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The complexity of sets

André Nies

in Computability and Randomness

Published in print:
2009
Published Online:
May 2009
ISBN:
9780199230761
eISBN:
9780191710988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230761.003.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter studies the complexity of sets of natural numbers. There are two interrelated types of complexity, these are computational and descriptive. In both cases, to understand the complexity of ... More


Introduction

Neil Tennant

in Changes of Mind: An Essay on Rational Belief Revision

Published in print:
2012
Published Online:
September 2012
ISBN:
9780199655755
eISBN:
9780191742125
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199655755.003.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This chapter distinguishes logic as a theory of belief‐statics from our sought account of belief dynamics. The various kinds of belief change are classified. These are: surrendering, adopting or ... More


Notes on Computational Complexity

Kumaraswamy Velupillai

in Computable Economics: The Arne Ryde Memorial Lectures

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198295273
eISBN:
9780191596988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198295278.003.0008
Subject:
Economics and Finance, Macro- and Monetary Economics

A characterization of dynamic choice attributable to Marshall is used as the vehicle through which to discuss the computational complexity of sequentially rational behaviour. Khachiyan's famous ... More


Computable Economics: The Arne Ryde Memorial Lectures

K. Velupillai

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198295273
eISBN:
9780191596988
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/0198295278.001.0001
Subject:
Economics and Finance, Macro- and Monetary Economics

In a discipline such as economics, increasingly devoted to its computational content, the mathematical underpinnings of the computability assumptions of economic fundamentals have not been ... More


PRELIMINARIES AND AGREEMENTS

Rolf Niedermeier

in Invitation to Fixed-Parameter Algorithms

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198566076
eISBN:
9780191713910
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566076.003.0002
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the basic mathematical formalism and discusses concepts used throughout the book. Among other things, it looks at decision problems vs optimization problems, Random Access ... More


A Formal Theory of Contraction

Neil Tennant

in Changes of Mind: An Essay on Rational Belief Revision

Published in print:
2012
Published Online:
September 2012
ISBN:
9780199655755
eISBN:
9780191742125
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199655755.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics, Metaphysics/Epistemology

This is the heart of the formal theory. Mathematically rigorous definitions are provided of all the formal notions that have been gently introduced in the earlier discussion. The main data type of a ... More


Martin-Löf randomness and its variants

André Nies

in Computability and Randomness

Published in print:
2009
Published Online:
May 2009
ISBN:
9780199230761
eISBN:
9780191710988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230761.003.0003
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter begins with a general background on how to introduce mathematical randomness notions. It then studies the central randomness notion of Martin–Löf, and gives some connections to ... More


Applications to Other Fields

Daniel L. Stein and Charles M. Newman

in Spin Glasses and Complexity

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691147338
eISBN:
9781400845637
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691147338.003.0007
Subject:
Sociology, Science, Technology and Environment

This chapter explores how spin glass concepts have found use in and, in some cases, further advanced areas such as computational complexity, combinatorial optimization, neural networks, protein ... More


Calculating, Reasoning, and “Doing Things with the Mind”

in Inside Man: The Discipline of Modeling Human Ways of Being

Published in print:
2011
Published Online:
June 2013
ISBN:
9780804773041
eISBN:
9780804777421
Item type:
chapter
Publisher:
Stanford University Press
DOI:
10.11126/stanford/9780804773041.003.0004
Subject:
Economics and Finance, Behavioural Economics

This chapter focuses on the models of human reasoning and the process of calculation. It explains that these models are based on the structure of the problems which a subject can attempt to solve and ... More


Ideas, Pioneers, and Precursors

Kumaraswamy Velupillai

in Computable Economics: The Arne Ryde Memorial Lectures

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198295273
eISBN:
9780191596988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198295278.003.0002
Subject:
Economics and Finance, Macro- and Monetary Economics

In this chapter, a first, tentative attempt is made to define the nature and scope of what is meant by computable economics. From a study of the way economics was mathematized in the modern era, – ... More


