Showing papers on "Recursively enumerable language published in 1972"

Matrix Grammars with a Leftmost Restriction

Abstract: The family of languages generated by matrix grammars with context-free (context-free λ -free) core productions and with a leftmost restriction on derivations equals the family of recursively enumerable (context-sensitive) languages.

Reversal-bounded multi-pushdown machines

Abstract: This paper presents several representations of the recursively enumerable (re) sets The first states that every re set is the homomorphic image of the intersection of two linear context-free languages Another states that every re set is accepted by an on-line Turing acceptor with two pushdown stores such that in every computation, each pushdown store can make at most one reversal (that is, one change from "pushing" to "popping") It is shown that this automatatheoretic representation cannot be strengthened by restricting the acceptors to be either deterministic multitape, nondeterministic one-tape, or nondeterministic multicounter acceptors An investigation of the properties of reversal-bounded computations suggests that reversal bounds are not a "natural" measure of computational complexity for multitape Turing acceptors The above results are used to obtain an independence theorem for full semi-AFLs and an undecidability result for effective families of languages

Grammars with structured vocabulary: A model for the algol-68 definition

Abstract: Three aspects of the definition of algol 68 ( van Wijngaarden et al., 1969 ) are modelled: (1) A style of grammar (vWg) with infinitely many productions and variables, corresponding to the algol -68 “syntax≓ is defined; (2) The passage from “strict language≓ to “representation language≓ is formalized, essentially as an inverse gsm mapping; (3) Another style of grammar (vWpg), obtained by applying the “property≓ idea of Stearns and Lewis (1969) to vWg, and corresponding to the algol -68 syntax with “context conditions,≓ is defined. It is shown that being a representation language of a vW [p]g is characteristic of the recursively enumerable sets. A length-nondecreasing [and nondisappearance of indices] restriction on vW[p]g is given, and it is shown that being a representation language of a vW [p]g satisfying that restriction is characteristic of the A-free context-sensitive languages.

Minimal Upper Bounds for Sequences of Recursively Enumerable Degrees

Abstract: G. E. Sacks [3; p. 171, q. 4] asked whether there is a uniformly recursively enumerable (r.e.) ascending sequence of r.e. degrees which has as one of its minimal upper bounds another r.e. degree. We show (see the Corollary below) that the answer i s \" yes \". a is said to be a string if it is the restriction A[n] of a characteristic function A to the first n +1 non-negative integers for some number n. a is said to be a beginning of A of length n +1. We use (j> to denote the empty string. Let { s} of finite approximations to {Oc} such that

Program size and economy of descriptions: Preliminary Report

Abstract: Restricted programming languages, for example primitive recursive definition schemes, are very often not nearly as succinct in describing primitive recursive functions as a general programming language [1]. We show that as one increases the power of programming languages, one can obtain economies in program size by any recursive amount for even very simple functions. This parallels a situation in the arithmetic hierarchy, where it is possible to get a recursively enumerable set whose smallest recursively enumerable index is much larger than the smallest index for the same set considered, say, as a set recursively enumerable in o'. These phenomena follow from the fact that the ability to write programs which refer to the universal functions of an enumeration enables one to decrease significantly the size of programs. The notation, when not defined is that of [4].

Some post canonical systems in one letter

Abstract: Post's Canonical systems in one letter are viewed as operations on natural numbers. It is shown that, in a certain sense, these systems are capable of producing arbitrary recursively enumerable sets. Also, certain special cases are examined.

Degrees of unsolvability associated with Markov algorithms

Abstract: The degree representations of the general halting, word, and confluence problems for Markov algorithms are investigated. Each of these problems is shown to represent every r.e. (recursively enumerable) many-one degree but not every r.e. one-one degree of unsolvability. In the course of proving this we also show that every total recursive function is computed by a Markov algorithm which always halts and that the immortality problem for the class of Markov algorithms is Σ2o-complete.

$\alpha$-Degrees of $\alpha$-Theories

Abstract: Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α . α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it ( α -recursion theory) by providing the generalized notions of α -recursively enumerable, α -recursive and α -finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9]. Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment ( A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Some concepts for languages more powerful than context-free

Abstract: Some new restrictive devices for grammars are introduced: standard control-sets, G-control-sets, checking state grammars, besides the wellknown ones: matrix-sets, regular control-sets, programs. All these concepts turn out to be equivalent with respect to their generative power. So we get further characterizations of the class of recursively enumerable sets. The generative power of context-freelabel-grammars and Chomsky-grammars both with these restrictive devices is found to be equal. New characterizations of the unconditional-transfer programmed-grammars are introduced, namely unconditional-transfer programs correspond to all the other devices mentioned above under so-called “full-checking possibility” — i.e. the whole set of labels is the checking-set.