language recognition (11.4) and turing machines (11.5) longin jan latecki temple university
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Language Recognition (11.4)and Turing Machines (11.5) Longin Jan LateckiTemple University
Based on slides by Costas Busch from the coursehttp://www.cs.rpi.edu/courses/spring05/modcomp/and …
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Three Equivalent Representations
Finite automata
Regularexpressions
Regular languages
Each can
describethe others
Kleene’s Theorem: For every regular expression, there is a deterministic finite-state automaton that defines the same language, and vice versa.
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EXAMPLE 1
Consider the language { ambn | m, n N}, which is represented by the regular expression a*b*.
A regular grammar for this language can
be written as follows:
S | aS | B B b | bB.
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Regular Expression
Regular Grammar
a* S | aS(a+b)* S | aS | bSa* + b* S | A | B
A a | aAB b | bB
a*b S b | aSba* S bA
A | aA(ab)* S | abS
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NFAs Regular grammarsThus, the language recognized by FSA
is a regular language Every NFA can be converted into a corresponding regular grammar and vice versa.Each symbol A of the grammar is associated with a non-terminal node of the NFA sA, in particular, start symbol
S is associated with the start state sS.
Every transition is associated with a grammar production: T(sA,a) = sB A aB.
Every production B is associated with final state sB.
See Ex. 3, p. 771, and Ex. 4, p. 772.
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Kleene’s Theorem
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
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LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
We will show:
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Proof - Part 1
r)(rL
For any regular expression the language is recognized by FSA (= is a regular language)
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
Proof by induction on the size of r
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Induction BasisPrimitive Regular Expressions: a,,
NFAs
)()( 1 LML
)(}{)( 2 LML
)(}{)( 3 aLaML
regularlanguages
a
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Inductive Hypothesis
Assume for regular expressions andthat and are regular languages
1r 2r
)( 1rL )( 2rL
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Inductive StepWe will prove:
1
1
21
21
*
rL
rL
rrL
rrL
Are regular Languages
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By definition of regular expressions:
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11
2121
2121
**
rLrL
rLrL
rLrLrrL
rLrLrrL
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)( 1rL )( 2rLBy inductive hypothesis we know: and are regular languages
Regular languages are closed under: *1
21
21
rLrLrL
rLrL Union Concatenation
Star
We need to show:
This fact is illustrated in Fig. 2 on p. 769.
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Therefore:
** 11
2121
2121
rLrL
rLrLrrL
rLrLrrL
Are regularlanguages
And trivially: ))(( 1rL is a regular language
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Proof - Part 2
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
Lr LrL )(
For any regular language there is a regular expression with
Proof by construction of regular expression
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Since is regular take the NFA that accepts it
LM
LML )(
Single final state
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From construct the equivalentGeneralized Transition Graph in which transition labels are regular
expressions
M
Example:
a
ba,
cM
a
ba
c
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Another Example:
ba a
b
b0q 1q 2q
ba,a
b
b0q 1q 2q
b
b
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Reducing the states:ba
ab
b0q 1q 2q
b
0q 2q
babb*
)(* babb
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Resulting Regular Expression:
0q 2q
babb*
)(* babb
*)(**)*( bbabbabbr
LMLrL )()(
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In GeneralRemoving states:
iq q jqa b
cde
iq jq
dae* bce*dce*
bae*
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The final transition graph:
0q fq
1r
2r
3r4r
*)*(* 213421 rrrrrrr
LMLrL )()(
The resulting regular expression:
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DFA - regular languagesPush down automata - Context-freeBounded Turing M’s - Context sensitiveTuring machines - Phrase-structure
Models of computing
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Foundations
The theory of computation and the practical application it made possible — the computer — was developed by an Englishman called Alan Turing.
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Alan Turing1912 (23 June): Birth, Paddington, London1931-34: Undergraduate at King's College, Cambridge University1932-35: Quantum mechanics, probability, logic1936: The Turing machine, computability, universal machine1936-38: Princeton University. Ph.D. Logic, algebra, number theory 1938-39: Return to Cambridge. Introduced to German Enigma cipher machine1939-40: The Bombe, machine for Enigma decryption
1939-42: Breaking of U-boat Enigma, saving battle of the Atlantic1946: Computer and software design leading the world.1948: Manchester University1949: First serious mathematical use of a computer1950: The Turing Test for machine intelligence1952: Arrested as a homosexual, loss of security clearance1954 (7 June): Death (suicide) by cyanide poisoning, Wilmslow, Cheshire.
