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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, the authors present a proof of the extension theorem in the context of maximal sets and automorphisms and maximal sets, and show that the maximal sets can be represented by maximal sets.
Abstract: Introduction . 80 ?1. Background material 82 S2. Automorphisms and maximal sets . 88 S3. Satisfying condition (2.2) and the hypotheses of the Extension Theorem . 92 S4. Proof of the Extension Theorem Part I: Motivation . ...... 97 S5. Proof of the Extension Theorem Part II: Covering ......... 105 S6. Proof of the Extension Theorem Part III: Mappings . ...... 114 S7. Conclusion 118
95 citations
Journal Article•
TL;DR: The theoretical proof of the possibility to design universal programmable computers with the structure of such a system is obtained, hence the existence of universal test tube distributed systems is obtained.
Abstract: We define a symbol processing mechanism with the components (test tubes) working as splicing schemes in the sense of T. Head and communicating by redistributing the contents of tubes (in a similar way to the separate operation of Lipton-Adleman). (These systems are similar to the distributed generative mechanisms called Parallel Communicating Grammar Systems.) Systems with finite initial contents of tubes and finite sets of splicing rules associated to each component are computationally complete, they characterize the family of recursively enumerable languages. The existence of universal test tube distributed systems is obtained on this basis, hence the theoretical proof of the possibility to design universal programmable computers with the structure of such a system.
93 citations
TL;DR: Very simple languages are introduced which play the same role for the class of recursively enumerable languages that the Dyck languages play for theclass of context-free languages.
Abstract: Fixed point languages and equality languages of homomorphisms and dgsm mappings are consid- ered. Some basic properties of these classes of languages are proved, and it is shown how to use them to represent recursively enumerable sets. In particular, very simple languages are introduced which play the same role for the class of recursively enumerable languages that the Dyck languages play for the class of context-free languages. Finally, a new type of acceptor for defining equality languages is introduced. KEY WOADS AND PHRASES: equality language, fLxed point language, recursively enumerable language, determin- istic sequential machine, Turing machine, Post correspondence problem, shuffle, AFL generator, representation of languages
93 citations
TL;DR: A new hierarchy of sets and operators which is called the REA hierarchy for “recursively enumerable in and above” is introduced by composing (possibly) transfinite sequences of the pseudo-jump operators considered in Jockusch and Shore [1983].
Abstract: In this paper we introduce a new hierarchy of sets and operators which we call the REA hierarchy for “recursively enumerable in and above”. The hierarchy is generated by composing (possibly) transfinite sequences of the pseudo-jump operators considered in Jockusch and Shore [1983]. We there studied pseudo-jump operators defined by analogy with the Turing jump as ones taking a set A to A ⊕ for some index e. We would now call these 1-REA operators and will extend them to α-REA operators for recursive ordinals α in analogy with the iterated Turing jump operators (A → A(α) for α < and Kleene's hyperarithmetic hierarchy. The REA sets will then, of course, be the results of applying these operators to the empty set. They will extend and generalize Kleene's H sets but will still be contained in the class of set singletons thus providing us with a new richer subclass of the set singletons which, as we shall see, is related to the work of Harrington [1975] and [1976] on the problems of Friedman [1975] about the arithmetic degrees of such singletons. Their degrees also give a natural class extending the class H of Jockusch and McLaughlin [1969] by closing it off under transfinite iterations as well as the inclusion of [d, d′] for each degree d in the class. The reason for the class being closed under this last operation is that the REA operators include all operators and so give a new hierarchy for them as well as the sets. This hierarchy also turns out to be related to the difference hierarchy of Ershov [1968], [1968a] and [1970]: every α-r.e. set is α-REA but each level of the REA hierarchy after the first extends all the way through the difference hierarchy although never entirely encompassing even the next level of the difference hierarchy.
91 citations
30 Oct 1989
Abstract: Two results on interactive proof systems with two-way probabilistic finite-state verifiers are proved. The first is a lower bound on the power of such proof systems if they are not required to halt with high probability on rejected inputs: it is shown that they can accept any recursively enumerable language. The second is an upper bound on the power of interactive proof systems that halt with high probability on all inputs. The proof method for the lower bound also shows that the emptiness problem for one-way probabilistic finite-state machines is undecidable. In the proof of the upper bound some results of independent interest on the rate of convergence of time-varying Markov chains to their halting states are obtained. >
91 citations