Showing papers on "Recursively enumerable language published in 1982"
TL;DR: Matijasevic's theorem implies the existence of a diophantine equation U such that for all x and v, x ∈ W v is also recursively enumerable, and the nonexistence of such an algorithm follows immediately from theexistence of r.e. nonrecursive sets.
Abstract: In 1961 Martin Davis, Hilary Putnam and Julia Robinson [2] proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the form Here P is a polynomial with integer coefficients and the variables range over positive integers. In 1970 Ju. V. Matijasevic used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevic proved [11] that the exponential relation y = 2 x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set W can be represented in the form From this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets. Now it is well known that the recursively enumerable sets W 1 , W 2 , W 3 , … can be enumerated in such a way that the binary relation x ∈ W v is also recursively enumerable. Thus Matijasevic's theorem implies the existence of a diophantine equation U such that for all x and v ,
98 citations
64 citations
TL;DR: It is shown that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes, and this method leads to a new class of recursively enumerable sets: r.e. generic sets.
Abstract: We show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0'. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice S of recursively enumerable sets with inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions. It is obvious that constructions of generic sets in set theory and certain con- structions of recursively enumerable (r.e) sets have something in common. It is typical for the construction of a generic set U that a sentence 0 about U is finally made true (i.e. M(U) I= 0) if one has unboundedly often the chance to make it true during the construction of U (i.e. Vp 3q E G}) are of incomparable Turing degree. The sentences 3x((G)o (x) # {e}(G) 1(x)) and 3x ((G)1(x) # {e}(G)o (x)) are finally made true for every e where one has infinitely often the chance to make them true without filling up the com- plement of (G)O, (G)1. Then we cannot have (G)o = {e}(G)1 because such an e
48 citations
32 citations
TL;DR: In this article, a relative recursive enumerability conjecture was first suggested by Cooper, which states that if a set C is a recursively enumerable (r.e., nonrecursive) set, then there is a set A such that A is r.e.
Abstract: Publisher Summary This chapter discusses relative recursive enumerability. It presents a theorem that was first suggested by a conjecture of Cooper. The theorem shows that let C be a recursively enumerable (r.e.), nonrecursive set. Then there is a set A such that A is r.e. in C, C is recursive in A, but A does not have r.e. degree. The chapter describes a single requirement for proving the theorem and presents a construction that meets this requirement. It outlines a method for meeting all the requirements, assuming that C is low, and describes the method for meeting the requirements without this assumption. The chapter further refinements of the theorem and also discusses the relationship of this discussion to that of Jockusch and Shore. The technique of replacing an infinite injury priority argument by a finite injury argument using lowness is due to Robinson.
25 citations
15 Aug 1982
TL;DR: A simple, yet comprehensive first order theory of lazy spaces relying on three axiom schemes asserting the principle of structural induction for finite objects, the existence of least upper bounds for directed sets, and the continuity of functions is developed.
Abstract: Since the publication of two influential papers on lazy evaluation in 1976 [Henderson and Morris, Friedman and Wise], the idea has gained widespread acceptance among language theoreticians—particularly among the advocates of “functional programming” [Henderson80, Backus78]. There are two basic reasons for the popularity of lazy evaluation. First, by making some of the data constructors in a functional language non-strict, it supports programs that manipulate “infinite objects” such as recursively enumerable sequences, which may make some applications easier to program. Second, by delaying evaluation of arguments until they are actually needed, it may speed up computations involving ordinary finite objects.First, there are several semantically distinct definitions of lazy evaluation that plausibly capture the intuitive notion.Second, non-trivial lazy spaces are similar in structure (under the approximation ordering) to universal domains (as defined by Scott [Scott81, Scott76]) such as the P
24 citations
TL;DR: In this paper, the exponential diophantine representation of recursively enumerable sets is discussed and several theorems and states that in the case of certain particular recursive sets one may delete a quantifier.
