cdt314 faber formal languages, automata and models of computation lecture 12
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CDT314
FABER
Formal Languages, Automata and Models of Computation
Lecture 12
Mälardalen University
2011
2
Content
Chomsky’s Language Hierarchy Turing Machines DeterminismHaltingTM ExamplesStandard TMComputing Functions with TMCombining TMsTuring Thesis
3
The Language Hierarchy
*aRegular Languages
Context-Free Languagesnnba Rww
nnn cba ww?
**ba
?
4
Regular Languages
**ba
Context-free languages
Recursively enumerable languages}0:{ ncba nnn
}0:{ nba nn
}},{:{ bawww
}*},{:{ bawwwR
}0:{ ! nan }0:{2
nba nn
5
*aFinite Automata
Push-down Automatannba Rww
nnn cba ww
**ba
Turing Machines
6
http://www.turing.org.uk/turing/
Who was Alan Turing?
Founder of Computability Theory,
mathematician,
philosopher,
codebreaker,
visionary man before his time.
http://www.cs.usfca.edu/www.AlanTuring.net/turing_archive/index.html- Jack
Copeland and Diane Proudfoot http://www.turing.org.uk/turing/ The Alan Turing Home Page
Andrew Hodges
7
Alan Turing
1912 (23 June): Birth, London
1926-31: Sherborne School
1930: Death of friend Christopher Morcom
1931-34: Undergraduate at King's College, Cambridge University
1932-35: Quantum mechanics, probability, logic
1935: Elected fellow of King's College, Cambridge
1936: The Turing machine, computability, universal machine
1936-38: Princeton University. Ph.D. Logic, algebra, number theory
1938-39: Return to Cambridge. Introduced to German Enigma cipher machine
1939-40: The Bombe, machine for Enigma decryption
1939-42: Breaking of U-boat Enigma, saving battle of the Atlantic
8
Alan Turing
1943-45: Chief Anglo-American crypto consultant. Electronic work.
1945: National Physical Laboratory, London
1946: Computer and software design leading the world.
1947-48: Programming, neural nets, and artificial intelligence
1948: Manchester University
1949: First serious mathematical use of a computer
1950: The Turing Test for machine intelligence
1951: Elected FRS. Non-linear theory of biological growth
1952: Arrested as a homosexual, loss of security clearance
1953-54: Unfinished work in biology and physics
1954 (7 June): Death (suicide) by cyanide poisoning, Wilmslow, Cheshire.
9
Hilbert’s Program, 1900
Hilbert’s hope was that mathematics would be reducible to finding proofs (manipulating the strings of symbols) from a fixed system of axioms, axioms that everyone could agree were true.
Can all of mathematics be made algorithmic, or will there always be new problems that outstrip any given algorithm, and so require creative acts of mind to solve?
10
Turing Machines and Computability
The question Hilbert raised was whether there could be a general method or process by which one could decide whether a mathematical proposition could be proved. But what exactly was meant by a 'method' or 'process'? People had already used the concept of a 'mechanical' process, and Turing had an idea which made this quite precise: computability. Turing wrote in his first sentences:
“The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means... a number is computable if its decimal can be written down by a machine. “
http://www.turing.org.uk/turing/scrapbook/machine.html
11
Turing Machines and Computability
The Turing machine concept involves specifying a very restricted set of logical operations, but Turing showed how other more complex mathematical procedures could be built out of these atomic components.
Turing argued that his formalism was sufficiently general to encompass anything that a human being could do when carrying out a definite method.
Turing's famous 1936-7 paper On computable numbers, with an application to the Entscheidungsproblem, which worked out the theory of Turing machines and the definition of computability, is available as a PDF file on-line:
http://www.cs.ox.ac.uk/activities/ieg/e-library/sources/tp2-ie.pdf
12
Turing Machines
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)
13
14
............Tape
Read-Write head
Control Unit
Turing Machine
15
............
Read-Write head
No boundaries -- infinite length
The head moves Left or Right
The Tape
16
............
