COSC 3340: Introduction to Theory of Computation

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COSC 3340 Introduction to Theory of Computation

• University of Houston
• Dr. Verma
• Lecture 17

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Definitions

• A configuration is a snapshot of the machine
  written uqv.
• Configuration C1 yields configuration C2 if the
  Turing machine can legally go from C1 to C2 in a
  single step.
• Suppose a, b, and c in ?, u and v in ? and
  states qi and qj. Then, uaqibv and uqjacv are two
  configurations.
• uaqibv yields uqjacv if ?(qi, b) (qj, c, L).
• handles the case where the TM moves leftward.
• For a rightward move, uaqibv yields uacqjv if
  ?(qi, b) (qj,c,R).

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Special Cases

• Special Cases occur when the head is at one of
  the ends of the configurations.
• For the left-hand end, the configurations qibv
  yields qjcv if the transition is left moving, and
  it yields cqjv for the right moving transition.
• For the right-hand end, the configuration uaqi is
  equivalent to uaqi? because we assume that blanks
  follow the part of the tape represented in the
  configuration. Thus we can handle this case as
  before, with the head no longer at the right hand
  end.

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More Definitions

• The start configuration of M on input w is the
  configuration q0w, which indicates that the
  machine is in the start state q0 with its head at
  the leftmost position on the tape
• In an accepting configuration the state of the
  configuration is qaccept
• In a rejecting configuration the state of the
  configurations is qreject
• Accepting and rejecting configurations are
  halting configurations and accordingly do not
  yield further configurations

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More Definitions

• A TM M accepts input w if a sequence of
  configurations C1,C2,,Ck exists where
• C1 is the start configuration of M on input w,
• Each Ci yields Ci1, and
• Ck is an accepting configuration.
• The collection of strings that M accepts is the
  language of M, denoted L(M).

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More Definitions

• - Turing-recognizable/acceptable
• TM M accepts/recognizes language L if L w M
  accepts w. Note 3 outcomes possible, either TM
  accepts, rejects, or loops.
• Turing-decidable TM M decides L if
• (i) w ? L, M writes a Yes on tape and halts
• (ii) w ? L, M writes a No on tape and halts.
• Every decidable language is Turing-recognizable
  but certain Turing-recognizable language are not
  decidable.

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Example of TM for 0n1n2n 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 right, else
  reject.
• Scan past 1s and zs until you reach 2
• If 2 found, change it to z and move left, else
  reject.
• Move left scanning past 0s, ys, zs and 1s
• If x found move right
• If 0 found, loop back to step 2.
• If 0 not found, scan past ys and zs and accept.

Head is on the left or start of the string.
x, y and z are just variables to keep track of
equality
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Example of TM for 0n1n2n n gt 0 contd.
In this case we are starting from the right or at
the end of a given string on the tape. Table will
be very similar if we start from the left.
State Symbol Next state action
q0 0 (q1, x, R)
q0 1 halt/reject
q0 2 halt/reject
q0 x halt/reject
q0 y (q4, y, R)
q0 z halt/reject
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Example of TM for 0n1n2n 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, R)
q1 2 halt/reject
q1 x halt/reject
q1 y (q1, y, R)
q1 z halt/reject
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Example of TM for 0n1n2n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q2 0 halt/reject
q2 1 (q2, 1, R)
q2 2 (q2, z, R)
q2 x halt/reject
q2 y halt/reject
q2 z (q2, z, R)
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Example of TM for 0n1n2n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q3 0 (q3, 0, L)
q3 1 (q3, 1, L)
q3 2 halt/reject
q3 x (q0, x, R)
q3 y (q3, y, L)
q3 z (q3, z, L)
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Example of TM for 0n1n2n n gt 0 contd.
Head is on the left or start of the string.
State Symbol Next state action
q4 0 halt/reject
q4 1 halt/reject
q4 2 halt/reject
q4 x halt/reject
q4 y (q4, y, L)
q4 z (q4, z, L)
q4 ? (q5, y, L)
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Example of TM for 0n1n2n n gt 0 contd.
State Symbol Next state action
q5 0 halt/reject
q5 1 halt/reject
q5 2 halt/reject
q5 x halt/reject
q5 y halt/reject
q5 z halt/reject
q5 ? halt/reject
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Example of TM for 0n1n2n n gt 0 contd.
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Different way of making 0n1n2n n gt 0