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
01 Jan 2010
TL;DR: Computation with advice as discussed by the authors is a generalization of both computation with discrete advice and Type-2 Nondeterminism, and it has been shown that correct solutions are guessable with positive probability.
Abstract: Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solutions being guessable with positive probability. In the framework of computation with advice, it is possible to define computational complexity for certain concepts of hypercomputation. Finally, some examples are given which illuminate the interplay of uniform and non-uniform techniques in order to investigate both computability with advice and the Weihrauch lattice.

18 citations

Posted Content
TL;DR: In this article, fixed points of functors and fibrations are used to model solution concepts abstractly, so that solving equations whose arguments are solution concepts can be solved abstractly.
Abstract: The present paper is structured around two main constructions, fixed points of functors and fibrations and sections of functors. Fixed points of functors are utilized to resolve problems of infinite regress that have recently appeared in economics. Fibrations and sections are utilized to model solution concepts abstractly, so that we can solve equations whose arguments are solution concepts. Most of the objects (games, solution concepts) that we consider can be obtained as some kind of limit of their finite subobjects. Some of the constructions preserve computability. The paper relies heavily on recent work on the semantics of program- ming languages.

18 citations

Proceedings ArticleDOI
01 Jun 2017
TL;DR: In this paper, the authors considered the problem of finding the minimum number of robots required to traverse a ring of arbitrary size in order to solve the perpetual exploration problem, where each node is required to be infinitely often visited by a robot.
Abstract: We consider systems made of autonomous mobile robots evolving in highly dynamic discrete environment i.e., graphs where edges may appear and disappear unpredictably without any recurrence, stability, nor periodicity assumption. Robots are uniform (they execute the same algorithm), they are anonymous (they are devoid of any observable ID), they have no means allowing them to communicate together, they share no common sense of direction, and they have no global knowledge related to the size of the environment. However, each of them is endowed with persistent memory and is able to detect whether it stands alone at its current location. A highly dynamic environment is modeled by a graph such that its topology keeps continuously changing over time. In this paper, we consider only dynamic graphs in which nodes are anonymous, each of them is infinitely often reachable from any other one, and such that its underlying graph (i.e., the static graph made of the same set of nodes and that includes all edges that are present at least once over time) forms a ring of arbitrary size. In this context, we consider the fundamental problem of perpetual exploration: each node is required to be infinitely often visited by a robot. This paper analyzes the computability of this problem in (fully) synchronous settings, i.e., we study the deterministic solvability of the problem with respect to the number of robots. We provide three algorithms and two impossibility results that characterize, for any ring size, the necessary and sufficient number of robots to perform perpetual exploration of highly dynamic rings.

18 citations

Proceedings ArticleDOI
05 Jul 2016
TL;DR: The main results relate Kolmogorov’s entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X, and offer some guidance towards suitable notions of complexity for higher types.
Abstract: We promote the theory of computational complexity on metric spaces: as natural common generalization of (i) the classical discrete setting of integers, binary strings, graphs etc. as well as of (ii) the bit-complexity theory on real numbers and functions according to Friedman, Ko (1982ff), Cook, Braverman et al.; as (iii) resource-bounded refinement of the theories of computability on, and representations of, continuous universes by Pour-El& Richards (1989) and Weihrauch (1993ff); and as (iv) computational perspective on quantitative concepts from classical Analysis: Our main results relate (i.e. upper and lower bound) Kolmogorov’s entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X. The upper bounds are attained by carefully crafted oracles and bit-cost analyses of algorithms perusing them. They all employ the same representation (i.e. encoding, as infinite binary sequences, of the elements) of such spaces, which thus may be of own interest. The lower bounds adapt adversary arguments from unit-cost Information-Based Complexity to the bit model. They extend to, and indicate perhaps surprising limitations even of, encodings via binary string functions (rather than sequences) as introduced by Kawamura&Cook (SToC’2010, §3.4). These insights offer some guidance towards suitable notions of complexity for higher types.

18 citations

Journal ArticleDOI
TL;DR: The tractability thesis as discussed by the authors identifies the intuitive class of computationally tractable problems with a precise class of problems whose solutions can be computed in polynomial time, and it has been widely accepted in computer science.
Abstract: According to Church’s thesis, we can identify the intuitive concept of effective computability with such well-defined mathematical concepts as Turing computability and partial recursiveness. The almost universal acceptance of Church’s thesis among logicians and computer scientists is puzzling from some epistemological perspectives, since no formal proof is possible of a thesis that involves an informal concept such as effectiveness. Elliott Mendelson has recently argued, however, that equivalencies between intuitive notions and precise notions need not always be considered unprovable theses, and that Church’s thesis should be accepted as true. I want to discuss a thesis that is nearly as important in current research in computer science as Church’s thesis. I call the newer thesis the tractability thesis, since it identifies the intuitive class of computationally tractable problems with a precise class of problems whose solutions can be computed in polynomial time. After briefly reviewing the theory of intractability, I compare the grounds for accepting the tractability thesis with the grounds for accepting Church's thesis. Intimately connected with the tractability thesis is the mathematical conjecture, whose meaning I shall shortly explain, that P≠NP. Unlike Church's thesis, this conjecture is precise enough to be capable of mathematical proof, but most computer scientists believe it even though no proof has been found. As we shall see below, understanding the grounds for acceptance of the conjecture that P≠NP has implications for general questions in the philosophy of mathematics and science, especially concerning the epistemological importance of explanatory and conceptual coherence.

18 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