scispace - formally typeset
Search or ask a question
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.


Papers
More filters
Book ChapterDOI
TL;DR: This chapter describes the lattice embeddings into the recursively enumerable (r.e.) degrees preserving and shows the embeddability of M5 into the r.e. degrees preserving 1, which is harder as the usual proof for embedding requires infinitary traces.
Abstract: Publisher Summary This chapter describes the lattice embeddings into the recursively enumerable (r.e.) degrees preserving. The characterization of the finite lattices embeddable into the r.e. degrees is important to recursion theorists for two reasons: it provides insight into the structure of the r.e. degrees; and it constitutes a crucial step in determining the decidability of the universal existential theory of the partial ordering of the r.e. degrees and of the existential theory of the r.e. degrees in the language of lattices, possibly with constant symbols for the least and/or greatest element. The full characterization of the lattices embeddable into the r.e. degrees remains open. All known lattice embeddings into the r.e. degrees preserve the least element, 0. Preserving the greatest element, 1, turned out to be quite a bit harder. The chapter shows the embeddability of M5 into the r.e. degrees preserving 1, which is harder as the usual proof for embedding requires infinitary traces. The modular nondistributive five-element lattice, M5, can be embedded into the r.e. degrees preserving the greatest element. The nonmodular nondistributive five-element lattice, N5, can be embedded into the r.e. degrees preserving the greatest element.

4 citations

Journal ArticleDOI
TL;DR: The existence of a recursively enumerable lambda theory where the number is always one or infinite is shown and it is shown that there are λ-theories such that some terms have only two fixed points.
Abstract: A natural question in the λ-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in λ-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are λ-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) λ-theory where the range property is violated. A second, more complex example of a non-r.e. λ-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.

4 citations

Journal ArticleDOI
TL;DR: It is proved here that the first-order theory of all separated distributive lattices is undecidable, and Rubin's result which made the undecidability proof very simple.
Abstract: It is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1, Y2 such that Y1 c X1, Y2 c X2 and Y1 U Y2 = X1 U X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated distributive lattices is undecidable. Introduction. A distributive lattice with 0 is separated if it satisfies the following separation property: for every x1, x2 there are Yi < x1 and Y2 < x2 such that Yil Y2 are disjoint (i.e. Yi A Y2 = 0) and Yi V Y2 x1 V x2. Alistair Lachlan introduced separated distributive lattices in [La] in connection with his study of the first-order theory of the lattice of recursively enumerable sets. He mentioned to me a question whether the first-order theory of separated distributive lattices is decidable. The answer is negative: in ?2 a known undecidable theory is interpreted in the firstorder theory of separated distributive lattices. The known undecidable theory is the first-order theory of the following structures: a Boolean algebra with a distinguished subalgebra. About undecidability of it see [Ru]. Actually the first version of the undecidability proof used the closure algebra CACD of Cantor Discontinuum, i.e. the Boolean algebra of subsets of Cantor Discontinuum with the closure operation. CACD is easily interpretable in the separated distributive lattice of functions f from Cantor Discontinuum into {0, 1, 2} such thatf1(2) is clopen. By [GS1] a finitely axiomatizable essentially undecidable arithmetic reduces to the first-order theory of CACD, hence to the first-order theory of the mentioned separated distributive lattice of functions, hence to the first-order theory of separated distributive lattices. The last step is somewhat complicated by the fact that [GS1] does not interpret the standard model N of arithmetic in CACD. (Even though [GS2] reduces the second-order theory of N to the first-order theory of CACD, [GS3] proves that N cannot be interpreted in CACD.) However the Boolean algebra of subsets of Cantor Discontinuum with a distinguished subalgebra of clopen (closed and open) sets is easily interpretable in CACD. This way I came to use Rubin's result which made the undecidability proof very simple. From the other side the cited result of [GS1] can be used to reprove Rubin's theorem and Received October 12 1980; revised August 30, 1981. 'The results were obtained and the paper was written during the Logic Year in the Institute for Advanced Studies of Hebrew University. ? 1983, Association for Symbolic Logic 0022-4812/83/4801-0020/$01.40

4 citations

Journal ArticleDOI
TL;DR: It is proved that any recursively enumerable language can be obtained as the intersection of a regular language and the language of simple eco–grammar systems where the active teams are organized according to different conditions of team constitution.
Abstract: In this paper we extend the conditions of dynamic team constitution in simple eco–grammar systems, motivated by the bottom–up–clustering algorithm. The relationships of simple eco–grammar systems formed according to the newly introduced conditions to each other as well as to certain language classes of the Chomsky hierarchy and L systems are established. We prove that any recursively enumerable language can be obtained as the intersection of a regular language and the language of simple eco–grammar systems where the active teams are organized according to different conditions of team constitution. We also propose some further research directions.

4 citations

Book ChapterDOI
12 Dec 2018
TL;DR: It is proved that any Turing machine can be simulated by a binary-state neural network extended with three analog neurons (3ANNs) having rational weights, with a linear-time overhead.
Abstract: The languages accepted online by binary-state neural networks with rational weights have been shown to be context-sensitive when an extra analog neuron is added (1ANNs). In this paper, we provide an upper bound on the number of additional analog units to achieve Turing universality. We prove that any Turing machine can be simulated by a binary-state neural network extended with three analog neurons (3ANNs) having rational weights, with a linear-time overhead. Thus, the languages accepted offline by 3ANNs with rational weights are recursively enumerable, which refines the classification of neural networks within the Chomsky hierarchy.

4 citations

Network Information
Related Topics (5)
Decidability
9.9K papers, 205.1K citations
91% related
Multimodal logic
6.1K papers, 178.8K citations
86% related
Type (model theory)
38.9K papers, 670.5K citations
86% related
Dynamic logic (modal logic)
6.6K papers, 210.3K citations
86% related
Mathematical proof
13.8K papers, 374.4K citations
84% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202220
202127
202022
201918
201823