Turing Machines

1
Turing Machines

• There are languages that are not context-free.
• What can we say about the most powerful automata
  and the limits of computation?.
• Alan Turing (1912 - 1954).

2
Standard Turing Machine
Tape
Read-write head
Control unit q0
3
Standard Turing Machine

• M (Q, ?, ?, ?, q0, , F)
• Q finite set of internal states
• ? finite set of symbols - tape alphabet
• ? ? blank
• ? ? ? - finite set of symbols - input
  alphabet
• ? Q ? ? ? Q ? ? ? L, R transition function
• q0 ? Q initial state
• F ? Q set of final states

4
Standard Turing Machine

• ? Q ? ? ? Q ? ? ? L, R

current symbol
head move direction
replacing symbol
5
Example

• ?(q0, a) (q1, d, R)

current symbol
head move to the right
replacing symbol
6
Halt State

• A state for which ? is not defined.
• Assume that all final states are halt states.

7
Example

• M (Q, ?, ?, ?, q0, , F)
• Q q0, q1 ?(q0, a) (q0, b, R)
• ? a, b ?(q0, b) (q0, b, R)
• ? a, b, ?(q0, ) (q1, , L)
• F q1

8
Example

• M (Q, ?, ?, ?, q0, , F)
• Q q0, q1 ?(q0, a) (q1, a, R)
• ? a, b ?(q0, b) (q1, b, R)
• ? a, b, ?(q0, ) (q1, , R)
• F ? ?(q1, a) (q0, a, L)
• ?(q1, b) (q0, b, L)
• ?(q1, ) (q0, , L)

9
Instantaneous Description

• a1a2 ... ak-1qakak1 ... an
• current state current symbol

10
Instantaneous Description

• move abq1cd ? abeq2d
• if ?(q1, c) ? (q2, e, R)

11
Instantaneous Description

• x1qix2 ?M y1qjy2
• x1qix2 ?M y1qjy2

12
Turing Machines as Language Accepters

• Let M (Q, ?, ?, ?, q0, , F) be a TM.
• L(M) w ? ? q0w ?M x1qfx2 where qf ? F and
  x1, x2 ? ?

13
Example

• L 0
• M (Q, ?, ?, ?, q0, , F) ?

14
Example

• L anbn n ? 1
• M (Q, ?, ?, ?, q0, , F) ?

15
Example

• L anbncn n ? 1
• M (Q, ?, ?, ?, q0, , F) ?

16
Turing Machines as Language Transducers

• q0w ?M qfw
• function w f(w)

17
Turing Machines as Language Transducers

• A function f with domain D is said to be
  Turing-computable if there exists some Turing
  machine M (Q, ?, ?, ?, q0, , F) such that
• q0w ?M qff(w) qf ? F
• for all w ? D.

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Example

• f(x, y) x y
• M (Q, ?, ?, ?, q0, , F) ?

19
Example

• f(w) ww w ? 1
• M (Q, ?, ?, ?, q0, , F) ?

20
Example

• f(x, y) true if x ? y
• or f(x, y) false otherwise
• M (Q, ?, ?, ?, q0, , F) ?

21
Combining Turing Machines

• x y if x ? y
• f(x, y)
• 0 if x ? y
• M (Q, ?, ?, ?, q0, , F) ?

22
Combining Turing Machines
x y
Adder A
x ? y
x, y
f(x, y)
Comparer C
x ? y
Eraser E
0
23
Combining Turing Machines

• qcw(x)0w(y) ?M qAw(x)0w(y) if x ? y
• qcw(x)0w(y) ?M qEw(x)0w(y) if x ? y
• qAw(x)0w(y) ?M qAfw(x y)0
• qEw(x)0w(y) ?M qEf0

24
Macroinstructions

• if a then qj else qk
• ?(qi, a) (qj0, a, R)
• ?(qi, b) (qk0, b, R)
• ?(qj0, c) (qj, c, L)
• ?(qk0, c) (qk, c, L)

25
Subprograms
Region separator


Workspace for A
Workspace for B
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Example

• f(x, y) x ? y
• For each 1 in x, create a 1-string of length y.

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Turing's Thesis

• Turing machine appears to be simple.
• Turing seems to approach a typical digital
  computer.

28
Turing's Thesis

• Any computation that can be carried out by
  mechanical means can be performed by some Turing
  machine.

29
Turing's Thesis

• Anything done by existing digital computers can
  be done by a Turing machine.
• No problem solvable by an algorithm cannot be
  solved by a Turing machine.
• No alternative mechanical computation model is
  more powerful than the Turing machine model.

30
Homework

• Exercises 2, 5, 8, 9, 16, 19 of Section 9.1.
• Exercises 1, 2, 3, 4, 9 of Section 9.2.
• Presentations
• Section 12.1 Computability and Decidability
  Halting Problem
• Section 13.1 Recursive Functions
• Post Systems Church's Thesis
• Section 13.2 Measures of Complexity
  Complexity Classes