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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04 Apr 2005TL;DR: This is the first work, to the authors' knowledge, that provides a complete axiomatization for weak equivalences in the presence of recursion and both nondeterministic and probabilistic choice.
Abstract: We study a process calculus which combines both nondeterministic and probabilistic behavior in the style of Segala and Lynch's probabilistic automata. We consider various strong and weak behavioral equivalences, and we provide complete axiomatizations for finite-state processes, restricted to guarded definitions in case of the weak equivalences. We conjecture that in the general case of unguarded recursion the “natural” weak equivalences are undecidable.
This is the first work, to our knowledge, that provides a complete axiomatization for weak equivalences in the presence of recursion and both nondeterministic and probabilistic choice.
40 citations
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01 Feb 1991TL;DR: It is proved that the existential fragment of the theory of ground term algebras modulo a congruence generated by a set E of equations such that there exists a finite, noetherian, confluent rewrite system S satisfying (P) with \(\mathop \leftrightarrow \limits^* S = \mathop | E\) is undecidable.
Abstract: We study the connections between recognizable tree languages and rewrite systems. We investigate some decision problems. Particularly, let us consider the property (P): a rewrite system S is such that, for every recognizable tree language F, the set of S-normal forms of terms in F is recognizable too. We prove that the property (P) is undecidable. We prove that the existential fragment of the theory of ground term algebras modulo a congruence \(\mathop \leftrightarrow \limits^* E\) generated by a set E of equations such that there exists a finite, noetherian, confluent rewrite system S satisfying (P) with \(\mathop \leftrightarrow \limits^* S = \mathop \leftrightarrow \limits^* E\) is undecidable. Nevertheless, we develop a decision procedure for the validity of linear formulas in a fiagment of such a theory.
39 citations
01 May 1991
TL;DR: The techniques used in the analysis provide a complete description of the complexity of deciding the equivalence of conjunctive queries (single-rule, nonrecursive programs), and tight undecidability results for the detection of program equivalence.
Abstract: A deductive database consists of a set of stored facts, and a set of logical rules (typically, recursive Horn clauses) that are used to manipulate these facts. A number of optimizations in such databases involve the transformation of sets of logical rules (programs) to simpler, more efficiently evaluable programs. We consider a class of optimizations in which the transformation is a simple syntactic restriction on the form of the original program, and in which the correctness of the transformation indicates the existence of a normal form for the proof trees generated by the program. For example, the existence of basis-linearizability in a nonlinear program indicates that the program is inherently linear, and permits the use of special-purpose query evaluators for linear recursions. The canonical example of a basis-linearizable program is the program that computes the transitive closure of a binary relation; the corresponding normal form for the proof trees is that of right-linearity. Similarly, if a program is sequencable, then it is conducive to a pipelined evaluation. In addition, the existence of k-boundedness in a program permits the elimination of recursion overhead in evaluating the program. We investigate the complexity of detecting such optimization opportunities, and provide correct (but not always complete) algorithms for this purpose.
Each of the problems that are mentioned above may be described in terms of the subtree-elimination problem, which we define and analyze. We relate the detection of basis-linearizability, sequencability and 1-boundedness to the complexity classes ${\cal NC}$, ${\cal P}$ and ${\cal NP}$, and show that the first two of these problems are, in general, undecidable. The techniques used in our analysis provide a complete description of the complexity of deciding the equivalence of conjunctive queries (single-rule, nonrecursive programs), and tight undecidability results for the detection of program equivalence.
39 citations
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TL;DR: It is proved that even in the presence of an unbounded set of nonces the secrecy problem is decidable for a reasonable subclass of protocols, which the authors call context-explicit protocols.
Abstract: An important problem in the analysis of security protocols is that of checking whether a protocol preserves secrecy, i.e., no secret owned by the honest agents is unintentionally revealed to the intruder. This problem has been proved to be undecidable in several settings. In particular, Durgin et al. prove the undecidability of the secrecy problem in the presence of an unbounded set of nonces, even when the message length is bounded. In this paper we prove that even in the presence of an unbounded set of nonces the secrecy problem is decidable for a reasonable subclass of protocols, which we call context-explicit protocols.
39 citations
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TL;DR: This work develops basic computability theory, starting from a few simple axioms, which never speaks of Turing machines and Godel encodings, but rather use familiar concepts from set theory and topology.
39 citations