—from Andrew Hodges http://www.turing.org.uk/turing/
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The Decision Problem
In 1928 the German mathematician, David Hilbert (1862-1943), asked whether there could be a mechanical way (i.e. by means of a fully specifiable set of instructions) of determining whether some statement in a formal system like arithmetic was provable or not.In 1936 Turing published a paper the aim of which was to show that there was no such method. “On computable numbers, with an application to the Entscheidungs problem.” Proceedings of the London Mathematical Society, 2(42):230-265.
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The Turing MachineIn order to argue for this claim, he needed a clear concept of “mechanical procedure.”His idea — which came to be called the Turing machine — was this:
(1) A tape of infinite length
(2) Finitely many squares of the tape have a single symbol from a finite language.
(3) Someone (or something) that can read the squares and write in them.
(4) At any time, the machine is in one of a finite number of internal states.
(5) The machine has instructions that determine what it does given its internal state and the symbol it encounters on the tape. It can change its internal state; change the symbol on
the square; move forward; move backward; halt (i.e. stop).
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01
1
1
1
Current state = 1
If current state = 1and current symbol = 0then new state = 10new symbol = 1move right
0
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11
1
1
1
Current state = 10
If current state = 1and current symbol = 0then new state = 10new symbol = 1move right
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11
1
1
1
Current state = 10
If current state = 1and current symbol = 0then new state = 10new symbol = 1move right
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FunctionsIt is essential to the idea of a Turing machine that it is not a physical machine, but an abstract one — a set of procedures.
It makes no difference whether the machine is embodied by a person in a boxcar on a track, or a person with a paper and pencil, or a smart and well-trained flamingo.
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Turing’s TheoremIn the 1936 paper Turing proved that there are “general-purpose” Turing machines that can compute whatever any other Turing machine.This is done by coding the function of the special-purpose machine as instructions of the other machine — that is by “programming” it. This is called Turing’s theorem.These are universal Turing machines, and the idea of a coding for a particular function fed into a universal Turing machine is basically our conception of a computer and a stored program. The concept of the universal Turing machine is just the concept of the computer as we know it.
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First computers: custom computing machines
1946 -- Eniac: the control is hardwired manually foreach problem.
Control
Input tape (read only)
Output tape (write only)
Work tape (memory)
1940: VON NEUMANN: DISTINCTION BETWEEN DATA AND INSTRUCTIONS
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Can Machines Think?In “Computing machinery and intelligence,” written in 1950, Turing asks whether machines can think.He claims that this question is too vague, and proposes, instead, to replace it with a different one.That question is: Can machines pass the “imitation game” (now called the Turing test)? If they can, they are intelligent. Turing is thus the first to have offered a rigorous test for the determination of intelligence quite generally.
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The Turing TestThe game runs as follows. You sit at a computer terminal and have an electronic conversation. You don’t know who is on the other end; it could be a person or a computer responding as it has been programmed to do.If you can’t distinguish between a human being and a computer from your interactions, then the computer is intelligent.Note that this is meant to be a sufficient condition of intelligence only. There may be other ways to be intelligent.
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Artificial Intelligence
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The Church-Turning ThesisTuring, and a logician called Alonzo Church (1903-1995), independently developed the idea (not yet proven by widely accepted) that whatever can be computed by a mechanical procedure can be computed by a Turing machine.This is known as the Church-Turing thesis.
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AI: The ArgumentWe’ve now got the materials to show that AI is possible:
P1: Any function that can be computed by a mechanical procedure can be computed by a Turing machine. (Church-Turing thesis)
P2: Thinking is nothing more than the computing of functions by mechanical procedures (i.e., thinking is symbol manipulation). (Functionalist-Computationalist thesis)
C1: Therefore, thinking can be performed by a Turing machine.
P3: Turing machines are multiply realizable. In particular, they can be realized by computers, robots, etc.
It is possible to build a computer, robot, etc. that can think. That is, AI is possible.
