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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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TL;DR: In this article, it was shown that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2,1) in Pw.
Abstract: Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. AlthoughRT is known to be structurally rich, a major source of frustration is that no specific, natural degrees inRT have been discovered, except the bottom and top degrees, 0 and 0′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (a.k.a., Muchnik degrees) of mass problems associated with nonempty Π01 subsets of 2 . It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here d is the weak degree of the diagonally nonrecursive functions, and rn is the weak degree of the n-random reals. It is known that r1 can be characterized as the maximum weak degree of a Π01 subset of 2 ω of positive measure. We now show that inf(r2,1) can be characterized as the maximum weak degree of a Π01 subset of 2 ω whose Turing upward closure is of positive measure. We exhibit a natural embedding of RT into Pw which is one-toone, preserves the semilattice structure of RT , carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2,1) in Pw.
55 citations
TL;DR: It is shown that many NP-complete problems can be solved in linear time in a quite uniform way (by systems which are very similar to each other), using only elementary membranes division (and not further ingredients, such as electrical charges).
55 citations
TL;DR: From semigroup theory it follows that the number of noncounting events of order @?1 is finite and the finite automata accepting such events over a fixed alphabet are homomorphic images of a universal automaton.
55 citations
TL;DR: This work introduces a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa relations between regions in topological spaces such as the real plane, and investigates the expressive power and computational complexity of logics obtained in this way.
Abstract: Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.
54 citations
TL;DR: The power of these operations and their relation to the usual AFL-operations is studied and it is shown that flow expressions, event expressions and even very restricted variants of them define all the recursively enumerable sets.
54 citations