CSCI 2670 Introduction to Theory of Computing

1. CSCI 2670Introduction to Theory of Computing October 13, 2004

2. Agenda • Yesterday • Variants of Turing machines • Allow “stay put” state • Multiple tapes • Nondeterministic • Today • Prove equivalence of deterministic and nondeterministic Turing machines • Enumerators October 13, 2004

3. Announcements • Quiz tomorrow • High-level description of TM • Trace through a TM’s operation • Tape notation & configuration notation • High-level description of equivalences • Reminder: tutorial sessions are back • Office hours return to normal • Tuesday 3:00 – 4:00 • Wednesday 3:00 – 4:00 October 13, 2004

4. Nondeterministic Turing machines • Same as standard Turing machines, but may have one of several choices at any point δ : Q × Г→ P(Q× Г × {L,R}) October 13, 2004

5. Equivalence of machines Theorem: Every nondeterministic Turing machine has an equivalent deterministic Turing machine Proof method: construction Proof idea: Use a 3-tape Turing machine to deterministically simulate the nondeterministic TM. First tape keeps copy of input, second tape is computation tape, third tape keeps track of choices. October 13, 2004

6. 0 1 0 1 ~ ~ ~ ~ M a a # 0 1 ~ ~ ~ 1 1 2 1 3 ~ ~ ~ Proof idea Input tape never changes Computation tape Decision path Try new decision paths until the string is accepted October 13, 2004

7. Intuition • Consider the nondeterministic calculations as a tree • Each node represents a configuration • A node for configuration C1 has one child for each configuration C2 such that C1 yields C2 • Root of tree is configuration q1w • A configuration may appear more than once in the tree October 13, 2004

8. Example • Nondeterministic TM that accepts {ww | w  {a,b}*} • Use 2 tapes • Copy input to tape 2 • Position heads at beginning of tapes • Move both heads right simultaneously October 13, 2004

9. Nondeterministic solution • Nondeterministically choose the midpoint • Mark this point on tape 2 and return tape 2’s head to beginning • Compare strings • If tape head points to ~ on tape 1 and midpoint marker on tape 2 then accept • Otherwise, if all possible midpoints have been tried then reject • Otherwise, try a new midpoint October 13, 2004

10. (a,~) → {a,a},{R,R} {b,~} → {b,b},{R,R} (a,x) → {a,a},{R,R} {b,x} → {b,b},{R,R} qreject (a,a) → (L,L) (b,b) → (L,L) (a,a) → (R,R) (b,b) → (R,R) (a,~)→ (a,a),(R,R) (b,~)→(b,b),(R,R) (~,~) → (L,L) (a,a) → (a,x),(S,Res) (b,b) → (b,x),(S,Res) (a,b), (b,a), (a,x), (b,x), (~,a), (~,b) → (Res,Res) (~,~) → (L,L) qaccept (~,x) → (S,S) Nondeterministic TM October 13, 2004

11. Move right Compare Right Accept Compare Reset Right Right Accept Reset Compare Compare … Tree representation October 13, 2004

12. Deterministic equivalent • Assume midpoint is at beginning • If so accept • If not … • Assume midpoint is after first symbol • If so accept • If not … • Assume midpoint is after second symbol • If so accept • If not … • etc. … October 13, 2004

13. How should it search the tree? • Breadth first search • Search all possibilities involving k steps before searching any possibilities involving (k+1) steps • What’s wrong with depth first search? • If some sequence of choices results in no halting, we will never get to the accept state October 13, 2004

14. When does it halt? • When it reaches an accept state or • Return accept October 13, 2004

15. Will it halt on strings in the language? • Yes if the TM accepts the input string • Let b be the largest number of children of any node • Can we be sure b is finite? • Let k be the minimum number of steps it takes to get to the accept state • This method will take at most bk steps to get to the accept state October 13, 2004

16. What about strings not in the language? • Won’t halt • That’s okay October 13, 2004

17. Equivalence of approaches Corollary: A language is Turing-recognizable if and only if some nondeterministic Turing machine recognizes it. October 13, 2004

18. Equivalence of approaches Corollary: A language is Turing-decidable if and only if some nondeterministic Turing machine decides it. Proof: Homework • Modify proof in the book October 13, 2004