CSCI 2670 Introduction to Theory of Computing

1. CSCI 2670Introduction to Theory of Computing October 5, 2004

2. Agenda • This week • Turing machines • Read section 3.1 October 5, 2004

3. Announcements • Tests will be returned tomorrow • Tutorial sessions are suspended until further notice • Extended office hours while tutorials are suspended • Monday 11:00 – 12:00 • Tuesday 3:00 – 4:00 • Wednesday 3:00 – 5:00 October 5, 2004

4. Recap to date • Finite automata (both deterministic and nondeterministic) machines accept regular languages • Weakness: no memory • Pushdown automata accept context-free grammars • Add memory in the form of a stack • Weakness: stack is restrictive October 5, 2004

5. Turing machines • Similar to a finite automaton • Unrestricted memory in the form of a tape • Can do anything a real computer can do! • Still cannot solve some problems October 5, 2004

6. Control a b a ~ Touring machine schematic • Initially tape contains the input string • Blanks everywhere else (denoted ~ in class … different symbol in book) • Machine may write information on the tape October 5, 2004

7. Control a b a ~ Touring machine schematic • Can move tape head to read information written to tape • Continues computing until output produced • Output values accept or reject October 5, 2004

8. Control a b a ~ Touring machine schematic • Turing machine results • Accept • Reject • Never halts October 5, 2004

9. Differences between TM and FA • TM has tape you can read from and write to • Read-write head can be moved in either direction • Tape is infinite • Accept and reject states take immediate effect October 5, 2004

10. Example • How can we design a Turing machine to find the middle of a string? • If string length is odd, return middle symbol • If string length is even, reject string • Make multiple passes over string Xing out symbols at end until only middle remains October 5, 2004

11. Processing input • Check if string is empty • If so, return reject • Write X over first and last non-X symbols • After this, the head will be at the second X • Move left one symbol • If symbol is an X, return reject (string is even in length) • Move left one symbol • If symbol is an X, return accept (string is even in length) • Go to step 2 October 5, 2004

12. Example • 00110~ • First check if string is empty • X first and last non-X symbols • X011X~ • Move left one symbol • X011X~ • Is symbol an X? No • Move left one symbol • X011X~ • Is symbol an X? No • Write X over first and last non-X symbols October 5, 2004

13. Example • XX1XX~ • Move left one symbol • XX1XX~ • Is symbol an X? No • Move left one symbol • XX1XX~ • Is symbol an X? Yes • Return accept October 5, 2004

14. Formal definition of a TM Definition: A Turing machine is a 7-tuple (Q,,,,q0,qaccept,qreject), where Q, , and  are finite sets and • Q is the set of states, •  is the input alphabet not containing the special blank symbol ~ •  is the tape alphabet, where ~ and , • : QQ{L,R} is the transition function, October 5, 2004

15. Formal definition of a TM Definition: A Turing machine is a 7-tuple (Q,,,,q0,qaccept,qreject), where Q, , and  are finite sets and • q0Q is the start state, • qacceptQ is the accept state, and • qrejectQ is the reject state, where qrejectqaccept October 5, 2004

16. Computing with a TM • M receives input w = w1w2…wn* on leftmost n squares of tape • Rest of tape is blank (all ~ symbols) • Head position begins at leftmost square of tape • Computation follows rules of  • Head never moves left of leftmost square of the tape • If  says to move L, head stays put! October 5, 2004

17. Completing computation • Continue following  transition rules until M reaches qaccept or qreject • Halt at these states • May never halt if the machine never transitions to one of these states! October 5, 2004

18. TM configurations • The configuration of a Turing machine is the current setting • Current state • Current tape contents • Current tape location • Notation uqv • Current state = q • Current tape contents = uv • Only ~ symbols after last symbol of v • Current tape location = first symbol of v October 5, 2004