Computability and Complexity Lecture Notes Winter Semester 2016/2017 Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria
[email protected] July 18, 2016 Contents List of Definitions5 List of Theorems7 List of Theses9 List of Figures9 Notions, Notations, Translations 12 1. Introduction 18 I. Computability 23 2. Finite State Machines and Regular Languages 24 2.1. Deterministic Finite State Machines...................... 25 2.2. Nondeterministic Finite State Machines.................... 30 2.3. Minimization of Finite State Machines..................... 38 2.4. Regular Languages and Finite State Machines................. 43 2.5. The Expressiveness of Regular Languages................... 58 3. Turing Complete Computational Models 63 3.1. Turing Machines................................ 63 3.1.1. Basics.................................. 63 3.1.2. Recognizing Languages........................ 68 3.1.3. Generating Languages......................... 72 3.1.4. Computing Functions.......................... 74 3.1.5. The Church-Turing Thesis....................... 79 Contents 3 3.2. Turing Complete Computational Models.................... 80 3.2.1. Random Access Machines....................... 80 3.2.2. Loop and While Programs....................... 84 3.2.3. Primitive Recursive and µ-recursive Functions............ 95 3.2.4. Further Models............................. 106 3.3. The Chomsky Hierarchy............................ 111 3.4. Real Computers................................