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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01 Apr 1960
TL;DR: Rankin et al. as mentioned in this paper studied the notion of compact connected sets in euclidean spaces and showed that they have a certain characteristic of compacted sets in a certain dimension.
Abstract: 2. A. H. Copeland, Admissible numbers in the theory of probability, Amer. J. Math, vol. 50 (1928) pp. 535-552. 3. F. Hausdorff, Mengenlehre, 3rd ed., Berlin, 1935. 4. B. Rankin, Computable probability spaces, Acta Math. vol. 103 (1960) pp. 89122. 5. H. Reichenbach, The theory of probability, Berkeley, 1949. 6. M. Sekanina, On a certain characteristic of compact connected sets in euclidean space, Casopia Pëst. Mat. vol. 82 (1957) pp. 129-136 (Czech) (reviewed in Math. Reviews vol. 19 (1958) p. 667). 7. J. Ville, Etude critique de la notion de collectif, Paris, 1939.
17 citations
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01 Jan 2019
TL;DR: The first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem is given, stating that every recursively enumerable problem -- in this case by a Minsky machine -- is Diophantine.
Abstract: We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem - in our case by a Minsky machine - is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer.
17 citations
01 Jun 1968
TL;DR: In this paper, the problem of recursively enumerable sets has been studied in the context of unary operations on sets of natural numbers, where the number of steps needed to compute a function does not depend on the size of the set.
Abstract: Suppose we wish to build up the class of recursively enumerable sets by starting with the set DI of natural numbers and constructing new sets from those already obtained using as little auxiliary machinery as possible. One way would be to start with a finite number of functions F1, * * *, Fk (of one variable, from and to 9t) such that every recursively enumerable set can be obtained from t by constructing new sets Fj [3 I where 5 is a previously obtained set. We can think of F1, * * *, Fk as unary operations on sets of natural numbers. Any set 8 obtained in this way is the range of a function F obtained by composition from F1, **, Fk. If we consider the values of F1, * * *, Fk as given, then the number of steps needed to compute Fn does not depend on n. Hence for all xe8, there exists a proof that xCS of bounded length in terms of F1, * * *, Fk (just as there is a one-step proof that a composite number is composite in terms of multiplication). We say a set of natural numbers is generated by F1, * * *, Fk if it is the range of a function obtained by composition from F1, * * * , Fk. Also a class e of sets is generated by F1, * * *, Fk if every nonempty set of e is generated by F1, l * , Fk and every set generated by Fl, @ , Fk is in e. EXAMPLE. Let Go, G1, * * * be the primitive recursive functions listed systematically so that the function G given by
16 citations
TL;DR: A variant of P systems is defined, namely, probabilistic rewriting P systems, where the selection of rewriting rules is probabilists, and it is shown that, with non-zero cut-point, probable rewriting rules are chosen.
Abstract: In this paper we define a variant of P systems, namely, probabilistic rewriting P systems, where the selection of rewriting rules is probabilistic. We show that, with non-zero cut-point, probabilis...
16 citations
TL;DR: The natural sets that can be enumerated by a computable function always seem to be either actually computable ( recursive ) or of the same complexity (with respect to Turing computability) as the Halting Problem, the complete r.e. sets.
Abstract: §1. Introduction. Natural sets that can be enumerated by a computable function (the recursively enumerable or r.e. sets) always seem to be either actually computable (recursive) or of the same complexity (with respect to Turing computability) as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K? Let be the r.e. degrees, i.e., the r.e. sets modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order (indeed, an uppersemilattice or usl)with least element 0, the degree (equivalence class) of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of . The (positive) solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved: Theorem 1.1 (Embedding theorem; Muchnik [1958], Sacks [1963]). Every countable partial ordering or even uppersemilattice can be embedded into . Theorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degree a there are r.e. degrees b, c < a such that b ∨ c = a. Theorem 1.3 (Sacks Density Theorem [1964]). For every pair of nonrecursive r.e. degrees a < b there is an r.e. degree c such that a < c < b.
16 citations