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Turing Machines
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The Language Hierarchy
*aRegular Languages
Context-Free Languagesnnba Rww
nnn cba ww?
**ba
?
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*aRegular Languages
Context-Free Languagesnnba Rww
nnn cba ww
**ba
Languages accepted byTuring Machines
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A Turing Machine
............Tape
Read-Write headControl Unit
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The Tape
............
Read-Write head
No boundaries -- infinite length
The head moves Left or Right
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............
Read-Write head
The head at each time step:
1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right
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............
Example:Time 0
............Time 1
1. Reads2. Writes
a a cb
a b k c
ak
3. Moves Left
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............Time 1
a b k c
............Time 2
a k cf
1. Reads2. Writes
bf
3. Moves Right
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The Input String
............
Blank symbol
head
a b ca
Head starts at the leftmost positionof the input string
Input string
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............
Blank symbol
head
a b ca
Input string
Remark: the input string is never empty
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States & Transitions
1q 2qLba ,
Read Write Move Left
1q 2qRba ,
Move Right
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Example:
1q 2qRba ,
............ a b ca
Time 1
1qcurrent state
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............ a b caTime 1
1q 2qRba ,
............ a b cbTime 2
1q
2q
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............ a b caTime 1
1q 2qLba ,
............ a b cbTime 2
1q
2q
Example:
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............ a b caTime 1
1q 2qRg,
............ ga b cbTime 2
1q
2q
Example:
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Determinism
1q
2qRba ,
Allowed Not Allowed
3qLdb ,
1q
2qRba ,
3qLda ,
No lambda transitions allowed
Turing Machines are deterministic
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Partial Transition Function
1q
2qRba ,
3qLdb ,
............ a b ca
1q
Example:
No transitionfor input symbol c
Allowed:
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Halting
The machine halts if there areno 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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Turing Machine Example
A Turing machine that accepts the language:*aa
0q
Raa ,
L,1q
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aaTime 0
0q
a
0q
Raa ,
L,1q
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aaTime 1
0q
a
0q
Raa ,
L,1q
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aaTime 2
0q
a
0q
Raa ,
L,1q
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aaTime 3
0q
a
0q
Raa ,
L,1q
65
aaTime 4
1q
a
0q
Raa ,
L,1q
Halt & Accept
66
Rejection Example
0q
Raa ,
L,1q
baTime 0
0q
a
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0q
Raa ,
L,1q
baTime 1
0q
a
No possible TransitionHalt & Reject
68
Infinite Loop Example
0q
Raa ,
L,1q
Lbb ,
A Turing machine for language *)(* babaa
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baTime 0
0q
a
0q
Raa ,
L,1q
Lbb ,
70
baTime 1
0q
a
0q
Raa ,
L,1q
Lbb ,
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baTime 2
0q
a
0q
Raa ,
L,1q
Lbb ,
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baTime 2
0q
a
baTime 3
0q
a
baTime 4
0q
a
baTime 50q
a
Infinite loop
73
Because of the infinite loop:
•The final state cannot be reached
•The machine never halts
•The input is not accepted
74
Another Turing Machine Example
Turing machine for the language }{ nnba
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
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0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
ba
0q
a bTime 0
76
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
bx
1q
a b Time 1
77
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
bx
1q
a b Time 2
78
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
2q
a b Time 3
79
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
2q
a b Time 4
80
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
0q
a b Time 5
81
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
1q
x b Time 6
82
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
1q
x b Time 7
83
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx x y
2q
Time 8
