Sequentiality in Real Number Computation

1 Introduction 31.1 Classes of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Countable Sets of Numbers: N, Z and Q . . . . . . . . . 41.1.2 Uncountable Sets of Numbers: R and C . . . . . . . . . . 71.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Computation on Countable Sets of Numbers . . . . . . . . . . . . 121.2.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . 131.2.2 Rational Numbers . . . . . . . . . . . . . . . . . . . . . 161.3 Computation on Real Numbers . . . . . . . . . . . . . . . . . . . 171.3.1 Floating Point Representation . . . . . . . . . . . . . . . 171.3.2 Cauchy Sequence Representation . . . . . . . . . . . . . 201.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Technical Background 292.1 λ-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Domains for Computation . . . . . . . . . . . . . . . . . . . . . 332.3 PCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.1 Operational Semantics of PCF . . . . . . . . . . . . . . . 392.3.2 Denotational Semantics of PCF . . . . . . . . . . . . . . 402.3.3 The Necessity of Parallel Operators . . . . . . . . . . . . 412.4 Sequentiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48i

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