Topic
Undecidable problem
About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.
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TL;DR: The following problem concerning any two finite state machines M and N that exchange messages via two 1-directional channels is considered, and some sufficient conditions for the problem to have a positive answer are discussed.
Abstract: We consider the following problem concerning any two finite state machines M and N that exchange messages via two 1-directional channels. “Is there a positive integer K such that the communication between M and N over K -capacity channels is guaranteed to progress indefinitely?” The problem is shown to be undecidable in general. For a practical class of communicating machines, the problem is shown to be decidable, and the decidability algorithm is polynomial. We also discuss some sufficient conditions for the problem to have a positive answer; these sufficient conditions can be checked for the given M and N in polynomial time. We apply the results to some practical protocols to show that their communications will progress indefinitely.
67 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
21 Aug 1995
TL;DR: This paper considers the problem of comparing an arbitrary Petri net against one whose places may contain only a bounded number of tokens, with respect to trace set inclusion and equivalence, as well as simulation and bisimulation, and finds that all the above are in fact decidable.
Abstract: In this paper we consider the problem of comparing an arbitrary Petri net against one whose places may contain only a bounded number of tokens (that is, against a regular behaviour), with respect to trace set inclusion and equivalence, as well as simulation and bisimulation. In contrast to the known result that language equivalence is undecidable, we find that all of the above are in fact decidable. We furthermore demonstrate that it is undecidable whether a given Petri net is either trace equivalent or simulation equivalent to any (unspecified) bounded net.
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.
66 citations
Journal Article•
TL;DR: In this article, the reachability problem for parametric flat counter automata is studied in relation with the satisfiability problem of three fragments of integer arithmetic, and the latter problem is shown to be decidable using a number-theoretic argument.
Abstract: In this paper we study the reachability problem for parametric flat counter automata, in relation with the satisfiability problem of three fragments of integer arithmetic. The equivalence between non-parametric flat counter automata and Presburger arithmetic has been established previously by Comon and Jurski [5].We simplify their proof by introducing finite state automata defined over alphabets of a special kind of graphs (zigzags). This framework allows one to express also the reachability problem for parametric automata with one control loop as the existence of solutions of a 1-parametric linear Diophantine systems. The latter problem is shown to be decidable, using a number-theoretic argument. Finally, the general reachability problem for parametric flat counter automata with more than one loops is shown to be undecidable, by reduction from Hilbert's Tenth Problem [9J.
66 citations