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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: In this article, the existence, uniqueness and computability of solutions for a class of discrete time recursive utilities models were studied, and it was shown that the natural iterative algorithm is convergent if and only if a solution exists.
Abstract: We study existence, uniqueness and computability of solutions for a class of discrete time recursive utilities models. By combining two streams of the recent literature on recursive preferences---one that analyzes principal eigenvalues of valuation operators and another that exploits the theory of monotone concave operators---we obtain conditions that are both necessary and sufficient for existence and uniqueness of solutions. We also show that the natural iterative algorithm is convergent if and only if a solution exists. Consumption processes are allowed to be nonstationary.
13 citations
01 Jan 2007
TL;DR: A rigorous mathematical definition of computability and algorithm generated new approaches to understanding a solution to a problem and new mathematical disciplines such as computer science, algorithmical complexity, linear programming, computational modeling and simulation databases and search algorithms.
Abstract: The intuitive notion of computability was formalized in the XXth century, which strongly affected the development of mathematics and applications, new computational technologies, various aspects of the theory of knowledge, etc A rigorous mathematical definition of computability and algorithm generated new approaches to understanding a solution to a problem and new mathematical disciplines such as computer science, algorithmical complexity, linear programming, computational modeling and simulation databases and search algorithms, automatical cognition, program languages and semantics, net security, coding theory, cryptography in open systems, hybrid control systems, information systems, etc
13 citations
08 Jun 2005
TL;DR: This is a survey of a century long history of interplay between Hilbert's tenth problem (about solvability of Diophantine equations) and different notions and ideas from the Computability Theory.
Abstract: This is a survey of a century long history of interplay between Hilbert's tenth problem (about solvability of Diophantine equations) and different notions and ideas from the Computability Theory.
13 citations
TL;DR: The notion of “semicomputability” is investigated, intended to generalize the notion of recursive enumerability of relations to abstract structures and leads to the formulation of a “Generalized Church-Turing Thesis” for definability of Relations on abstract structures.
Abstract: We investigate the notion of “semicomputability,” intended to generalize the notion of recursive enumerability of relations to abstract structures. Two characterizations are considered and shown to be equivalent: one in terms of “partial computable functions” (for a suitable notion of computability over abstract structures) and one in terms of definability by means of Horn programs over such structures. This leads to the formulation of a “Generalized Church-Turing Thesis” for definability of relations on abstract structures.
12 citations
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TL;DR: A new proof of the fundamental theorem on ordinal computability is presented that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory.
Abstract: The notion of ordinal computability is defined by generalising standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. The fundamental theorem on ordinal computability states that a set $x$ of ordinals is ordinal computable from ordinal parameters if and only if $x$ is an element of the constructible universe $\mathbf{L}$. In this paper we present a new proof of this theorem that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory. The theory SO and the standard Zermelo–Fraenkel axiom system ZFC can be canonically interpreted in each other. The proof of the fundamental theorem is based on showing that the class of sets that are ordinal computable from ordinal parameters forms a model of SO.
12 citations