Read-Write head
1. Reads a symbol
2. Writes a symbol
3. Moves Left or Right
The head at each time step:
17
ExampleTime 0
............ a a cb
Time 1............ a b k c
1. Reads a2. Writes k3. Moves Left
18
Time 1............ a b k c
Time 2............ a k cf
1. Reads b2. Writes f
3. Moves Right
19
Head starts at the leftmost position
of the input string
............
Blank symbol
head
a b ca
Input string
The Input String
#####
20
1q 2qLba ,
Read WriteMove Left
1q 2qRba ,
Move Right
States & Transitions
21
Example
1q 2qRba ,
............ a b ca
Time 1
1qcurrent state
#####
22
............ # a b caTime 1
1q 2qRba ,
............ a b cbTime 2
1q
2q
# # # #
# # # # #
23
............ a b caTime 1
1q 2qLba ,
............ a b cbTime 2
1q
2q
Example
# # # # #
# # # # #
24
............ a b caTime 1
1q 2qRg,#
............ ga b cbTime 2
1q
2q
Example
# # # # #
# # # #
25
Determinism
1q
2qRba ,
Allowed Not Allowed
3qLdb ,
1q
2qRba ,
3qLda ,
No lambda transitions allowed in TM!
Turing Machines are deterministic
26
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.
27
Partial Transition Function
1q
2qRba ,
3qLdb ,
............ a b ca
1q
Example
No transition
for input symbol c
Allowed
# # # # #
28
Halting
The machine halts if there are
no possible transitions to follow
29
Example
............ a b ca
1q
1q
2qRba ,
3qLdb ,
No possible transition
HALT!
# # # # #
30
Final States
1q 2q Allowed
1q 2q Not Allowed
• Final states have no outgoing transitions
• In a final state the machine halts
31
Acceptance
Accept InputIf machine halts
in a final state
Reject Input
If machine halts
in a non-final state
or
If machine enters
an infinite loop
32
Turing Machine Example
A TM that accepts the language *aa
0q
Raa ,
L,##1q
33
Time 0 aa
0q
a
0q
Raa ,
1qL,##
# # # #
34
Time 1 aa
0q
a
0q
Raa ,
1q
# # # #
L,##
35
Time 2 aa
0q
a
0q
Raa ,
1q
# # # #
L,##
36
Time 3aa
0q
a
0q
Raa ,
1qL,##
# # # #
37
Time 4 aa
1q
a
0q
Raa ,
1q
Halt & Accept
L,##
# # # #
38
Rejection Example
Time 0 ba
0q
a
0q
Raa ,
1qL,##
# # # #
39
0q
Raa ,
1q
Time 1 ba
0q
a
No possible Transition
Halt & Reject
L,##
# # # #
40
Infinite Loop Example
0q
Raa ,
1q
Lbb ,
Another TM for language *aa
L,##
41
Time 0 ba
0q
a
0q
Raa ,
1q
Lbb ,
L,##
# # # #
42
Time 1 ba
0q
a
0q
Raa ,
1q
Lbb ,
L,##
# # # #
43
Time 2 ba
0q
a
0q
Raa ,
1q
Lbb ,
L,##
# # # #
44
Time 2 ba
0q
a
baTime 3
0q
a
baTime 4
0q
a
baTime 5
0q
a
... infinite loop…
# # # #
# # # #
# # # #
# # # #
45
Because of the infinite loop:
•The final state cannot be reached
•The machine never halts
•The input is not accepted
46
Another Turing Machine Example
Turing machine for the language }{ nnba
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
47
Time 0 ba
0q
a b
Ryy ,
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
4q
L,##
# # #}{ nnba
48
bx
1q
a bTime 1
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
49
bx
1q
a bTime 2
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
50
yx
2q
a bTime 3
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
51
yx
2q
a bTime 4
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
52
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
yx
0q
a bTime 5
L,##
# # #}{ nnba
53
yx
1q
x bTime 6
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
54
yx
1q
x bTime 7
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
55
yx x y
2q
Time 8
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
56
yx x y
2q
Time 9
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
57
Ryy ,
yx
0q
x yTime 10
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
4q
L,##
# # #}{ nnba
58
yx
3q
x yTime 11
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
59
yx
3q
x yTime 12
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,##
# # #}{ nnba
60
