## 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