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TURING MACHINES “Turing’s "Machines". These machines are humans who calculate.” (Wittgenstein) “A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.” (Turing) 3
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IDT Open Seminar
ALAN TURING AND HIS LEGACY100 Years Turing
celebration
Gordana Dodig Crnkovic, Computer Science and Network Department
Mälardalen UniversityMarch 8th 2012
http://www.mrtc.mdh.se/~gdc/work/TuringCentenary.pdf
http://www.mrtc.mdh.se/~gdc/work/TuringMachine.pdf
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*aFinite Automata
Push-down Automatannba Rww
nnn cba ww
**ba
Turing Machines
Chomsky Language Hyerarchy
TURING MACHINES
“Turing’s "Machines". These machines are humans who calculate.” (Wittgenstein)
“A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.” (Turing)
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............Tape
Read-Write headControl Unit
Turing Machine
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............
Read-Write head
No boundaries -- infinite length
The head moves Left or Right
The Tape
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............
Read-Write head
1. Reads a symbol2. Writes a symbol3. Moves Left or Right
The head at each time step:
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Head starts at the leftmost positionof the input string
............
Blank symbol
head
a b ca
Input string
The Input String
#####
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Determinism
1q
2qRba ,
Allowed Not Allowed
3qLdb ,
1q
2qRba ,
3qLda ,
No lambda transitions allowed in TM!
Turing Machines are deterministic
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Determinism
Note the difference between state indeterminismwhen not even possible future states are known in advance.
and choice indeterminismwhen possible future states are known,but we do not know which state will be taken.
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Halting
The machine halts if there are no possible transitions to follow
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Example
............ a b ca
1q
1q
2qRba ,
3qLdb ,
No possible transition
HALT!
# # # # #
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Final States
1q 2q Allowed
1q 2q Not Allowed
• Final states have no outgoing transitions• In a final state the machine halts
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Acceptance
Accept Input If machine halts in a final state
Reject Input
If machine halts in a non-final state or If machine enters an infinite loop
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Formal Definitions for
Turing Machines
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Transition Function
1q 2qRba ,
),,(),( 21 Rbqaq
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1q 2qLdc ,
),,(),( 21 Ldqcq
Transition Function
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Turing Machine
),#,,,,,( 0 FqQM
Transitionfunction
Initialstate
blank
Finalstates
States
Inputalphabet
Tapealphabet
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For any Turing Machine M
}:{)( 210 xqxwqwML f
Initial state Final state
The Accepted Language
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Standard Turing Machine
• Deterministic
• Infinite tape in both directions
•Tape is the input/output file
The machine we described is the standard:
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Computing Functionswith
Turing Machines
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)(0 wfqwq f
Initial Configuration
FinalConfiguration
Dw DomainFor all
A function is computable ifthere is a Turing Machine such that
fM
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Example (Addition)
The function yxyxf ),( is computable
Turing Machine:
Input string: yx0 unary
Output string: 0xy unary
yx, are integers
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Start
Finish 0
fq
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yx
11
final state
0
0q
1 11 1
x y
1
initial state
# #
# #
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0q 1q 2q 3qL,## L,01
L,11
R,##
R,10
R,11
4q
R,11
Turing machine for function yxyxf ),(
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Execution Example:
11x
11y
Time 0
0
0q
1 11 1x y
Final Result
0
4q
1 11 1yx
(2)
(2)
# #
# #
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Time 0 0
0q
1 11 1
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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0q
01 11 1Time 1
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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0
0q
1 11 1Time 2
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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1q
1 11 11Time 3
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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1q
1 11 11Time 4
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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1q
1 11 11Time 5
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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2q
1 11 11Time 6
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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3q
1 11 01Time 7
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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3q
1 11 01Time 8
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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3q
1 11 01Time 9
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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3q
1 11 01Time 10
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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3q
1 11 01Time 11
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
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4q
1 11 01
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
HALT & accept
Time 12 # #
L,##
R,##
yxyxf ),(
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Universal Turing Machine
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A limitation of Turing Machines:
Turing Machines are “hardwired”
they executeonly one program
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Solution: Universal Turing Machine
• Reprogrammable machine
• Simulates any other Turing Machine
Characteristics:
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Universal Turing Machine
simulates any other Turing Machine M
Input to Universal Turing Machine:
• Description of transitions ofM• Initial tape contents of M
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Universal Turing Machine
Description of Three tapes
MTape Contents of
Tape 2
State of M
Tape 3
M
Tape 1
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We describe Turing machine as a string of symbols:
We encode as a string of symbols
M
M
Description of M
Tape 1
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Alphabet Encoding
Symbols: a b c d
Encoding: 1 11 111 1111
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State Encoding
States: 1q 2q 3q 4q
Encoding: 1 11 111 1111
Head Move Encoding
Move:
Encoding:
L R
1 11
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Transition Encoding
Transition: ),,(),( 21 Lbqaq
Encoding: 10110110101
separator
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Machine Encoding
Transitions:
),,(),( 21 Lbqaq
Encoding:
10110110101
),,(),( 32 Rcqbq
110111011110101100
separator
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Tape 1 contents of Universal Turing Machine:
encoding of the simulated machine as a binary string of 0’s and 1’s
M
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As Turing Machine is described with a binary string of 0’s and 1’s
the set of Turing machines forms a language:
Each string of the language isthe binary encoding of a Turing Machine.
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Language of Turing Machines
L = { 010100101,
00100100101111,
111010011110010101, …… }
(Turing Machine 1)
(Turing Machine 2)
……
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Question:Do Turing machines have the same power with a digital computer?
Intuitive answer: Yes
There was no formal proof of Church-Turing thesis until 2008!
CHURCH TURING THESIS
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Dershowitz, N. and Gurevich, Y. A Natural Axiomatization of Computability and Proof of Church's Thesis, Bulletin of Symbolic Logic, v. 14, No. 3, pp. 299-350 (2008)
This formal proof of Church-Turing thesis relies on an axiomatization of computation that excludes randomness, parallelism and quantum computing and thus corresponds to the idea of computing that Church and Turing had.
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Turing’s thesis
Any computation carried outby algorithmic meanscan be performed by a Turing Machine. (1930)
http://www.engr.uconn.edu/~dqg/papers/myth.pdf The Origins of the Turing Thesis Myth Goldin & Wegner
