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.

Splitting an -recursively enumerable set

Abstract: We extend the priority method in a-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible ca. THEOREM: Let C be a regular car.e. set and D be a nonrecursive cs-r.e. set. Then there are regular or-r.e. sets A and B such that A U B = C, A n B=0, A, B

Some theorems on classes of recursively enumerable sets

Abstract: (2) to start a classification of recursively enumerable classes which parallels the classification of recursively enumerable sets initiated by Post in [15]. A mapping of numbers (or ordered n-tuples of numbers) onto numbers is called a function. Numbers and functions are denoted by small Latin letters, sets by small Greek letters and classes by capital Latin letters. 'a-,B' stands for the set of all numbers which belong to ax, but not to ,B, while 'A -B' stands for the class of all sets which belong to A, but not to B. We denote the range of the function f(x) by 'pf(x)' or 'pf'. If a is the range of the everywhere defined function f(x), we say that a can be generated by f(x). Let g(n, x) be defined for every ordered pair of numbers and let A be the class of all sets which occur at least once in the sequence pg(O, x), pg(l, x), We say that A can be generated by g(n, x). A set is recursively enumerable (r.e.) if it is empty or it can be generated by a recursive function of one variable. Similarly, a class of r.e. sets is recursively enumerable, if it is empty or consists only of the empty set, or the class of its nonempty members can be generated by a recursive function of two variables. Let 'F' stand for the class of all r.e. sets and 'Q' for the class of all finite sets. Then Q is a proper subclass of F and it is easily seen that both F and Q are r.e. classes [2, T2.2 ]. A set is called decidable or recursive if there exists a recursive procedure which enables us to test membership in the set, i.e., if its characteristic function is recursive. A decidable set is said to have a solvable decision problem, and an undecidable set an unsolvable decision problem. The class of all recursive sets is denoted by 'E'; this class properly includes Q and is properly

Automorphisms of the Lattice of Recursively Enumerable Sets

Abstract: Introduction The extension theorem revisited The high extension theorems The proof of the high extension theorem I The proof of the high extension theorem II Lowness notions in the lattice of r.e. sets Bibliography.

Minimal pairs and high recursively enumerable degrees

Abstract: A. H. Lachlan [2] and C. E. M. Yates [4] independently showed that minimal pairs of recursively enumerable (r.e.) degrees exist. Lachlan and Richard Ladner have shown (unpublished) that there is no uniform method for producing a minimal pair of r.e. degrees below a given nonzero r.e. degree. It is not known whether every nonzero r.e. degree bounds a r.e. minimal pair, but in the present paper it is shown (uniformly) that every high r.e. degree bounds a r.e. minimal pair. (A r.e. degree is said to be high if it contains a high set in the sense of Robert W. Robinson [3].) Theorem. Let a be a recursively enumerable degree for which a ′ = 0″. Then there are recursively enumerable degrees b 0 and b 1 such that 0 < bi < a for each i ≤ 1, and b 0 ⋂ b 1 = 0. The proof is based on the Lachlan minimal r.e. pair construction. For notation see Lachlan [2] or S. B. Cooper [1]. By Robinson [3] we can choose a r.e. representative A of the degree a , with uniformly recursive tower {A s, ∣ s ≥ 0} of finite approximations to A, such that C A dominates every recursive function where We define, stage by stage, finite sets B i,s , i ≤ 1, s ≥ 0, in such a way that B i, s + 1 ⊇ B i,s for each i, s, and { B i,s ∣ i ≤ 1, s ≥ 0} is uniformly recursive.