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Summary
Author Biography
Table of Contents
Preface | p. xi |
Mathematical Foundations | p. 1 |
Sets and Logic; Naïvely | p. 1 |
A Detour via Logic | p. 2 |
Sets and their Operations | p. 27 |
Alphabets, Strings and Languages | p. 39 |
Relations and Functions | p. 40 |
Big and Small Infinite Sets; Diagonalization | p. 51 |
Induction from a User's Perspective | p. 61 |
Complete, or Course-of-Values, Induction | p. 61 |
Simple Induction | p. 64 |
The Least Principle | p. 65 |
The Equivalence of Induction and the Least Principle | p. 65 |
Why Induction Ticks | p. 68 |
Inductively Defined Sets | p. 69 |
Recursive Definitions of Functions | p. 78 |
Additional Exercises | p. 85 |
Algorithms, Computable Functions and Computations | p. 91 |
A Theory of Computability | p. 91 |
A Programming Framework for Computable Functions | p. 92 |
Primitive Recursive Functions | p. 103 |
Simultaneous Primitive Recursion | p. 116 |
Pairing Functions | p. 118 |
Iteration | p. 123 |
A Programming Formalism for the Primitive Recursive Functions | p. 125 |
PR vs. L | p. 135 |
Incompleteness of PR | p. 139 |
URM Computations and their Arithmetization | p. 141 |
A Double Recursion that Leads Outside the Primitive Recursive Function Class | p. 147 |
The Ackermann Function | p. 148 |
Properties of the Ackermann Function | p. 149 |
The Ackermann Function Majorizes All the Functions of PR | p. 153 |
The Graph of the Ackermann Function is in PR* | p. 155 |
Semi-computable Relations; Unsolvability | p. 158 |
The Iteration Theorem of Kleene | p. 172 |
Diagonalization Revisited; Unsolvability via Reductions | p. 175 |
More Diagonalization | p. 176 |
Reducibility via the S-m-n Theorem | p. 183 |
More Dovetailing | p. 196 |
Recursive Enumerations | p. 202 |
Productive and Creative Sets | p. 209 |
The Recursion Theorem | p. 212 |
Applications of the Recursion Theorem | p. 214 |
Completeness | p. 217 |
Unprovability from Unsolvability | p. 221 |
Supplement: ¿x(x) ↑ is Expressible in the Language of Arithmetic | p. 229 |
Additional Exercises | p. 234 |
A Subset of the URM Language; FA and NFA | p. 241 |
Deterministic Finite Automata and their Languages | p. 243 |
The Flow-Diagram Model | p. 243 |
Some Closure Properties | p. 251 |
How to Prove that a Set is Not Acceptable by a FA; Pumping Lemma | p. 253 |
Nondeterministic Finite Automata | p. 257 |
From FA to NFA and Back | p. 260 |
Regular Expressions | p. 266 |
From a Regular Expression to NFA and Back | p. 268 |
Regular Grammars and Languages | p. 277 |
From a Regular Grammar to a NFA and Back | p. 282 |
Epilogue on Regular Languages | p. 285 |
Additional Exercises | p. 287 |
Adding a Stack to a NFA: Pushdown Automata | p. 293 |
The PDA | p. 294 |
PDA Computations | p. 295 |
ES vs AS vs ES+AS | p. 300 |
The PDA-acceptable Languages are the Context Free Languages | p. 305 |
Non Context Free Languages; Another Pumping Lemma | p. 312 |
Additional Exercises | p. 322 |
Computational Complexity | p. 325 |
Adding a Second Stack; Turing Machines | p. 325 |
Turing Machines | p. 330 |
NP-Completeness | p. 338 |
Cook's Theorem | p. 342 |
Axt, Loop Program, and Grzegorczyk Hierarchies | p. 350 |
Additional Exercises | p. 370 |
Bibliography | p. 375 |
Index | p. 379 |
Table of Contents provided by Ingram. All Rights Reserved. |
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