scispace - formally typeset
Search or ask a question
Topic

Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.


Papers
More filters
Journal ArticleDOI
Xian Liu1
TL;DR: This paper proposes a new filled function that needs only one parameter and does not include exponential terms, and has better computability than the traditional ones.
Abstract: The Filled Function Method is an approach to finding global minima of multidimensional nonconvex functions. The traditional filled functions have features that may affect the computability when applied to numerical optimization. This paper proposes a new filled function. This function needs only one parameter and does not include exponential terms. Also, the lower bound of weight factor a is usually smaller than that of one previous formulation. Therefore, the proposed new function has better computability than the traditional ones.

98 citations

Journal ArticleDOI
Hisao Yamada1
TL;DR: As an attempt to investigate a general theory of real-time computability in digital computers, a subclass of Turing machines is formally introduced together with some classes of functions that are computable by them in real time.
Abstract: As an attempt to investigate a general theory of real-time computability in digital computers, a subclass of Turing machines is formally introduced together with some classes of functions that are computable by them in real time. Then the existence is established of a class of recursive functions that are not computable in real time by use of a class of machines, no matter how general we make the machines subject to a given constraint.

97 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied computability properties of Julia sets of quadratic polynomials and showed that a set is computable if, roughly speaking, its image can be generated by a computer with an arbitrary precision.
Abstract: Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most naive work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor's distance-estimator algorithm [Mil] uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show:

97 citations

Journal Article
TL;DR: This work compares transition super-cell systems with classic mechanisms in formal language theory, context-free and matrix grammars, E0L and ET0L systems, interpreted as generating mechanisms of number relations (the authors take the Parikh image of the usual language generated by these mechanisms rather than the language).
Abstract: We continue the investigation of the power of the computability models introduced in [Gh. Paun, Computing with membranes, TUCS Report 208, November 1998] under the name of transition super-cell systems. We compare these systems with classic mechanisms in formal language theory, context-free and matrix grammars, E0L and ET0L systems, interpreted as generating mechanisms of number relations (we take the Parikh image of the usual language generated by these mechanisms rather than the language). Several open problems are also formulated.

96 citations

01 Jan 2000
TL;DR: In this article, the authors explore aspects of computable analysis and topology in the framework of relative realizability, and demonstrate how to develop computable topology and analysis in the logic of modest sets.
Abstract: In this dissertation, I explore aspects of computable analysis and topology in the framework of relative realizability. The computational models are partial combinatory algebras with subalgebras of computable elements, out of which categories of modest sets are constructed. The internal logic of these categories is suitable for developing a theory of computable analysis and topology, because it is equipped with a computability predicate and it supports many constructions needed in topology and analysis. In addition, a number of previously studied approaches to computable topology and analysis are special cases of the general theory of modest sets. In the first part of the dissertation, I present categories of modest sets and axiomatize their internal logic, including the computability predicate. The logic is a predicative intuitionistic first-order logic with dependent types, subsets, quotients, inductive and coinductive types. The second part of the dissertation investigates examples of categories of modest sets. I focus on equilogical spaces, and their relationship with domain theory and Type Two Effectivity (TTE). I show that domains with totality embed in equilogical spaces, and that the embedding preserves both simple and dependent types. I relate equilogical spaces and TTE in three ways: there is an applicative retraction between them, they share a common cartesian closed subcategory that contains all countably based T0-spaces, and they are related by a logical transfer principle. These connections explain why domain theory and TTE agree so well. In the last part of the dissertation, I demonstrate how to develop computable analysis and topology in the logic of modest sets. The theorems and constructions performed in this logic apply to all categories of modest sets. Furthermore, by working in the internal logic, rather than directly with specific examples of modest sets, we argue abstractly and conceptually about computability in analysis and topology, avoiding the unpleasant details of the underlying computational models, such as Godel encodings and representations by sequences.

96 citations


Network Information
Related Topics (5)
Finite-state machine
15.1K papers, 292.9K citations
86% related
Mathematical proof
13.8K papers, 374.4K citations
86% related
Model checking
16.9K papers, 451.6K citations
85% related
Time complexity
36K papers, 879.5K citations
85% related
Concurrency
13K papers, 347.1K citations
85% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202344
2022119
202189
202098
2019111
201897