Showing papers on "Computability published in 1971"

Computability by probabilistic Turing machines

Abstract: In the present paper, the definition of probabilistic Turing machines is extended to allow the introduction of relative computability. Relative computable functions, predicates and sets are discussed and their operations investigated. It is shown that, despite the fact that randomness is involved, most of the conventional results hold in the probabilistic case. Various classes of ordinary functions characterizable by computable random functions are introduced, and their relations are examined. Perhaps somewhat unexpectedly, it is shown that, in some sense, probabilistic Turing machines are capable of computing any given function. Finally, a necessary and sufficient condition for an ordinary function to be partially recursive is established via computable probabilistic Turing machines.

Computability Over the Continuum

Abstract: Publisher Summary This chapter discusses the computability over the continuum Classical descriptive set theory was concerned with classifying sets of real numbers according to the complexity of their definitions from a point of view that considered individual reals as given and thus not subject to analysis Modern hierarchy theory as an outgrowth of recursion theory takes as given only the (potentially) infinite sequence of natural numbers and analyses the complexity of functions and relations of both natural and real numbers The chapter provides several characterizations of the class of hyperprojective functions The abstract theory of hyperprojective relations on arbitrary structures applies directly to any type and leads to the hyper-order n-projective relations

Theory of computing in computer science education

Abstract: Theory of computing means the abstract study of the nature of computation and computing devices. By convention, the terminology is usually applied in a narrower sense to exclude numerical analysis. Thus, theory of computing includes the theory of finite automata, formal languages, computability, computational complexity, and some aspects of switching circuit theory and logical design. The deviation from the literal meaning of the term may have occurred because numerical analysis was already a well-established subject when the other components of this area were in their infancy. On the other hand, it may be a reflection of the emphasis on discrete mathematics by workers in theory of computing, in contrast to the preponderance of continuous methods in numerical analysis.

Equivalence of partial recursivity and computability by algorithms without concluding formulas

Abstract: The present note gives a proof of the equivalence of partial recursivity and computability by normal algorithms withou concluding formulas.