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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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01 Jan 1995
TL;DR: In this paper, the issues of computability and constructivity in the mathematics of physics are discussed, and the sorts of questions to be addressed are those which might be expressed, roughly, as: are the mathematical foundations of our current theories unavoidably non-constructive: or, are the laws of physics computable?
Abstract: In this paper, the issues of computability and constructivity in the mathematics of physics are discussed. The sorts of questions to be addressed are those which might be expressed, roughly, as: Are the mathematical foundations of our current theories unavoidably non-constructive: or, Are the laws of physics computable?
15 citations
TL;DR: A survey based upon the talk at the satellite QQQ conference to ECM6, 3Quantum: Algebra Geometry Information, Tallinn, July 2012, describes three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.
Abstract: This paper is a survey dedicated to the analogy between the notions of {\it complexity} in theoretical computer science and {\it energy} in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.
15 citations
TL;DR: By giving an appropriate representation to objects, based on a hierarchical coding of information, this work exemplifies how it is remarkably easy to compute complex objects.
Abstract: How best to quantify the information of an object, whether natural or artifact, is a problem of wide interest. A related problem is the computability of an object. We present practical examples of a new way to address this problem. By giving an appropriate representation to our objects, based on a hierarchical coding of information, we exemplify how it is remarkably easy to compute complex objects. Our algorithmic complexity is related to the length of the class of objects, rather than to the length of the object.
15 citations
TL;DR: In this article, the authors studied Σ-definability in hereditarily finite superstructures over algebraic systems, and proved the Σ definability criterion, which was then used as a basis for establishing the reduction theorem for regular theories and obtaining a characterization of simple theories.
Abstract: Σ-definability in hereditarily finite superstructures over algebraic systems is studied. We prove the Σ-definability criterion, which is then used as a basis for establishing the reduction theorem for regular theories and for obtaining a characterization of simple theories. The idea of a nonstandard recursion theory is developed using subfields of the field of reals as an example. A partial algebraic description is given for a distributive upper semilattice of mΣ-degrees in hereditarily finite superstructures over models of simple theories.
15 citations
22 Jan 2016
TL;DR: This book provides an overview of the confluence of ideas in Turings era and work and examines the impact of his work on mathematical logic and theoretical computer science.
Abstract: This book provides an overview of the confluence of ideas in Turings era and work and examines the impact of his work on mathematical logic and theoretical computer science. It combines contributions by well-known scientists on the history and philosophy of computability theory as well as on generalised Turing computability. By looking at the roots and at the philosophical and technical influence of Turings work, it is possible to gather new perspectives and new research topics which might be considered as a continuation of Turings working ideas well into the 21st century.
14 citations