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Fall 2003 Costas Busch - RPI 1
Turing Machines (TMs)
Linear Bounded Automata(LBAs)
Fall 2003 Costas Busch - RPI 2
a b c d e
Input string
Working space in tape
Turing Machine (TM)
a b c d e
Infinite Tape
Finite State Control Unit
Fall 2003 Costas Busch - RPI 3
[ ]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)
Finite State Control Unit
Fall 2003 Costas Busch - RPI 4
We define LBA’s as NonDeterministic
Open Problem:
NonDeterministic LBA’shave same power withDeterministic LBA’s ?
Fall 2003 Costas Busch - RPI 5
Example languages accepted by LBAs:
}{ nnn cbaL
}{ !naL
LBA’s have more power than NPDA’s
LBA’s have also less power than Turing Machines
Fall 2003 Costas Busch - RPI 6
Linear Bounded Automata (LBAs)are the same as Turing Machineswith one difference:
The input string tape spaceis the only tape space allowed to use
Fall 2003 Costas Busch - RPI 7
The Chomsky Hierarchy
Fall 2003 Costas Busch - RPI 8
Unrestricted Grammars:
Productionsvu
String of variablesand terminals
String of variablesand terminals
Fall 2003 Costas Busch - RPI 9
Example unrestricted grammar:
dAc
cAaB
aBcS
Fall 2003 Costas Busch - RPI 10
A language is recursively enumerable(r.e.) if and only if is generated by anunrestricted grammar
LL
Theorem:
S is r.e. if there exists an algorithm A that enumerates the members of S (A need not necessarily terminate for non-members of S)
S is recursive if there exists a decision algorithm that determines if x is a member of S
Fall 2003 Costas Busch - RPI 11
Context-Sensitive Grammars:
and: |||| vu
Productionsvu
String of variablesand terminals
String of variablesand terminals
Fall 2003 Costas Busch - RPI 12
The language }{ nnn cba
is context-sensitive:
aaAaaaB
BbbB
BbccAc
bAAb
aAbcabcS
|
|
Fall 2003 Costas Busch - RPI 13
A language is context sensitive if and only if is accepted by a Linear-Bounded AutomatonL
LTheorem:
Fall 2003 Costas Busch - RPI 14
Non-recursively enumerable
Recursively-enumerable
Recursive
Context-sensitive
Context-free
Regular
The Chomsky Hierarchy
