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Introduction to Turing Machines and Variants - CSCI 2670 Theory of Computing

This lecture covers key concepts in the theory of computing, focusing on Turing machines and their variants. Topics include an overview of Turing machines, their definitions, and operations, as well as the exploration of multitape and nondeterministic Turing machines. The class will engage in hands-on group projects that involve designing Turing machines to solve specific language problems, such as copying strings and recognizing various language constructs. Students are reminded of upcoming homework and quizzes related to the course material.

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Introduction to Turing Machines and Variants - CSCI 2670 Theory of Computing

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  1. CSCI 2670Introduction to Theory of Computing October 4, 2005

  2. Agenda • Last week • Turing machines • This week • Sections 3.2 & 3.3 • Variants of Turing machines and the definition of algorithm October 4, 2005

  3. Announcements • Homework due next Tuesday • 3.7, 3.15 b & c, 3.16 d • 3.7 and 3.15 are same in both texts • 3.16 d – Show the collection of Turing-recognizable languages is closed under the operation of intersection. • Quiz tomorrow • Turing machine definition, calculation, and design October 4, 2005

  4. Group project 1 • Design a Turing machine to accept any string in {a,b}* after making a copy of it on the tape • The tape will start with w • After TM processes the string, the tape should read ww October 4, 2005

  5. Group project 2 • Write a Turing machine that accepts the language {w  {a,b}* | |w| is even} October 4, 2005

  6. Group project 3 • Write a Turing machine that accepts the language {anbm | nm and nm} October 4, 2005

  7. Group project 4 • Write a Turing machine that accepts the language {anbman+m | n0 and m1} October 4, 2005

  8. Group project 5 • Write a Turing machine that accepts the language {wwR | w{a,b}*} October 4, 2005

  9. Group project 6 • Design a Turing machine that accepts the language {w{a,b}* | w has more a’s than b’s} October 4, 2005

  10. Group project 1 • Design a Turing machine to accept any string in {a,b}* after making a copy of it on the tape • The tape will start with w • After TM processes the string, the tape should read ww October 4, 2005

  11. TM design • Read first symbol • If a, replace with x, move to end and write x • If b, replace with y, move to end and write y • Move left to first x or y • Move left to next x or y • If x, replace with a and move right • If y, replace with b and move right October 4, 2005

  12. TM design (cont.) • Read symbol at tape head • If x, replace with a and accept • If y, replace with b and accept • If neither, loop to beginning October 4, 2005

  13. States • Start state q1 • Move right • Write a at end, write b at end, write x at end, write y at end • Move left to first x or y • Move left to second x or y & replace with a or b • Replace a or b with x or y October 4, 2005

  14. qreject a → R b → R ~ → a,R a → L b → L ~ → L a → L b → L a,b,x,y → R a→ x,R ~ → x,R x → a,R y → b,R x → L y → L a → R b → R a→ x,R b→ y,R b→ y,R ~ → y,R a,b,x,y → R x → a,R y → b,R ~ → L ~ → b,R qaccept Copy machine October 4, 2005

  15. Variants of Turing machines • Robustness of model • Varying the model does not change the power • Unlike finite automata (making finite automata nondeterministic changes the class of languages accepted) • Simple variant of TM model • Add “stay put” direction • Other variants • More tapes • Nondeterministic October 4, 2005

  16. Multitape Turing machines • Same as standard Turing machine, but have several tapes • TM definition changes only in definition of δ δ : Q × Гk→ Q× Гk × {L,R}k October 4, 2005

  17. Equivalence of machines Theorem: Every multitape Turing machine has an equivalent single tape Turing machine Proof method: construction October 4, 2005

  18. 0 1 ~ ~ ~ ~ ~ ~ a a a ~ ~ ~ ~ ~ M a b ~ ~ ~ ~ ~ ~ # 0 1 # a a a # a b # ~ ~ S Equivalent machines October 4, 2005

  19. Simulating k-tape behavior • Single tape start string is #w#~#...#~# • Each move proceeds as follows: • Start at leftmost slot • Scan right to (k+1)st # to find symbol at each virtual tape head • Make second pass making updates indicated by k-tape transition function • When a virtual head moves onto a #, shift string to right October 4, 2005

  20. Corollary Corollary: A language is Turing-recognizable if and only if some multitape Turing machine recognizes it. October 4, 2005

  21. Example • Using 2-tape Turing machine, write a copy machine • Copy tape 1 to tape 2 • Move tape 1 to beginning • Copy tape 1 to tape 2 • Accept October 4, 2005

  22. qreject {a,~}→ {a,a},{R,R} {b,~}→ {b,b},{R,R} {a,~}→ {L,S} {b,~}→ {L,S} {a,~}→ {a,a},{R,R} {b,~}→ {b,b},{R,R} {a,~}→ {x,a},{R,R} {b,~}→ {y,b},{R,R} ~,~ → {L,S} {x,~} → {a,a},{R,R} {y,~} → {b,b},{R,R} {~,~} → {L,L} {~, ,~}→ {R,R} qaccept Copy machine October 4, 2005

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