Showing papers on "Recursively enumerable language published in 1988"

On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines

Abstract: A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis. >

1-Genericity in the Enumeration Degrees

Abstract: The structure of the Turing degrees of generic and n -generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]). In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree. Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree a e which is minimal-like ; that is, all functions in degrees below a e have partial recursive extensions.

Σ2-collection and the infinite injury priority method

Abstract: We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P−) together with Σ2-collection (BΣ2). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither lown nor highn-1, while true, cannot be established in P− + BΣn+1. Consequently, no bounded fragment of first order arithmetic establishes the facts that the highn and lown jump hierarchies are proper on the recursively enumerable degrees.

Numberings of R 1 ∪F

Abstract: It is a fundamental result of computability theory that the set of the computable total (one-place) functions (denoted by R1) is ~ recursively enumerable (r.e.), i.e. it does not have a computable universal function. Therefore, as a technical necessity, all programming languages must contain programs for partial functions, though in general a programmer seeks to avoid them. It seems to be an interesting question to determine necessary and sufficient properties of families F of partial functions such that R 1UF becomes r.e. This theme was explicitly considered by W. Menzel and V. Sperschneider [MS]. They investigated the families F of finite functions defined on an initial segment of o~ which yield an r.e. family when adjoined to R 1. If F is canonically enumerable a quite complete characterization can be given, however, in general F may have pathological features, e.g. it needn't be r.e. An interesting special case is the restriction to families F=Fin(M), M.~z~o; here Fin(M) denotes the set of all finite functions defined on an initial segment of length le M. The "length problem" of V. Sperschneider [Sp] asks for a characterization of the sets M such that R1uFin(M ) is r.e. Of particular interest is the case when M is co-r.e. ; V. Sperschneider has shown that R1uFin(MC ) is r.e. if M is r.e., co-infinite and not hypersimple. It was an open

Time varying pushdown automata

Abstract: Time varying pushdown automata (PDA) are defined and equivalence between two modes of acceptance shown. It is seen that periodically time varying pushdown automata accept exactly the class of context-free languages. Time varying generalized PDA are defined and their equivalence to terminal weighted context free grammars in GNF shown. It is shown that TVGPDA can be simulated by TVPDA. Thus TVPDA give another machine characterization of recursively enumerable sets.

Infima of recursively enumerable truth table degrees

Abstract: Etude des infima des degres de tables de verite recursivement enumerables. Reponse par la negative a une question d'Odifreddi sur la reductibilite

Trading reversals for alternation

Abstract: The relation between reversals and alternation is studied in two simple models of computation: the two-counter machine with a one-way input tape whose counters make only one reversal (1-reversal 2CM) and the one-way pushdown automation whose pushdown store makes only one reversal (1-reversal PDA). It is known that nondeterministic 1-reversal 2CMs (and, more generally, 1-reversal mCMs when there are m counters, m>0) can be simulated by a log n space-bounded nondeterministic TMs, and nondeterministic 1-reversal PDAs accept exactly the linear context-free languages. When nondeterministic is generalized to alternating, it is shown that alternating 1-reversal 2CMs accept all recursively enumerable languages and that alternating 1-reversal PDAs accept exactly the languages accepted by exponential time-bonded deterministic TMs. Since deterministic 2CMs with unrestricted counters accept all recursively enumerable languages, the first results show that reversals can be traded for alternation. >

The existence of minimal honest polynomial degree below and recursively enumerable degrees

Abstract: In [1] Homer introduced the honest polynomial reducibility and proved that under this new reducibility a set of minimal degree below O″ is constructed under the assumption thatP=NP. In this paper we will prove that under the same assumption a set of minimal degree can be constructed below any recursively enumerable degrees. So under the honest polynomial reducibility a set of low minimal degree does exist.

Splitting recursively enumerable subalgebras in recursive Boolean algebras

Abstract: A Boolean algebraB=\(\left\langle {B, \wedge , \vee , eg } \right\rangle \) is recursive ifB is a recursive subset of ω and the operations Λ, v and ┌ are partial recursive. A subalgebraC ofB is recursive an (r.e.) ifC is a recursive (r.e.) subset of B. Given an r.e. subalgebraA, we sayA can be split into two r.e. subalgebrasA1 andA2 if (A1 ∪A2)*=A andA1 ∩A2={0, 1}. In this paper we show that any nonrecursive r.e. subalgebra ofB can be split into two nonrecursive r.e. subalgebras ofB. This is a natural analogue of the Friedberg's splitting theorem in ω recursion theory.