Abstract: Publisher Summary This chapter discusses the exponential diophantine representation of recursively enumerable sets The chapter presents several theorems and states that in the case of certain particular recursive sets one may be able to delete a quantifier The chapter shows that this is the case for primes, Mersenne primes, perfect numbers, and certain other recursive sets occurring in classical number theory These sets are all particular examples of Kalmar Elementary Relations The results are essentially the same as those of Jones–Matijasevic
17 citations
TL;DR: This note defines a complete set to be natural if all other recursively enumerable sets can be reduced to it by computationallysimple reductions and shows that these natural complete sets are all isomorphic under the same type of computationally simple mappings.
13 citations
TL;DR: The recursion-theoretic study of mathematical structures other than ω, two such structures which have been studied for their effective contents, begins with a foundational inquiry into effectiveness in topological spaces.
Abstract: The recursion-theoretic study of mathematical structures other than ω is now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombe's inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabin's study of computable fields [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]–[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others. Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological space X with a countable basis ⊿ for the topology. The space is fully effective , that is, the basis elements are coded into ω and the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to , the lattice of recursively enumerable (r.e.) subsets of ω , the collection of r.e. open subsets of X forms a lattice ℒ(X) under the usual operations of union and intersection.
13 citations
TL;DR: These complexity theoretic notions are shown to be equivalent to various recursion theory notions and are used to relate the complexity properties of an r.e. set A to its algebraic structure in the appropriate lattice and to the information it encodes.
Abstract: We survey a variety of recent notions and results for classifying the computational complexity of a recursively enumerable (r.e.) set. These complexity theoretic notions are shown to be equivalent to various recursion theoretic notions and are used to relate the complexity properties of an r.e. set A to its algebraic structure in the appropriate lattice and to the information it encodes.
12 citations
TL;DR: It is shown that synchronized concurrent expressions with three signal/wait operations are universal in the sense that they can simulate any semaphore controlled concurrent expressions and they can describe the class of recursively enumerable sets.
TL;DR: It is proved that every infinite set of random strings is not recursively enumerable, which asserts that in a strong sense random strings are not constructable.
Abstract: We prove that every infinite set of random strings is not recursively enumerable. In particular, the set of all random strings is not recursively enumerable. This property asserts that in a strong sense random strings are not constructable.
TL;DR: In this article, the Δ03-Automorphism method and non-invariant classes of degrees of degrees are discussed, as well as automorphisms of the Lattice of Recursively enumerable Sets.
Abstract: Robert I. Soare; d-Simple Sets, Small Sets, and Degree Classes by Manuel Lerman; Robert I. Soare; Automorphisms of the Lattice of Recursively Enumerable Sets by Peter Cholak; The Δ03-Automorphism Method and Noninvariant Classes of Degrees by Leo Harrington; Robert I. Soare Review by: Rod Downey The Journal of Symbolic Logic, Vol. 62, No. 3 (Sep., 1997), pp. 1048-1055 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2275592 . Accessed: 31/03/2014 17:38
TL;DR: The existence of a System function is shown: a kind of Combinatorial System defined by Cleave such that arbitrary distinct recursively enumerable one-one degrees can be represented by distinct decision problems for this System function.
12 Jul 1982
TL;DR: Homomorphic equality and inverse homomorphic equality operations provide simple and uniform characterizations of the recursively enumerable sets in terms of the regular sets, and of classes H(L λ MR) in Terms of L, which resemble the Chomsky-Schutzenberger theorem for context-free languages.
Abstract: The operations of a homomorphic equality and an inverse homomorphic equality are introduced. These operations are obtained from n-tuples of homomorphisms, incorporating the notion of an equality set. For one-tuples they are a homomorphism and an inverse homomorphism. Homomorphic equality and inverse homomorphic equality operations provide simple and uniform characterizations of the recursively enumerable sets in terms of the regular sets, and of classes H(L λ MR) in terms of L. These characterizations resemble the Chomsky-Schutzenberger theorem for context-free languages.
