1. Costas Busch - RPI1 Decidability

2. Costas Busch - RPI2 Consider problems with answer YES or NO Examples: Does Machine have three states ? Is string a binary number? Does DFA accept any input?

3. Costas Busch - RPI3 A problem is decidable if some Turing machine decides (solves) the problem Decidable problems: Does Machine have three states ? Is string a binary number? Does DFA accept any input?

4. Costas Busch - RPI4 Turing Machine Input problem instance YES NO The Turing machine that decides (solves) a problem answers YES or NO for each instance of the problem

5. Costas Busch - RPI5 The machine that decides (solves) a problem: If the answer is YES then halts in a yes state If the answer is NO then halts in a no state These states may not be final states

6. Costas Busch - RPI6 YES states NO states Turing Machine that decides a problem YES and NO states are halting states

7. Costas Busch - RPI7 Difference between Recursive Languages and Decidable problems The YES states may not be final states For decidable problems:

8. Costas Busch - RPI8 Some problems are undecidable: which means: there is no Turing Machine that solves all instances of the problem A simple undecidable problem: The membership problem

9. Fall 2006Costas Busch - RPI9 Undecidable Problems (unsolvable problems)

10. Fall 2006Costas Busch - RPI10 Undecidable Languages there is no Turing Machine which accepts the language and makes a decision (halts) for every input string undecidable language = not decidable language There is no decider: (machine may make decision for some input strings)

11. Fall 2006Costas Busch - RPI11 For an undecidable language, the corresponding problem is undecidable (unsolvable): there is no Turing Machine (Algorithm) that gives an answer (yes or no) for every input instance (answer may be given for some input instances)

12. Fall 2006Costas Busch - RPI12 We have shown before that there are undecidable languages: Decidable Turing-Acceptable is Turing-Acceptable and undecidable

13. Fall 2006Costas Busch - RPI13 We will prove that two particular problems are unsolvable: Membership problem Halting problem