The Church-Turing Thesis

2 Example M 1

3 Definition 3.1

4 Configuration 1011q q7q ___...

5 Computation

6 Recognizer and Decider

7 Example 3.4 M 2

8 Example 3.6 M 3

9 Example 3.7 [Element distinctness problem] M 4

10 Multitape Turing Machines

11 Theorem 3.8 Every multitape Turing machine M has an equivalent single tape Turing machine S. S #01010#aaa_... #ba# M 01010_ aaa_ ba_

12 Simulation

13 Nondeterministic Turing Machines

14 Theorem 3.10 Every nondeterministic Turing machine has an equivalent deterministic Turing machine. D 0010__... xx#0 12_ input tape 1x__ _ simulation tape address tape

15 Proof

16 Corollary 3.11 A language is Turing recognizable if and only if some nondeterministic Turing machine recognizes it.

17 Enumerators

18 Theorem 3.13 A language is Turing-recognizable if and only if some enumerator enumerates it.

19 To avoid depth-first loops s1s1 s2s2 s3s3 s4s4 s5s5 s6s6

20 Hilbert’s 10 th Problem

21 Problem 3.18

22 Example 3.14 An algorithm