Effective Playability in Arithmetical Games

Kumaraswamy Velupillai

in Computable Economics: The Arne Ryde Memorial Lectures

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198295273
eISBN:
9780191596988
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198295278.003.0007
Subject:
Economics and Finance, Macro- and Monetary Economics

Class games, called Arithmetical Games, are defined and recursion theoretic questions such as effective playability, diophantine complexity, etc. are posed and formally answered. In the process, ... More


Introduction to combinatorial optimization

Marc Mézard and Andrea Montanari

in Information, Physics, and Computation

Published in print:
2009
Published Online:
September 2009
ISBN:
9780198570837
eISBN:
9780191718755
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570837.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter provides an elementary introduction to some basic concepts in theoretical computer science. It includes basic notions of graph theory and an informal introduction to computational ... More


Quantum algorithms

Vlatko Vedral

in Introduction to Quantum Information Science

Published in print:
2006
Published Online:
January 2010
ISBN:
9780199215706
eISBN:
9780191706783
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199215706.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

As computers get faster and faster, the size of the circuitry imprinted onto silicon chips decreases. The size of the circuitry becomes so small that its behavior is governed by the laws of quantum ... More


The Basics

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the complexity of a problem by thinking about the best possible algorithm, or computer program, that solves it. It shows that computational complexity theory is not about how ... More


Memory, Paths, and Games

Cristopher Moore and Stephan Mertens

in The Nature of Computation

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199233212
eISBN:
9780191775079
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233212.003.0008
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Both time and memory are limited resources in the real world, but a fundamental difference between the two is that the latter can be reused. The gap between determinism and nondeterministic ... More


Theoretical versus Practical Security

Keith M. Martin

in Everyday Cryptography: Fundamental Principles and Applications

Published in print:
2017
Published Online:
July 2017
ISBN:
9780198788003
eISBN:
9780191829956
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198788003.003.0003
Subject:
Mathematics, Computational Mathematics / Optimization, Logic / Computer Science / Mathematical Philosophy

In this chapter, we consider the relationship between theoretical and practical security of a cryptosystem. The concept of perfect secrecy is introduced, and the one-time pad is exhibited as a ... More


Computational complexity of abstractive morphology

Vito Pirrelli, Marcello Ferro, and Claudia Marzi

in Understanding and Measuring Morphological Complexity

Published in print:
2015
Published Online:
May 2015
ISBN:
9780198723769
eISBN:
9780191791109
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198723769.003.0008
Subject:
Linguistics, Syntax and Morphology, Theoretical Linguistics

Blevins’ distinction between abstractive and constructive approaches to word structure has proved to be of general theoretical interest, taking stock of considerably different descriptive statements ... More


Mathematical foundations

Max A. Little

in Machine Learning for Signal Processing: Data Science, Algorithms, and Computational Statistics

Published in print:
2019
Published Online:
October 2019
ISBN:
9780198714934
eISBN:
9780191879180
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198714934.003.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy, Mathematical Physics

Statistical machine learning and signal processing are topics in applied mathematics, which are based upon many abstract mathematical concepts. Defining these concepts clearly is the most important ... More


Computational complexity, phase transitions, and message-passing for community detection

Cristopher Moore

in Statistical Physics, Optimization, Inference, and Message-Passing Algorithms: Lecture Notes of the Les Houches School of Physics: Special Issue, October 2013

Published in print:
2015
Published Online:
March 2016
ISBN:
9780198743736
eISBN:
9780191803802
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198743736.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter provides a whirlwind tour of problems and techniques at the boundary of computer science and statistical physics. It starts with a brief description of computational complexity and P, ... More


Zero-Sum Matrix Games

João P. Hespanha

in Noncooperative Game Theory: An Introduction for Engineers and Computer Scientists

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175218
eISBN:
9781400885442
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175218.003.0003
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter discusses a number of key concepts for zero-sum matrix games. A zero-sum matrix game is played by two players, each with a finite set of actions. Player 1 wants to minimize the outcome ... More


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