Transcript Chomsky`s Hierarchy

The Chomsky Hierarchy
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Linear-Bounded Automata:
Same as Turing Machines with one difference:
the input string tape space
is the only tape space allowed to use
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Linear Bounded Automaton (LBA)
Input string
[ a b c d e ]
Left-end
marker
Working space
in tape
Right-end
marker
All computation is done between end markers
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We define LBA’s as NonDeterministic
Open Problem:
NonDeterministic LBA’s
have same power as
Deterministic LBA’s ?
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Example languages accepted by LBAs:
L  {a b c }
n n n
L  {a }
n!
LBA’s have more power than PDA’s
(pushdown automata)
LBA’s have less power than Turing Machines
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Unrestricted Grammars:
Productions
u v
String of variables
and terminals
String of variables
and terminals
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Example unrestricted grammar:
S  aBc
aB  cA
Ac  d
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Theorem:
A language L is Turing-Acceptable
if and only if L is generated by an
unrestricted grammar
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Context-Sensitive Grammars:
Productions
u v
String of variables
and terminals
and:
String of variables
and terminals
|u|  |v|
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The language
n n n
{a b c }
is context-sensitive:
S  abc | aAbc
Ab  bA
Ac  Bbcc
bB  Bb
aB  aa | aaA
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Theorem:
A language
L is context sensistive
if and only if
it is accepted by a Linear-Bounded automaton
Observation:
There is a language which is context-sensitive
but not decidable
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The Chomsky Hierarchy
Non Turing-Acceptable
Turing-Acceptable
decidable
Context-sensitive
Context-free
Regular
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