ICS 241

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ICS 241

• Discrete Mathematics II
• William Albritton, Information and Computer
  Sciences Department at University of Hawaii at
  Manoa
• For use with Kenneth H. Rosens Discrete
  Mathematics Its Applications (5th Edition)

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Section 11.5 Turing Machines

• Background of Turing machines
• Invented in 1935 by Alan Turing
• Solution to the decision problem of Hilbert
  (German mathematician)
• Is there a general method that can be applied to
  any assertion to determine whether the assertion
  is true?
• A model that roughly describes the capability of
  todays computers
• Different from the Turing test a computer is
  thinking if it cannot be distinguished from a
  person

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Definition

• A Turing machine T (S, I, f, s0 )
• S a finite set of n states (s0, s1, s2, sn)
• I an alphabet (0, 1, and blank symbol B)
• f a partial function from S I to S I R,
  L
• A partial function may be undefined for some
  inputs (such as the division function for 0)
• R, L move to the right (or left) one cell
• s0 the starting state

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Definition (in plain English)

• A Turing machine T consists of two parts
• A long tape that is infinite in both directions
• The tape is divided into cells, which contains a
  zero (0), one (1), or the blank symbol (B)
• A control unit that
• Is in a particular state
• Reads the contents of one cell from the tape
• Changes to a new state, based on this input
• Writes a 0, 1, or B to the same cell
• Moves one cell to the left or right
• Repeats steps 1-5, until it halts

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Defining a Turing Machine

• Define T with a 5-tuple set (s, x, s', x', d)
• s current state
• x current tape symbol
• s' next state
• x' next tape symbol
• d R to move right, or L to move left
• At beginning of operation, Turing machine is
• Starting at initial state s0
• Positioned over the leftmost nonblank symbol on
  tape

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Example T That Flips Bits

• Turing machine T defined by the three 5-tuples
• (s0, 0, s0, 1, R)
• (s0, 1, s0, 0, R)
• (s0, B, s1, B, R)
• s1 is designated final state
• Will take these steps to create the final tape
• BBBs00110BBB
• BBB1s0110BBB
• BBB10s010BBB
• BBB100s00BBB
• BBB1001s0BBB
• BBB1001Bs1BB (halt)

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Example T That Recognize Sets

• Turing machine that will recognize
  (01)(01)1(01)
• A bit string of at least length 3 that begins
  with two bits of any combination of 0s and 1s,
  then a 1, ending with any length and any
  combination of 0s and 1s
• (s0, 0, s1, 0, R), (s1, 0, s2, 0, R), (s2, 0, s3,
  0, R),
• (s0, 1, s1, 1, R), (s1, 1, s2, 1, R), (s2, 1, s4,
  1, R),
• (s0, B, s3, B, R), (s1, B, s3, B, R), (s2, B, s3,
  B, R)
• Valid string will halt in designated final state
  s4
• Otherwise will halt in nonfinal state s3

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Example T That Recognize Sets

• Steps for valid string
• BBBs00110111BBB
• BBB0s1110111BBB
• BBB01s210111BBB
• BBB011s40111BBB (halt)
• Steps for invalid string
• BBBs0010011BBB
• BBB0s110011BBB
• BBB01s20011BBB
• BBB010s3011BBB (halt)

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Class Exercise

• Exercise 1.c. (p. 782)
• Each pair of students should use only one sheet
  of paper while solving the class exercises

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Computing Functions

• Give string x as input, and halt with string y on
  the tape
• T(x) y
• Domain of T set of strings for which T halts
• if T does not halt, T(x) is undefined
• x, y nonnegative integers (N, natural numbers)
• Use unary representations of integers 0 a
  single 1, 1 two 1s, 2 three 1s,
• i.e., n n 1 ones
• For example, numeral 5 111111 (six ones)

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Representing Integers

• Input x and output y nonnegative integers
• Use unary representations of integers 0 a
  single 1, 1 two 1s, 2 three 1s,
• i.e., n n 1 ones
• For example, numeral 5 111111 (six ones)
• Separate integers with asterisks ()
• For example, T(0, 1, 4) is represented
  byBBB11111111BBB

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Example T That Computes Functions

• Function that adds two integers
• T(n1, n2) n1 n2
• Input n1, n2 n1 1 ones, followed by an
  asterisk (), followed by n2 1 ones
• Output n1 n2 n1 n2 1 ones
• Erase the leftmost two 1s, halt if n1 1,
  replace with 1, then halt
• (s0, 1, s1, B, R), (s1, 1, s2, B, R), (s2, 1, s2,
  1, R)
• (s1, , s3, B, R), (s2,
  , s3, 1, R)
  

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Example T That Computes Functions

• Steps to add 2 4 6
• BBBs011111111BBB
• BBBBs11111111BBB
• BBBBBs2111111BBB
• BBBBB1s211111BBB
• BBBBB11s311111BBB (halt)

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Class Exercise

• Exercise 17. (p. 782)
• Each pair of students should use only one sheet
  of paper while solving the class exercises