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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 is shown that the class of languages generated by generalized context-free grammars (gcfg's) introduced by Pollard is exactly the same class of recursively enumerable sets.
Abstract: It is shown that the class of languages generated by generalized context-free grammars (gcfg's) introduced by Pollard is exactly the class of recursively enumerable sets. Next, a subclass of gcfg's called multiple context-free grammars (mcfg's) is introduced and it is shown that the class of languages generated by mcfg's properly contains the class of context-free languages and is properly contained in the class of context-sensitive languages. In mcfg's, it is possible to account for structures involving discontinuous constituents in a particularly simple manner. Such concepts as phrase structure and derivation tree in context-free grammars (cfg's) can be extended naturally in mcfg's. Furthermore, the class of languages generated by mcfg's enjoys the formal language-theoretic closure properties that the class of context-free languages does.
8 citations
TL;DR: An undecidability result for products of matrices is deduced which can be viewed as a variant of Rice's theorem stating that all nontrivial properties of recursively enumerable sets are undecidable.
8 citations
TL;DR: This paper gives a precise characterization for the complexity of the problem of proving equal two streams defined with a finite number of equations: 0.
Abstract: This paper gives a precise characterization for the complexity of the problem of proving equal two streams defined with a finite number of equations: 0. Since the 0 class includes properly both the recursively enumerable and the co-recursively enumerable classes, this result implies that one can find no mechanical procedure to say when two streams are equal, as well as no procedure to say when two streams are not equal. In particular, there is no complete proof system for equality of streams and no complete system for dis-equality of streams.
8 citations
Dissertation•
01 Jan 2007
TL;DR: In this paper, the equivalence of recursively enumerable and diophantine sets was shown for function fields of curves over valued fields in characteristic zero under some conditions on the curve and the valuation.
Abstract: In 1900, the German mathematician David Hilbert proposed a list of 23 unsolved mathematical problems. In his Tenth Problem, he asked to find an algorithm to decide whether or not a given diophantine equation has a solution (in integers). Hilbert's Tenth Problem has a negative solution, in the sense that such an algorithm does not exist. This was proven in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson. Actually, this result was the consequence of something much stronger: the equivalence of recursively enumerable and diophantine sets (we will refer to this result as "DPRM"). The first new result in the thesis is about Hilbert's Tenth Problem for function fields of curves over valued fields in characteristic zero. Under some conditions on the curve and the valuation, we have undecidability for diophantine equations over the function field of the curve. One interesting new case are function fields of curves over formal Laurent series. The proof relies on the method with two elliptic curves as developed by K. H. Kim and F. Roush and generalised by K. Eisentrager. Additionally, the proof uses the theory quadratic forms and valuations. And especially for non-rational function fields there is some algebraic geometry coming in. The second type of results establishes the equivalence of recursively enumerable and diophantine sets in certain polynomial rings. The most important is the one-variable polynomial ring over a finite field. This is the first generalisation of DPRM in positive characteristic. My proof uses the structure of finite fields and in particular the properties of cyclotomic polynomials. In the last chapter, this result for polynomials over finite fields is generalised to polynomials over recursive algebraic extensions of a finite field. For these rings we don't have a good definition of "recursively enumerable" set, therefore we consider sets which are recursively enumerable for every recursive presentation. We show that these are exactly the diophantine sets. In addition to infinite extensions of finite fields, we also show the analogous result for polynomials over a ring of integers in a recursive totally real algebraic extension of the rationals. This generalises results by J. Denef and K. Zahidi.
8 citations
Metz1
TL;DR: It is shown that tissue P systems with symports having 3 cells and symport/antiport rules of minimal weight generate all recursively enumerable sets of numbers.
Abstract: We show that tissue P systems with symport/antiport having 3 cells and symport/antiport rules of minimal weight generate all recursively enumerable sets of numbers. Constructed systems simulate register machines and have purely deterministic behaviour. Moreover, only 2 symport rules are used and all symbols of any system are present in finite number of copies (except for symbols corresponding to registers of the machine). At the end of the article some open problems are formulated.
8 citations