TL;DR: It is shown that any infinite recursively enumerable set can be split into two sets each of which has the property under consideration and that there are recursive sets with no optimal order of enumeration.
Abstract: It is known from work of P. Young that there are recursively enumerable sets which have optimal orders for enumeration, and also that there are sets which fail to have such orders in a strong sense. It is shown that both these properties are widespread in the class of recursively enumerable sets. In fact, any infinite recursively enumerable set can be split into two sets each of which has the property under consideration. A corollary to this result is that there are recursive sets with no optimal order of enumeration.
TL;DR: It is demonstrated that every context-free language is a homomorphic image of the intersection of two DOS languages and that every recursively enumerable language is the homomorphic images of the intersections of three DOS languages.
TL;DR: In this article, the upper semilattices of recursively enumerable n~-degrees and truth table degrees are compared, and the relation between complete sets and complete sets is discussed.
Abstract: Nauk, 34, No. 3, 137-168 (1979). 4. Yu. L. Ershov, "Positive equivalences," Algebra Logika, i0, 620-650 (1971). 5. G. N. Kobzev, "The complete ~#-degree," Algebra Logika, 13, 22-25 (1974). 6. G. N. Kobzev, "The semilattice of ~-degrees," Soobshcho Akad. Nauk Gruz. SSR, 90, 281283 (1978). 7. S. S. Marchenkov, "On the comparison of the upper semilattices of recursively enumerable n~-degrees and truth-table degrees," Mat. Zametki, 20, 19-26 (1976). 8. S. S. Marchenkov, "The tabular degrees of maximal sets," Mat. Zametki, 20, 373-381 (1976). 9. T. P. Cleave, "Some properties of recursively unseparable sets," Z. Math. Logik Grundl. Math., 16, 187-200 (1970). I0. A. H. Lachlan, "Two theorems on many-one degrees of recursively enumerable sets," Algebra Logika, II, 216-229 (1972). 11. D. A. Martin, "Classes of recursively enumerable sets and degrees of unsolvability," Z. Math. Logik Grundl. Math., 12, 295-310 (1966). 12. A. N. Degtev, "Relations between complete sets," Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 26-31 (1981).
TL;DR: In this article, the 1- and the reducibilities between m- and tt-reducibility have been studied, and it has been shown that if an r- is such a reducibility, let L r be the upper semilattice of recursively enumerable (r. e.) r-degrees and Th(L r ) the elementary theory of L r.
Abstract: Publisher Summary This chapter considers the 1- and the reducibilities between m- and tt-reducibility. It discusses that if an r- is such a reducibility, let L r be the upper semilattice of recursively enumerable (r. e.) r-degrees and Th(L r ) the elementary theory of L r . An m-degree is called undissolvable, if it contains only one one-degree. It has been noticed that every m-degree is undissolvable or contains an infinite chain of one-degrees. It has been proved that every non-recursive tt-degree contains at least two btt-degrees and if a T-degree a is such that a’≽O”, then it has no minimal r.e. tt-degrees.
Journal Article•
TL;DR: In this paper, it was shown that the class of languages accepted in real time by non-deterministic reversal-bounded multitape Turing machines, NP and recursively enumerable sets are closed under bi-language form operations when the homomorphisms are linear erasing, polynomial erasing and arbitrary respectively.
Abstract: Let H be a language over alphabet Ω and L a language over alphabet Σ, each symbol in Ω being a homomorphism or an anti-homomorphism on L. The set H(L) = {X(w)\Xe.H, WeL} is said to be a bi-language form. In this paper it is shown that the class of language accepted in real time by nondeterministic reversal-bounded multitape Turing machines, NP and the class of the recursively enumerable sets are closed under bi-language form operations when the homomorphisms are linear-erasing, polynomial-erasing and arbitrary respectively.