Ausblick: Turing-Maschinen und linear beschränkte Automaten vs. allgemeine und Kontextsensitive Grammatiken

Turing Machines (TMs) Linear Bounded Automata

(LBAs)

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

[ a _b c _d e ]

Left-end marker

Input string

Right-end marker

Working space in tape

All computation is done between end markers Linear Bounded Automaton (LBA)

Finite State Control Unit

We define LBA’s as NonDeterministic

Open Problem:

NonDeterministic LBA’s have same power with Deterministic LBA’s ?

Example languages accepted by LBAs:

} { a

b

c

L 

} { a

L 

LBA’s have more power than NPDA’s

LBA’s have also less power than Turing Machines

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

The Chomsky Hierarchy

Unrestricted Grammars:

Productions

v u 

String of variables

and terminals String of variables and terminals

Example unrestricted grammar:

d Ac

cA aB

aBc S







A language is recursively enumerable (r.e.) if and only if is generated by an unrestricted grammar

L L

Theorem:

S is r.e. if there

exists an algorithm A that enumerates the members of S (A need not necessarily

S is recursive if there exists a decision

algorithm that

determines if x is a member of S

Context-Sensitive Grammars:

and:

| u |  | v |

Productions

v u 

String of variables

and terminals String of variables and terminals

The language

{ a

b

c

}

is context-sensitive:

aaA aa

aB

Bb bB

Bbcc Ac

bA Ab

aAbc abc

S

|

|











A language is context sensitive if and only if

is accepted by a Linear-Bounded Automaton

L L

Theorem:

Non-recursively enumerable Recursively-enumerable

Recursive

Context-sensitive Context-free

The Chomsky Hierarchy