Showing papers on "Recursively enumerable language published in 1970"

On Some Games Which Are Relevant to the Theory of Recursively Enumerable Sets

Abstract: In ? 1 we shall describe a class of two-person infinite games called basic games which may be applied to the elementary theory T(9Z) of recursively enumerable (r.e.) sets. The theory T(ER) will be defined in ? 1 and has been studied previously by the author in [5] and [6]. A good source for background material is Rogers [9]. Our reason for studying basic games is that every theorem of T(ER) known at the present time can be proved by constructing an effective winning strategy for a suitable basic game. This contention will be supported in ? 2 by a number of examples. In ? 3 we discuss briefly the solution of a particular kind of basic game. In ? 4 we show that two natural subclasses of the class of basic games are not adequate for deriving all theorems of T(9I). In ? 5 we have summarized our reasons for thinking that the gametheoretic approach to recursion theory is a useful one, and have listed some open questions. I am grateful to the referee for many valuable suggestions regarding the format of this paper.

Periodically time-variant context-free grammars

Abstract: For context-free grammars, all of the following restrictive devices are equivalent with respect to generative power: a regular control language, programming, a set of matrices, and periodic time-variance. This leads to several new characterizations of the family of recursively enumerable languages. Eg., every recursively enumerable language is generated by a context-free grammar, where the set of productions available at the ith step of a derivation is a periodic function of i.

The degree hierarchy of undecidable problems of formal grammars

Abstract: After a brief discussion of historical matters in §1, twenty-seven predicates of formal grammers are introduced in §2. The next two sections discuss recursively enumerable predicates and nonrecursively enumerable predicates, respectively. These results show that the degree of unsolvability of a predicate is determined by its domain of definition. The paper concludes with a degree diagram and suggestions for further development. From a more comprehensive point of view these results complement the computational complexity classification of solvable problems and extend that classification to unsolvable problems based on their degree of unsolvability.

Abstract families of languages generated by bounded languages

Abstract: : An AFL is defined to be bounded if it is generated by a set of bounded languages. It is shown that the smallest full AFL containing a bounded AFL is itself a bounded AFL. An AFL is exhibited which is contained in a bounded AFL but which is not itself bounded. It is shown that a bounded AFL containing a non-regular set cannot be closed under substitution. If g is any set of bounded languages, it is shown that the smallest intersection closed AFL containing g cannot contain all recursively enumerable languages. In contrast, a one-letter language L is exhibited for which the smallest intersection closed full AFL containing L does contain all recursively enumerable languages. A set g of languages is independent if every proper subset of g fully generates a smaller full AFL then g does. It is shown that there exist infinite independent sets of languages. A representation in terms of multitape sequential transducers is given for each language in the smallest intersection closed (full) AFL containing a given language. (Author)

Bases and α -dimensions of countable vector spaces with recursive operations

Abstract: This paper is based on the notions originally described by Dekker [2], [3], and the reader is referred to these for explanation of notation etc. Briefly, we are concerned with a countably infinite dimensional countable vector space Ū with recursive operations, regarded as being coded as a set of natural numbers. Necessarily, then, Ū must be a vector space over a field which itself is in some sense recursively enumerable and has recursive operations.

The state complexity of Turing machines

Abstract: Given a number of tape symbols, we define the state complexity of a partial-recursive function f as the minimal number of states necessary for a Turing machine that computes f. A similar definition gives us the state complexity of recursively enumerable sets and hence of abstract languages. We can show that all complexity classes are non-empty and finite. If a certain value of complexity is surpassed, then every language-complexity class contains at least one language of each of the following types: finite, nonfinite regular, nonregular context-free, context-sensitive but not context-free, recursive but not context-sensitive, enumerable but not recursive. The state complexity of a function or language is closely related to the amount of description or information needed to define the function or language.

Commutativity and common fixed points in recursion theory

Abstract: Let N be the set of nonnegative integers and, for eEN, let 45e be the partial recursive function of one argument having index e. In 1938 [1, The Recursion Theorem] Kleene showed that if f is any recursive function then, for some number c, /,c-qfCf(,). It follows that if We (the recursively enumerable (r.e.) set with index e) is defined as the domain of 0e, then W= Wf(,). Call a number-theoretic function h well-defined on the r.e. sets if, for all m, nEN, Wm Wn*Wh(m) -Wh(n). In this paper we show that if f, g are recursive functions which are well-defined on the r.e. sets and which commute as maps of the r.e. sets (i.e., for all n&VN, Wf(g(n)) = Wg(f(n))), then they have a common fixed point (i.e., for some e=N, We = Wf(e) = Wg()) We also give an example which shows that the assumption of well-definedness cannot be eliminated. First we prove a lemma related to the Myhill-Sheplherdson Theorem [2, p. 359, Theorem XXIX (6) ]. From now on, wheneverf is well defined on the r.e. sets and W is an r.e. set, we shall write f(W) for Wf(e) where e is any number such that W We.