Showing papers on "Computability published in 1986"

Theory of computation

Abstract: Part 1 Introduction: Preliminaries Languages and Computation. Part 2 Models: Finite Automata Regular Expressions Context-Free Grammars Pushdown Automata Turing Machines Functions, Relations, and Translations. Part 3 Properties: Family Relationships Closure Properties Decision Problems. Part 4 Onward: Further Topics.

Computability and physical theories

Abstract: The familiar theories of physics have the feature that the application of the theory to make predictions in specific circumstances can be done by means of an algorithm. We propose a more precise formulation of this feature—one based on the issue of whether or not the physically measurable numbers predicted by the theory are computable in the mathematical sense. Applying this formulation to one approach to a quantum theory of gravity, there are found indications that there may exist no such algorithms in this case. Finally, we discuss the issue of whether the existence of an algorithm to implement a theory should be adopted as a criterion for acceptable physical theories. “Can it then be that there is... something of use for unraveling the universe to be learned from the philosophy of computer design?” —J. A. Wheeler

Systolic array synthesis: computability and time cones

Abstract: : Many important algorithms in signal and image processing, speech and pattern recognition of matrix computations consist of coupled systems of recurrence equations. Systolic arrays are regular networks of tightly coupled simple processors with limited storage that provide cost effective high throughput implementations of many such algorithms. While there are some mathematical techniques for finding efficient schedules for uniform recurrence equations, there is no general theory for more general systems of recurrence equations. The first elements of such a theory are presented in this paper and constitute a significant step towards establishing a complete methodology that determines systolic array implementations for a very general class of coupled systems of recurrence equations; these implementations exhibit provably optimal computation time while satisfying various user-specified constraints.

An automatic method for generating random variates with a given characteristic function

Abstract: An automatic method is developed for the computer generation of random variables with a characteristic function satisfying certain regularity conditions. The method is based upon a generalization of the rejection method and exploits the duality between densities and their Fourier transforms. It takes finite time almost surely, does not use approximations or inversions, and does not require explicit knowledge of the characteristic function (only its computability is assumed—hence the adjective “automatic”). As a by-product, we show how the sum of n independent random variables with common density f can be generated in time essentially independent of n, at least when its characteristic function satisfies the above mentioned regularity conditions.

Computational complexity of real functions and real numbers

Abstract: Computability and complexity of real functions and real numbers are studied in a model where methods of recursive function theory as well as methods of numerical analysis can be used at a very low level of complexity. Topological properties turn out to be important for computational questions as well as for questions of existence of complexity bounds. As an example of application the computational complexity of the roots of real functions is studied with respect to the analytic properties of the functions.

Are Recursion Theoretic Arguments Useful in Complexity Theory

Abstract: Publisher Summary Recursion theory is that area of mathematical logic where one studies the qualitative aspects of computability. In complexity theory, which is part of computer science, one studies in addition quantitative aspects of computations. A number of open problems about the structure of NP where one can prove that even under the assumption PI≠NP recursion theoretic arguments will not suffice. The chapter presents polynomial time approximation schemes for some strongly NP-complete problems that arise—for example, in robotics. The chapter presents a survey for a proof of optimal lower bounds for two tapes versus one on deterministic and nondeterministic Turing machines. Results that show a substantial superiority of nondeterminism over determinism resp. co-nondeterminism over nondeterminism for one-tape Turing machines are given in the chapter. The chapter presents the proof of the desired result as the construction of a winning strategy for a two-person game.