Algorithms: Main Ideas and Applications

Table of Contents

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   Book Overview
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   Chapter 1 Introduction
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   Chapter 2 Notation and Terminology
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   Chapter 3 Preliminary notions of the theory of algorithms: constructive objects and aggregates; local properties and local actions
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   Chapter 4 The general notion of an algorithm as an independent (separate) concept
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   Chapter 5 Representative computational models
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   Chapter 6 The general notion of a calculus as an independent (separate) concept
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   Chapter 7 Representative generating models
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   Chapter 8 Interrelations between algorithms and calculuses
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    Chapter 9 Time and Space as complexities of computation and generation
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    Chapter 10 Computable functions and generable sets; decidable sets; enumerable sets
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    Chapter 11 The concept of a μ -recursive function
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    Chapter 12 Possibility of an arithmetical and even Diophantine representation of any enumerable set of natural numbers
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    Chapter 13 Construction of an undecidable generable set
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    Chapter 14 Post’s reducibility problem
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    Chapter 15 The concept of a relative algorithm, or an oracle algorithm
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    Chapter 16 The concept of a computable operation
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    Chapter 17 The concept of a program; programs as objects of computation and generation
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    Chapter 18 The concept of a numbering and the theory of numberings
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    Chapter 19 First steps in the invariant, or machine-independent, theory of complexity of computations
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    Chapter 20 The theory of complexity and entropy of constructive objects
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    Chapter 21 Convenient computational models
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    Chapter 22 Investigations of mass problems
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    Chapter 23 Applications to the foundations of mathematics: constructive semantics
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    Chapter 24 Applications to mathematical logic: formalized languages of logic and arithmetic
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    Chapter 25 Computable analysis
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    Chapter 26 Numbered structures
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    Chapter 27 Applications to probability theory: definitions of a random sequence
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    Chapter 28 Applications to information theory: the algorithmic approach to the concept of quantity of information
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    Chapter 29 Complexity bounds for particular problems
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    Chapter 30 Influence of the theory of algorithms on algorithmic practice