Turing Machines (TMs)
Linear Bounded Automata
(LBAs)
Fall 2003
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Turing Machine (TM)
Input string
a b c d e
Finite State
Control Unit
Infinite Tape
a b c d e
Working space
in tape
Fall 2003
Costas Busch - RPI
2
Linear Bounded Automaton (LBA)
Input string
[ a b c d e ]
Left-end
marker
Working space
in tape
Finite State
Control Unit
Right-end
marker
All computation is done between end markers
Fall 2003
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We define LBA’s as NonDeterministic
Open Problem:
NonDeterministic LBA’s
have same power with
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 NPDA’s
LBA’s have also less power
than Turing Machines
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Linear Bounded Automata (LBAs)
are the same as Turing Machines
with one difference:
The input string tape space
is the only tape space allowed to use
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The Chomsky Hierarchy
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Unrestricted Grammars:
Productions
u v
String of variables
and terminals
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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 recursively enumerable
(r.e.) if and only if L is generated by an
unrestricted grammar
S is r.e. if there
exists an algorithm A
that enumerates the
members of S (A need
not necessarily
terminate for nonmembers of S)
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S is recursive if there
exists a decision
algorithm that
determines if x is a
member of S
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Context-Sensitive Grammars:
Productions
u v
String of variables
and terminals
and:
Fall 2003
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
L is context sensitive
if and only if
is accepted by a Linear-Bounded Automaton
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The Chomsky Hierarchy
Non-recursively enumerable
Recursively-enumerable
Recursive
Context-sensitive
Context-free
Regular
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