COSC 3340: Introduction to Theory of Computation

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COSC 3340 Introduction to Theory of Computation

• University of Houston
• Dr. Verma
• Lecture 16

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Turing Machine (TM)
. . .
Bi-direction Read/Write
Finite State control
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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.

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Turing Machine (contd.)

• We can design TMs for computing functions from
  strings to strings
• We can design TMs to decide languages
• using special states accept/reject or by writing
  Y/N on tape.
• We can design TMs to accept languages.
• if TM halts string is accepted
• Note there is a big difference between language
  decision and acceptance!

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Example of TM for 0n1n n gt 0

• English description of how the machine works
• Look for 0s
• If 0 found, change it to x and move right, else
  reject
• Scan past 0s and ys until you reach 1
• If 1 found, change it to y and move left, else
  reject.
• Move left scanning past 0s and ys
• If x found move right
• If 0 found, loop back to step 2.
• If 0 not found, scan past ys and accept.

Head is on the left or start of the string.
x and y are just variables to keep track of
equality
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Example of TM for 0n1n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q0 0 (q1, x, R)
q0 1 halt/reject
q0 x halt/reject
q0 y (q3, y, R)
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Example of TM for 0n1n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q1 0 (q1, 0, R)
q1 1 (q2, y, L)
q1 x halt/reject
q1 y (q1, y, R)
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Example of TM for 0n1n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q2 0 (q2, 0, L)
q2 1 halt/reject
q2 x (q0, x, R)
q2 y (q2, y, L)
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Example of TM for 0n1n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q3 0 halt/reject
q3 1 halt/reject
q3 x halt/reject
q3 y (q3, y, R)
q3 ? (q4, ?, R)
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Example of TM for 0n1n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q4 0 illegal i/p
q4 1 illegal i/p
q4 x illegal i/p
q4 y illegal i/p
q4 ? halt/accept
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Example of TM for 0n1n n gt 0 contd.
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JFLAP SIMULATION
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Formal Definition of TM

• Formally a TM M (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.