general, this boundedness is decidable for an important subclass of formulas. As far as expressive power is concerned, alignment logic includes previous proposals for querying string databases, and gives full Turing computability. The language can be restricted to define exactly regular sets and sets in the polynomial hierarchy.
TL;DR: This work considers a constraint language in which one may express equations and functional dependencies over complex object types, and presents a sound and complete axiomatization of the constraint language for the case in which interpretations are permitted to be infinite.
Abstract: Virtually all semantic or object-oriented data models assume that objects have an identity separate from any of their parts, and allow users to define complex object types in which part values may be any other objects. This often results in a choice of query language in which a user can express navigating from one object to another by following a property value path. We consider a constraint language in which one may express equations and functional dependencies over complex object types. The language is novel in the sense that component attributes of individual constraints may correspond to property paths. The kind of equations we consider are also important, because they are a natural abstraction of the class of conjunctive queries for query languages that support property value navigation. In our introductory comments, we give an example of such a query and outline two applications of the constraint theory to problems relating to a choice of access plan for the query. We present a sound and complete axiomatization of the constraint language for the case in which interpretations are permitted to be infinite, where interpretations themselves correspond to a form of directed labeled graph. Although the implication problem for our form of equational constraint alone over arbitrary schema is undecidable, we present decision procedures for the implication problem for both kinds of constraints when the problem schema satisfies a stratification condition, and when all input functional dependencies are keys. >
11 Jul 1994
TL;DR: The main result states that every Turing Machine can be realized by a dynamical system with piecewise-constant derivatives in a 3-dimensional space and thus the reachability problem for such systems is undecidable for 3 dimensions.
Abstract: In this paper we define a precise notion of abstraction relation between continuous dynamical systems and discrete state-transition systems. Our main result states that every Turing Machine can be realized by a dynamical system with piecewise-constant derivatives in a 3-dimensional space and thus the reachability problem for such systems is undecidable for 3 dimensions. A decision procedure for 2-dimensional systems has been recently reported by Maler and Pnueli. On the other hand we show that some non-deterministic finite automata cannot be realized by any continuous dynamical system with less than 3 dimensions.
26 Jun 1994
TL;DR: This work shows that the unification of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and finitary.
Abstract: Second-order unification is undecidable in general. Miller showed that unification of so-called higher-order patterns is decidable and unitary. We show that the unification of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and finitary. A few extensions of this unification problem are still decidable: unifying two second-order terms, where one term is linear, is undecidable if the terms contain bound variables but decidable if they don't.
TL;DR: A model of belief is proposed, which combines features from both possible-world semantics and relevance logic, and the decidability result follows from a close connection between this notion of belief and an existing decidable form of first-order entailment.
04 Jul 1994
TL;DR: It is proved that it is undecidable whether a given intersection type is non-empty, i.e., whether there exists a closed term of this type.
Abstract: We prove that it is undecidable whether a given intersection type is non-empty, i.e., whether there exists a closed term of this type. >
01 Dec 1994
TL;DR: This article presents a much more simple tile set which can play the same role as the surjectivity problem for 2D cellular automata and proves it undecidable.
Abstract: The surjectivity problem for 2D cellular automata was proved undecidable in 1989 by Jarkko Kari. The proof consists in a reduction of a problem concerning finite tilings into the previous one. This reduction uses a special and very sophisticated tile set. In this article, we present a much more simple tile set which can play the same role.
15 Feb 1994
TL;DR: Curry′s system for F-deducibility, the basis for static type inference algorithms for programming languages such as ML, becomes undecidable, but complete relative to a variety of model classes if a natural "preservation of types by conversion" rule is added to it.
Abstract: Curry′s system for F-deducibility is the basis for static type inference algorithms for programming languages such as ML. If a natural "preservation of types by conversion" rule is added to Curry′s system, it becomes undecidable, but complete relative to a variety of model classes. We show completeness for Curry′s system itself, relative to an extended notion of model that validates reduction but not conversion. Two proofs are given: one uses a term model and the other a model built from type expressions. Extensions to systems with polymorphic or intersection types are also considered.
TL;DR: It is shown that the following nine problems are undecidable, for ranges L1 and L2 of arbitrary two deterministic, nondeleting and finite copying top-down tree transducers, and whether the set of elements irreducible with respect to R derivable from L is recognizable or not.
01 Jan 1994
TL;DR: This paper provides here the first full proof, and shows that the general snake problem is actually PSPACE-complete, and establishes a resemblance between snake problems and classical tiling problems, considering the corresponding bounded, unbounded and recurring cases.
TL;DR: A modified proof of Conway's result that the general problem of whether the Collatz type functions converge is undecidable is presented, which enables to state new undecidability properties concerning functional equations.
TL;DR: In this paper, a polynomial dynamical system where one cannot decide whether a Hopf bifurcation occurs was shown to be incompleteness, and the same incompletteness result was shown for axiomatized versions of dynamical systems theory.
Abstract: We exhibit a polynomial dynamical system where one cannot decide whether a Hopf bifurcation occurs. Therefore one cannot decide whether there will be parameter values such that a stable fixed point becomes an unstable one. Related incompleteness results for previously described axiomatized versions of dynamical systems theory are also discussed.
15 Dec 1994
TL;DR: A generalization of a unification problem for term rewriting systems, named a unification problems under constrained substitutions, is investigated, and a procedure to solve the problem with the help of tree automata is presented.
Abstract: A generalization of a unification problem for term rewriting systems, named a unification problem under constrained substitutions is investigated, and a procedure to solve the problem with the help of tree automata is presented. Since the problem is undecidable in general, there are rewriting systems for which our procedure does not terminate. We clarify a sufficient condition undet which the procedure always terminates, and review some classes of rewriting systems that satisfy the condition. These classes include a known decidable class introduced in [6].