84
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx x y
2q
Time 9
85
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
0q
x y Time 10
86
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
3q
x y Time 11
87
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
3q
x y Time 12
88
0q 1q 2q3q Rxa ,
Raa ,Ryy ,
Lyb ,
Laa ,Lyy ,
Rxx ,
Ryy ,
Ryy ,4q
L,
yx
4q
x y
Halt & Accept
Time 13
89
If we modify the machine for the language }{ nnba
we can easily construct a machine for the language }{ nnn cba
Observation:
90
Formal Definitionsfor
Turing Machines
91
Transition Function
1q 2qRba ,
),,(),( 21 Rbqaq
92
1q 2qLdc ,
),,(),( 21 Ldqcq
Transition Function
93
Turing Machine:
),,,,,,( 0 FqQM
States
Inputalphabet
Tapealphabet
Transitionfunction
Initialstate
blank
Finalstates
94
Configuration
ba
1q
a
Instantaneous description:
c
baqca 1
95
yx
2q
a bTime 4
yx
0q
a bTime 5
A Move: aybqxxaybq 02
96
yx
2q
a bTime 4
yx
0q
a bTime 5
bqxxyybqxxaybqxxaybq 1102
yx
1q
x bTime 6
yx
1q
x bTime 7
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bqxxyybqxxaybqxxaybq 1102
bqxxyxaybq 12Equivalent notation:
98
Initial configuration: wq0
ba
0q
a b
wInput string
99
The Accepted Language
For any Turing Machine M
}:{)( 210 xqxwqwML f
Initial state Final state
100
Standard Turing Machine
• Deterministic
• Infinite tape in both directions
•Tape is the input/output file
The machine we described is the standard:
101
Computing Functionswith
Turing Machines
102
A function )(wf
Domain: Result Region:
has:
D
Dw
S
Swf )()(wf
103
A function may have many parameters:
yxyxf ),(
Example: Addition function
104
Integer Domain
Unary:
Binary:
Decimal:
11111
101
5
We prefer unary representation:
easier to manipulate with Turing machines
105
Definition:A function is computable ifthere is a Turing Machine such that:
fM
Initial configuration Final configuration
Dw Domain
0q
w
fq
)(wf
final stateinitial state
For all
106
)(0 wfqwq f
Initial Configuration
FinalConfiguration
A function is computable ifthere is a Turing Machine such that:
fM
In other words:
Dw DomainFor all
107
Example
The function yxyxf ),( is computable
Turing Machine:
Input string: yx0 unary
Output string: 0xy unary
yx, are integers
108
0
0q
1 1 1 1
x y
1 Start
initial state
The 0 is the delimiter that separates the two numbers
109
0
0q
1 1 1 1
x y
1
0
fq
1 1
yx
11
Start
Finish
final state
initial state
110
0
fq
1 1
yx
11Finish
final state
The 0 helps when we usethe result for other operations
111
0q
Turing machine for function
1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
yxyxf ),(
112
Execution Example:
11x
11y 0
0q
1 1 1 1
Time 0x y
Final Result
0
4q
1 1 1 1yx
(2)
(2)
113
0
0q
1 1Time 0
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
1 1
114
0q
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
01 11 1Time 1
115
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
0
0q
1 1 1 1Time 2
116
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
1q
1 11 11Time 3
117
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
1q
1 1 1 11Time 4
118
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
1q
1 11 11Time 5
119
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
2q
1 1 1 11Time 6
120
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
3q
1 11 01Time 7
121
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
3q
1 1 1 01Time 8
122
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
3q
1 11 01Time 9
123
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
3q
1 1 1 01Time 10
124
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
3q
1 11 01Time 11
125
0q 1q 2q 3qL, L,01
L,11
R,
R,10
R,11
4q
R,11
4q
1 1 1 01
HALT & accept
Time 12
126
Another Example
The function xxf 2)( is computable
Turing Machine:
Input string: x unary
Output string: xx unary
x is integer
127
0q
1 1
x
1
1
fq
1 1
x2
11
Start
Finish
final state
initial state
128
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
129
0q 1q 2q
3q
R,1$
L,1
L,
R$,1 L,11 R,11
R,
Turing Machine for xxf 2)(
130
0q 1q 2q
3q
R,1$
L,1
L,
R$,1 L,11 R,11
R,
Example
0q
1 1
3q
1 11 1
Start Finish
131
Another Example
The function ),( yxf
is computable 0
1 yx
yx
if
if
132
Turing Machine for
Input: yx0
Output: 1 0or
),( yxf0
1 yx
yx
if
if
133
Turing Machine Pseudocode:
Match a 1 from with a 1 from x y
• Repeat
Until all of or is matchedx y
• If a 1 from is not matched erase tape, write 1 else erase tape, write 0
x)( yx
)( yx
134
Combining Turing Machines
135
Block Diagram
TuringMachineinput output
136
Example:
),( yxf0
yx yx
yx
if
if
Comparer
Adder
Eraser
yx,
yx,
yx
yx
yx
0
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