Showing papers on "Computability published in 1978"

On the Tape Complexity of Deterministic Context-Free Languages

Abstract: Let DSPACE(L(n)) denote the family of languages recognized by deterministic L(n)-tape bounded Turmg machines The pnnopal result described m this paper is the equivalence of the following statements (l) The determtmsttc context-free language L~ 2) (described m the paper) is m DSPACE(Iog(n)) (2) The simple LL(I) languages are m DSPACE(tog(n)) (3) The simple precedence languages are in DSPACE(Iog(n)). (4) DSPACE(Iog(n)) is identical to the famdy of languages recogmzed by deterministic two-way multlhead pushdown automata m polynomml tmae These results are obtained by constructing a determlmstlc context-free language L~ 2~ which is log(n)-complete for the family of determlmstlc context-free languages In other words, a tape hardest deterministic context-free language is described The best upper bound known on the tape complexity of (deterministic) context-free languages is (log(n)) 2

Logic for Mathematicians

Abstract: Preface 1. Informal statement calculus 2. Formal statement calculus 3. Informal predicate calculus 4. Formal predicate calculus 5. Mathematical systems 6. The Godel incompleteness theorem 7. Computability, unsolvability, undecidability Appendix Hints and solutions to selected exercises References and further reading Glossary of symbols Index.

Machines, languages, and computation

Abstract: In what case do you like reading so much? What about the type of the machines languages and computation book? The needs to read? Well, everybody has their own reason why should read some books. Mostly, it will relate to their necessity to get knowledge from the book and want to read just to get entertainment. Novels, story book, and other entertaining books become so popular this day. Besides, the scientific books will also be the best reason to choose, especially for the students, teachers, doctors, businessman, and other professions who are fond of reading.

Elementary bounded languages

Abstract: The POL be the class of polynomials having nonnegative integer coefficients and EXP the class of exponential functions. We call the closure of POL ∪ EXP under superposition, primitive recursion, and exponentiation, the class of elementary functions ( EF ). We have obtained that every elementary bounded language (i.e., language in the form { w 1 f 1 ( n ) ⋯ w t f t ( n | n ࢠ N k , w i words f i ࢠ EF }) is context-sensitive. A concept for the computability of the functions usinggrammars is given, and it is shown that every function from EF is computable inthis manner, using context-sensitive grammars. By considering a new unaryoperation with respect to languages, called polynomial iteration (which is ageneralization of star closure) we prove that the class of context-sensitivelanguages is closed with respect to this operation but neither the class of contextfreenor the class of regular languages is closed.

On the Foundation of General Recursion Theory: Computations Versus Inductive Definability.

Abstract: Publisher Summary General recursion theory can be approached in a number of different ways. One starting point is to analyze the relation, which expresses that the “computing device” named or coded by a and acting on the input sequence ϭ = (x l ,……,x n ) gives z as an output. The history of this notion goes back to the very foundation of the theory of general recursion in the mid 1930's. It can be traced from the theory of Turing and other types of idealized machine computability. In 1959, this relation was taken as the basic in developing the theory of recursion in higher types subsequently it was adopted in the study of prime and search computability over more general domains. Indexing was also behind various other abstract approaches. “Computations” is not the only possible way of doing general recursion theory. The purpose of this note is to set out briefly some other approaches based on inductive definability and fixed point operators, and to compare them to the theories based on computations.

Computability and cognition

Abstract: Linguists conceive of grammar in much the same manner as all of us perceive the ambiguous figure of the rabbit/duck. Looked at in one way, grammars are abstract mathematical objects, statable within axiomatic systems, and susceptible to the techniques of logical analysis. There are as many different grammars as there are sets of primitive expressions, formation rules, and rules of interpretation. As with our knowledge of other abstract objects, the thoughts of human beings may be radically imperfect characterizations of the languages represented in thought. It happens that a tiny fraction of these abstract objects are used by members of our species to communicate. The role of human activity is not to determine the nature of a language so much as to determine which language it is that human beings use. Languages, like numbers, exist independently of thought. Yet linguists can reorient their vision, and see the duck's bill of grammar as a bunny's ear. Thus transformed, grammars are psychological entities whose natures are determined by the mental contents of certain human beings. Given this connection between thought and language, it is impossible that competent speakers of English should have internalized rules that only imperfectly represent the grammar of English. 1 If speakers of English were never subject to memory lapses, slips of the tongue, limitations of attention and computation space (that is, if we abstract from those factors that linguists lump together under the rubric of performance), then English would simply be what English speakers can do. The relationship between these two kinds of approaches is old and unfriendly. Platonism and conceptualism are alternative and incompatible theories in many problem areas; the dispute among philosophers of mathematics over the nature of number provides an obvious example. Within philosophy of language itself, Frege formulated a now classic argument for shunning the psychological interpretation of grammar. For Frege, the nature of a language must be something that all speakers of that language have in common, and Frege thought that at the level of psychology, they have nothing of interest in common. Frege's variability argument encapsulates a

How to show something is not: Proofs in formal language and computability theory

Abstract: Most introductory courses in theoretical computer science (formal language theory or computability theory) start with a seemingly endless series of definitions, including what it means for a grammar or language to be regular, context-free, etc., or what it means for a function to be recursive, primitive recursive, or partial recursive. Bright students immediately ask two questions. First, what are examples of languages or functions that belong to one class but not the other? Second, is some particular language context-free, or is a particular function recursive?We must develop new techniques which allow us to give a negative answer to question two (and thus to answer question one as well). In this note we will discuss some of the methods that are often used in elementary proofs in formal language theory and computability theory.

Computer Science And Recursion Theory

Abstract: One of the basic concerns of classical mathematical logic has been a rigorous definition of “mathematical proof". A proof may be discovered by luck, genius or accident. But once discovered it is mechanically checkable. Thus the most general type of mechanically checkable procedure provides a definition of the most general kind of proof. An investigation of “mechanical checkability" leads naturally to the notion of “computable process". Recursion theory is that branch of mathematical logic which studies computability theory. From its very inception recursion theory has been so closely associated with theoretical computer science that it is sometimes difficult to tell where one begins and the other ends. Both appear to have a common origin in the 1930's and 1940's - when the fundamental results of Turing, Post, Herbrand-Godel, Kleene and Church on computability were obtained [52,53,35,36,19,25,26,11,12,13]. Although the work of each of these authors is based on a different intuition as to the nature of computability, they all share a common feature. In each, the approach is discrete and atomic. No attempt was made to use machines with continuously varying parameters - e.g. analog computers - as a basis for understanding computability. The author finds this regrettable and is trying to remedy the situation [39]. At all events, all of these discrete, atomic definitions have been proved to be equivalent. It is now widely accepted that the concept “computable function of natural numbers" is correctly identified with “partial recursive function". Lack of space forces us to be selective rather than exhaustive. We discuss briefly five areas of interest to the computer scientist. The purpose of this paper is to explore the interaction of recursion theory with computer science in these areas. We consider, not only similarities between the two fields, but also points of difference. We discuss how recursion theory supplied computer science with specific mathematical definitions and techniques. We also consider how the computer scientist shaped (and is still shaping) this recursion-theoretic material to his own needs. The five areas are not mutually exclusive: there are numerous connections between them. In the interest of clarity and brevity, reference is made only to those works which are easily understandable to the beginner. No attempt is made to make the bibliography complete or up-to-date.