Topic
Recursively enumerable language
About: Recursively enumerable language is a research topic. Over the lifetime, 1508 publications have been published within this topic receiving 32382 citations.
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15 Oct 1969
TL;DR: An n-dimensional bug-automation is generalization of a finite state acceptor to n-dimensions and associates the language L(B) which is the set of top rows of the n- dimensional rectangular arrays accepted by B with each bug B.
Abstract: An n-dimensional bug-automation is generalization of a finite state acceptor to n-dimensions. With each bug B, we associate the language L(B) which is the set of top rows of the n-dimensional rectangular arrays accepted by B. One-dimensional bugs define trivially the regular sets. Twodimensional bugs define precisely the context-sensitive languages, while bugs of dimension 3 or greater define all the recursively enumerable sets. We consider also finite state acceptors with n two-way non-writing input tapes. For each such machine M, let domain (M) be the set of all strings which are the first component of some n-tuple of tapes accepted by M. For any n ≥ l, the domains of n-tape two-way finite state acceptors are precisely the same as the languages definable by n-dimensional bugs, so as a corollary, the domains of two-tape two-way finite state acceptors are precisely the context-sensitive languages.
26 citations
TL;DR: In this article, a relative recursive enumerability conjecture was first suggested by Cooper, which states that if a set C is a recursively enumerable (r.e., nonrecursive) set, then there is a set A such that A is r.e.
Abstract: Publisher Summary This chapter discusses relative recursive enumerability. It presents a theorem that was first suggested by a conjecture of Cooper. The theorem shows that let C be a recursively enumerable (r.e.), nonrecursive set. Then there is a set A such that A is r.e. in C, C is recursive in A, but A does not have r.e. degree. The chapter describes a single requirement for proving the theorem and presents a construction that meets this requirement. It outlines a method for meeting all the requirements, assuming that C is low, and describes the method for meeting the requirements without this assumption. The chapter further refinements of the theorem and also discusses the relationship of this discussion to that of Jockusch and Shore. The technique of replacing an infinite injury priority argument by a finite injury argument using lowness is due to Robinson.
25 citations
TL;DR: It is proved that there are two incomplete d.r.e. degrees such that every non‐zero recursively enumerable degree cups at least one of them to 0′, the greatest recursive enumerable (Turing) degree.
Abstract: We prove that there are two incomplete dre\ degrees (the Turing degrees of differences of two recursively enumerable sets) such that every non-zero recursively enumerable degree cups at least one of them to , the greatest recursively enumerable (Turing) degree
25 citations
TL;DR: This article studies the computational complexity of the agent design problem for tasks that are of the form “achieve this state of affairs” or “maintain thisstate of affairs,” and considers three general formulations of these problems (in both non-deterministic and deterministic environments).
Abstract: The agent design problem is as follows: given a specification of an environment, together with a specification of a task, is it possible to construct an agent that can be guaranteed to successfully accomplish the task in the environment? In this article, we study the computational complexity of the agent design problem for tasks that are of the form "achieve this state of affairs" or "maintain this state of affairs." We consider three general formulations of these problems (in both non-deterministic and deterministic environments) that differ in the nature of what is viewed as an "acceptable" solution: in the least restrictive formulation, no limit is placed on the number of actions an agent is allowed to perform in attempting to meet the requirements of its specified task. We show that the resulting decision problems are intractable, in the sense that these are non-recursive (but recursively enumerable) for achievement tasks, and non-recursively enumerable for maintenance tasks. In the second formulation, the decision problem addresses the existence of agents that have satisfied their specified task within some given number of actions. Even in this more restrictive setting the resulting decision problems are either pspace-complete or np-complete. Our final formulation requires the environment to be history independent and bounded. In these cases polynomial time algorithms exist: for deterministic environments the decision problems are nl-complete; in non-deterministic environments, p-complete.
25 citations
TL;DR: In this article, it was shown that any a-finite partially ordered set is order-isomorphic to a subset of an a-recursive function, i.e. a-degrees, if and only if it asserts a given finite set of order relationships between a finite number of elements.
25 citations