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Exploring Turing Machines: Theory and Applications in Computation

This lecture from COSC 3340 at the University of Houston, led by Dr. Verma, delves into the fundamentals of Turing Machines (TMs), a pivotal concept in computational theory. Proposed by Alan Turing in 1936, the lecture covers the operational mechanics of TMs, including state changes and symbol manipulations. It further explains TM design for function computation and language decision. Through a practical example of TMs processing binary strings, students will gain insight into the theoretical underpinnings of computational functions. Finally, a JFLAP simulation demonstrates these concepts in action.

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Exploring Turing Machines: Theory and Applications in Computation

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  1. COSC 3340: Introduction to Theory of Computation University of Houston Dr. Verma Lecture 16 UofH - COSC 3340 - Dr. Verma

  2. Turing Machine (TM) . . . Bi-direction Read/Write Finite State control UofH - COSC 3340 - Dr. Verma

  3. Historical Note • Proposed by Alan Turing in 1936 in: On Computable Numbers, with an application to the Entscheidungsproblem, Proc. Lond. Math. Soc. (2) 42 pp 230-265 (1936-7); correction ibid. 43, pp 544-546 (1937). UofH - COSC 3340 - Dr. Verma

  4. Turing Machine (contd.) • Based on (q, ), q – current state, – symbol scanned by head, in one move, the TM can: (i) change state (ii) write a symbol in the scanned cell (iii) move the head one cell to the left or right • Some (q, ) combinations may not have any moves. In this case the machine halts. UofH - COSC 3340 - Dr. Verma

  5. Turing Machine (contd.) • We can design TM’s for computing functions from strings to strings • We can design TM’s to decide languages • using special states accept/reject or by writing Y/N on tape. • We can design TM’s to accept languages. • if TMhalts string is accepted Note: there is a big difference between language decision and acceptance! UofH - COSC 3340 - Dr. Verma

  6. Example of TM for {0n1n | n> 0} • English description of how the machine works: • Look for 0’s • If 0 found, change it to x and move right, else reject • Scan past 0’s and y’s until you reach 1 • If 1 found, change it to y and move left, else reject. • Move left scanning past 0’s and y’s • If x found move right • If 0 found, loop back to step 2. • If 0 not found, scan past y’s and accept. Head is on the left or start of the string. x and yare just variables to keep track of equality UofH - COSC 3340 - Dr. Verma

  7. Example of TM for {0n1n | n > 0} contd. Head is on the left or start of the string. UofH - COSC 3340 - Dr. Verma

  8. Example of TM for {0n1n | n > 0} contd. Head is on the left or start of the string. UofH - COSC 3340 - Dr. Verma

  9. Example of TM for {0n1n | n > 0} contd. Head is on the left or start of the string. UofH - COSC 3340 - Dr. Verma

  10. Example of TM for {0n1n | n > 0} contd. Head is on the left or start of the string. UofH - COSC 3340 - Dr. Verma

  11. Example of TM for {0n1n | n > 0} contd. Head is on the left or start of the string. UofH - COSC 3340 - Dr. Verma

  12. Example of TM for {0n1n | n 0} contd. UofH - COSC 3340 - Dr. Verma

  13. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  14. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  15. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  16. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  17. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  18. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  19. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  20. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  21. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  22. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  23. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  24. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  25. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  26. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  27. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  28. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  29. JFLAP SIMULATION UofH - COSC 3340 - Dr. Verma

  30. Formal Definition of TM • Formally a TMM = (Q, , , , s) where, • Q – a finite set of states • – input alphabet not containing the blank symbol # • – the tape alphabet of M • s in Q is the start state •  : Q X Q X  X {L, R} is the (partial) transition function. • Note: (i) We leave out special states. (ii) The model is deterministic but we just say TM instead of DTM. UofH - COSC 3340 - Dr. Verma

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