Showing papers on "Recursively enumerable language published in 1969"

Classes of computable functions defined by bounds on computation: Preliminary Report

Abstract: The structure of the functions computable in time or space bounded by t is investigated for recursive functions t. The t-computable classes are shown to be closed under increasing recursively enumerable unions; as a corollary the primitive recursive functions are shown to equal the t-computable functions for a certain recursive t. Any countable partial order can be isomorphically embedded in the family of t-computable classes partially ordered by set inclusion. For any recursive t, there is a recursive t' which is (approximately) equal to an actual running time such that the t-computable functions equal the t'-computable functions.

Recursively enumerable degrees and the conjugacy problem

Abstract: The principal result obtained is the theorem tha t for every recursively enumerable degree of unsolvability, there exists a finitely presented group whese con jugacy problem has tha t degree. (Parts I, I I , I I I and IV.) In Par t V this result is generalised to the theorem tha t certain complexes of recursively enumerable degrees of unsolvabil i ty m a y be obtained as the degrees of a complex of problems concerning conjugacy in a finitely presented group. I t is a pleasure to acknowledge the encouragement and inspiration provided by Professor William Boone during this work.

Two characterizations of the context-sensitive languages

Abstract: An n-dimensional bug-automation is generalization of a finite state acceptor to n-dimensions. With each bug B, we associate the language L(B) which is the set of top rows of the n-dimensional rectangular arrays accepted by B. One-dimensional bugs define trivially the regular sets. Twodimensional bugs define precisely the context-sensitive languages, while bugs of dimension 3 or greater define all the recursively enumerable sets. We consider also finite state acceptors with n two-way non-writing input tapes. For each such machine M, let domain (M) be the set of all strings which are the first component of some n-tuple of tapes accepted by M. For any n ≥ l, the domains of n-tape two-way finite state acceptors are precisely the same as the languages definable by n-dimensional bugs, so as a corollary, the domains of two-tape two-way finite state acceptors are precisely the context-sensitive languages.

Applications of Strict Pi 1 1 Predicates to Infinitary Logic.

Abstract: Consider the predicate of natural numbers defined by: where R is recursive. If, as usual, the variable ƒ ranges over ω ω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π 1 1 sets. If, however, ƒ ranges over 2 ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable. 3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-Konig Infinity Lemma.

Index Sets of Finite Classes of Recursively Enumerable Sets

Abstract: Let q0, q1,… be a standard enumeration of all partial recursive functions of one variable. For each i, let wi = range qi and for any recursively enumerable (r.e.) set α, let θα = {n | wn = α}. If A is a class of r.e. sets, let θA = the index set of A = {n | wn ∈ A}. It is the purpose of this paper to classify the possible recursive isomorphism types of index sets of finite classes of r.e. sets. The main theorem will also provide an answer to the question left open in [2] concerning the possible double isomorphism types of pairs (θα, θβ) where α ⊂ β.

On the representation of formal languages using automata on networks

Abstract: A new model of abstract automata is presented employing the concept of finite automata on a network Each normal network n provided with a one-way input tape determines a family of languages nl A representation theorem, analogous to the Chomsky-Schutzenberger representation theorem for context free languages1, is proved for the class nl One consequence is that nl is a principal full AFL generated by a closed set (one that contains all its prefixes) The converse is also proved, thereby establishing an equivalence between families of languages defined by normal networks and principal full AFLs generated by closed sets The representation theorem is applied to the push-down store and Turing machine networks to obtain a stronger version of the Ginsburg, Greibach, and Harrison representation theorem for recursively enumerable sets6

Abstract families of processors

Abstract: A “processor” is a Turing-like automaton with auxiliary storage. An “abstract family” of processors (AFP) consists of all processors that use the storage in the same way. Properties common to all AFP are derived. For a family of operations to be the output functions of some AFP, it is necessary and sufficient that certain word-sets representing its members form a full AFL (in the sense of Ginsburg and Greibach) closed under intersection and iterated finite substitution.2 For a family of word-sets to be the accepted languages of some AFP, it is necessary and sufficient that it be a full AFL closed under intersection and iterated finite substitution. The smallest full AFL of this kind is the family of all recursively enumerable sets.