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Fall 2006 2Costas Busch - RPI
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Fall 2006Costas Busch - RPI3
Same as Turing Machines with one difference:
the input string tape spaceis the only tape space allowed to use
Linear-Bounded Automata:
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Fall 2006Costas Busch - RPI4
[ ]a b c d e
Left-endmarker
Input string
Right-endmarker
Working space in tape
All computation is done between end markers
Linear Bounded Automaton (LBA)
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Fall 2006Costas Busch - RPI5
We define LBA’s as NonDeterministic
Open Problem:
NonDeterministic LBA’shave same power asDeterministic LBA’s ?
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Fall 2006Costas Busch - RPI6
Example languages accepted by LBAs:
}{ nnn cbaL }{ !naL
LBA’s have more power than PDA’s (pushdown automata)
LBA’s have less power than Turing Machines
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Fall 2006Costas Busch - RPI7
Unrestricted Grammars:
Productionsvu
String of variablesand terminals
String of variablesand terminals
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Fall 2006Costas Busch - RPI8
Example unrestricted grammar:
dAc
cAaB
aBcS
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Fall 2006Costas Busch - RPI9
A language is Turing-Acceptableif and only if is generated by anunrestricted grammar
LL
Theorem:
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Fall 2006Costas Busch - RPI10
Context-Sensitive Grammars:
and: |||| vu
Productionsvu
String of variablesand terminals
String of variablesand terminals
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Fall 2006Costas Busch - RPI11
The language }{ nnn cba
is context-sensitive:
aaAaaaB
BbbB
BbccAc
bAAb
aAbcabcS
|
|
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Fall 2006Costas Busch - RPI12
A language is context sensistive if and only ifit is accepted by a Linear-Bounded automaton
LTheorem:
There is a language which is context-sensitivebut not decidable
Observation:
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Fall 2006Costas Busch - RPI13
Non Turing-Acceptable
Turing-Acceptable
decidable
Context-sensitive
Context-free
Regular
The Chomsky Hierarchy
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