yx
4q
x y
0q 1q 2q3qRxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
Halt & Accept
Time 13
L,##
# # #}{ nnba
61
If we modify the
machine for the language }{ nnba
we can easily construct
a machine for the language }{ nnn cba
Observation
62
Formal Definitions for
Turing Machines
63
Transition Function
1q 2qRba ,
),,(),( 21 Rbqaq
64
1q 2qLdc ,
),,(),( 21 Ldqcq
Transition Function
65
Turing Machine
),#,,,,,( 0 FqQM
Transition
functionInitial
stateblank
Final
states
States
Input
alphabetTape
alphabet
66
Instantaneous description:
ba
1q
ac
baqca 1
Configuration
# # # #
67
Time 4 Time 5
yx
2q
a b yx
0q
a b
A move:aybqxxaybq 02
# # # # # #
68
Time 4 Time 5
yx
2q
a b yx
0q
a b
bqxxyybqxxaybqxxaybq 1102
Time 6 Time 7
yx
1q
x b yx
1q
x b
# # # # # #
# # # # # #
69
bqxxyxaybq 12
Equivalent notation:
bxxqybxxqaybxqxaybq 1102
70
Initial configuration: wq0
ba
0q
a b
w
Input string
# # #
71
For any Turing Machine M
}:{)( 210 xqxwqwML f
Initial state Final state
The Accepted Language
72
Standard Turing Machine
• Deterministic
• Infinite tape in both directions
•Tape is the input/output file
The machine we described is the standard:
73
Computing Functionswith
Turing Machines
74
A function )(wf has:
Domain
(domän):
DDw
Range
(värdemängd):
S Swf )(
75
A function may have many parameters:
yxyxf ),(
Example: Addition function
76
Integer Domain
Binary: 101
Decimal: 5
We prefer unary representation:
easier to manipulate
Unary: 11111
77
Definition
A function is computable if
there is a Turing Machine such that:
fM
Initial configuration Final configuration
Dw Domain
fq
)(wf
final state0q
w
initial state
For all
# # # #
78
)(0 wfqwq f
Initial
Configuration
Final
Configuration
In other words
Dw DomainFor all
A function is computable if
there is a Turing Machine such that
fM
79
Example (Addition)
The function yxyxf ),( is computable
Turing Machine:
Input string: yx0 unary
Output string: 0xy unary
yx, are integers
80
Start
Finish 0
fq
11
yx
11
final state
0
0q
1 11 1
x y
1
initial state
# #
# #
81
0q 1q 2q 3qL,## L,01
L,11
R,##
R,10
R,11
4q
R,11
Turing machine for function yxyxf ),(
82
Execution Example:
11x
11y
Time 0
0
0q
1 11 1x y
Final Result
0
4q
1 11 1
yx
(2)
(2)
# #
# #
83
Time 0 0
0q
1 11 1
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
84
0q
01 11 1Time 1
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
85
0
0q
1 11 1Time 2
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
86
1q
1 11 11Time 3
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
87
1q
1 11 11Time 4
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
88
1q
1 11 11Time 5
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
89
2q
1 11 11Time 6
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
90
3q
1 11 01Time 7
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
91
3q
1 11 01Time 8
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
92
3q
1 11 01Time 9
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
93
3q
1 11 01Time 10
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
94
3q
1 11 01Time 11
0q 1q 2q 3qL,01
L,11
R,10
R,11
4q
R,11
# #
L,##
R,##
yxyxf ),(
95
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 ),(
96
Another Example (Multiplication)
The function xxf 2)( is computable
Turing Machine
Input string: x unary
Output string: xx unary
x is integer
97
Start
Finish 1
fq
11
x2
11
final state
0q
11
x
1
initial state
# #
# #
98
Turing Machine Pseudocode for xxf 2)(
• Replace every 1 with $
• Repeat:
• Find rightmost $, replace it with 1
• Go to right end, insert 1
Until no more $ remain
99
0q 1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
Turing Machine for xxf 2)(
100
0q
11
3q
1 11 1
Start Finish
# # # #
0q 1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
