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Computability

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


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TL;DR: Using nested singularities (which are built), it is shown how to decide higher levels of the corresponding arithmetical hierarchies and not only is Zeno effect possible but it is used to unleash the black hole power.
Abstract: The so-called Black Hole model of computation involves a non Euclidean space-time where one device is infinitely "accelerated" on one world-line but can send some limited information to an observer working at "normal pace". The key stone is that after a finite duration, the observer has received the information or knows that no information was ever sent by the device which had an infinite time to complete its computation. This allows to decide semi-decidable problems and clearly falls out of classical computability. A setting in a continuous Euclidean space-time that mimics this is presented. Not only is Zeno effect possible but it is used to unleash the black hole power. Both discrete (classical) computation and analog computation (in the understanding of Blum, Shub and Smale) are considered. Moreover, using nested singularities (which are built), it is shown how to decide higher levels of the corresponding arithmetical hierarchies.

25 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that for any theory T which is given by an L!1;! sentence, T has less than 2 2 @ 0 many countable models if and only if we have that, for every X 2 2! on a cone of Turing degrees, every X-hyperarithmetic model of T has an X-computable copy.

25 citations

Journal ArticleDOI
TL;DR: A new deductive system CL12 is introduced and its soundness and completeness with respect to the semantics of CL are proved and it is shown that this system presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories.
Abstract: Computability logic (CL) (see this http URL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, "truth" means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and completeness with respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logic-based Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings -- an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems.

25 citations

Journal ArticleDOI
TL;DR: In this article, countable free groups of different ranks were considered and the complexity of index sets within the class of free groups and within the classes of all groups was investigated. But the complexity was not considered for a computable free group of infinite rank.
Abstract: We consider countable free groups of different ranks. For these groups, we investigate computability theoretic complexity of index sets within the class of free groups and within the class of all groups. For a computable free group of infinite rank, we consider the difficulty of finding a basis.

25 citations

Journal ArticleDOI
Thomas Strahm1
TL;DR: It is shown that Cook and Urquhart's system PVω is directly contained in a natural applicative theory of polynomial strength, and these theories can be regarded as applicative analogues of traditional systems of bounded arithmetic.
Abstract: Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the 1970s In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: self-application is meaningful, but not necessarily total It has turned out that theories with self-application provide a natural setting for studying notions of abstract computability, especially from a proof-theoretic perspective This paper is concerned with the study of (unramified) bounded applicative theories which have a strong relationship to classes of computational complexity We propose new applicative systems whose provably total functions coincide with the functions computable in polynomial time, polynomial space, polynomial time and linear space, as well as linear space Our theories can be regarded as applicative analogues of traditional systems of bounded arithmetic We are also interested in higher-type features of our systems; in particular, it is shown that Cook and Urquhart's system PVω is directly contained in a natural applicative theory of polynomial strength

24 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202344
2022119
202189
202098
2019111
201897