xxf 2)(
101
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
0q
1Time 0 # # # #
0q
1 xxf 2)(
102
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
0q
1$Time 1 # # # #
0q
xxf 2)(
103
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
0q
Time 2 # # # #
0q
$ $ xxf 2)(
104
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
1q
Time 3 # # # #
0q
$ $ xxf 2)(
105
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
2q
1Time 4 # # # #
0q
$ xxf 2)(
106
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
1q
1 1Time 5 # # #
0q
$ xxf 2)(
107
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
1q
1 1Time 6 # # #
0q
$ xxf 2)(
108
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
2q
1 1Time 7 # # #
0q
xxf 2)( 1
109
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
2q
1 1Time 8 # # #
0q
xxf 2)( 1
110
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
2q
1 1Time 9 # # #
0q
xxf 2)( 1
111
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
11Time 10 # #
0q
1q
1 1 xxf 2)(
112
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
11Time 11 # #
0q
1q
1 1 xxf 2)(
113
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
11Time 12 # #
0q
1q
1 1 xxf 2)(
114
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
11Time 13 # #
0q
1q
1 1 xxf 2)(
115
1q 2q
3q
R,1$
L,1#
L,##
R$,1 L,11 R,11
R,##
11Time 14 # #
0q
3q
1 1 xxf 2)(
116
Another Example
The function is computable
),( yxf0
1 yx
yx
if
if
Turing Machine for
Output: 1 0or
Input: yx0
117
Turing Machine Pseudocode
Match a 1 from with a 1 from x y
• Repeat
Until all of or are matchedx y
• If a 1 from is not matched
erase tape, write 1
else
erase tape, write 0
x
118
Combining Turing Machines
119
Block Diagram
Turing
Machineinput output
120
Example
),( yxf0
yx yx
yx
if
if
Comparer
Adder
Eraser
yx, yx
yx
yx
0
121
Turing’s Thesis
122
Do Turing machines have
the same power with
a digital computer?
Intuitive answer: Yes
There is no formal answer!
Question:
123
Turing’s thesis
Any computation carried out
by mechanical means
can 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
124
And, the other way round:
A computation is mechanical
if and only if
it can be performed by a Turing Machine
Strong Turing Thesis: There is no known model of computation
more powerful than Turing Machines
Wrong! Asynchronous parallel computation is more powerful than TM model, and cannot be reduced to TM.
125
Definition of Algorithm
An algorithm for function
is a Turing Machine which computes
)(wf)(wf
126
When we say
There exists an algorithm
Algorithms are Turing Machines
It means
There exists a Turing Machine
127
Language Grammar Machine
Chomsky Hierarchy
Non-computable ??
Recursively enumerable (RE)
Unrestricted Turing machines 0
Recursive
Turing machines that always halt 0
Context-sensitive
Context-sensitive grammar
Linear-bounded automata 1
Context-free
Context-free grammar
Non-deterministic pushdown automata 2
Regular
Regular expressions Finite state automata 3
properly inclusive
Languages, Grammars, Machines
128
Regular Languages
FINITE AUTOMATA
**ba
Context-free languages
PUSH-DOWN AUTOMATA
Unrestricted grammar languages
TURING MACHINES
}0:{ ncba nnn
}0:{ nba nn
}},{:{ bawww
}*},{:{ bawwwR
}0:{ ! nan }0:{2
nba nn
Interesting further reading
Computational and evolutionary aspects of language
http://www.nature.com/nature/journal/v417/n6889/full/nature00771.html
Turing’s thesis
http://www.ams.org/notices/200610/fea-feferman.pdf
http://www.mathcomp.leeds.ac.uk/turing2012/give-page.php?408
2012 The Alan Turing World
https://sites.google.com/site/naturalcomputingaisbiacap2012 Symposium on Natural/Unconventional Computing AISB/IACAP World Congress
129
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