1. Fall 2006Costas Busch - RPI1 The Chomsky Hierarchy

2. Fall 2006Costas Busch - RPI2 Same as Turing Machines with one difference: the input string tape space is the only tape space allowed to use Linear-Bounded Automata:

3. Fall 2006Costas Busch - RPI3 Left-end marker Input string Right-end marker Working space in tape All computation is done between end markers Linear Bounded Automaton (LBA)

4. Fall 2006Costas Busch - RPI4 We define LBA’s as NonDeterministic Open Problem: NonDeterministic LBA’s have same power as Deterministic LBA’s ?

5. Fall 2006Costas Busch - RPI5 Example languages accepted by LBAs: LBA’s have more power than PDA’s (pushdown automata) LBA’s have less power than Turing Machines

6. Fall 2006Costas Busch - RPI6 Unrestricted Grammars: Productions String of variables and terminals String of variables and terminals

7. Fall 2006Costas Busch - RPI7 Example unrestricted grammar:

8. Fall 2006Costas Busch - RPI8 A language is Turing-Acceptable if and only if is generated by an unrestricted grammar Theorem:

9. Fall 2006Costas Busch - RPI9 Context-Sensitive Grammars: and: Productions String of variables and terminals String of variables and terminals

10. Fall 2006Costas Busch - RPI10 The language is context-sensitive:

11. Fall 2006Costas Busch - RPI11 A language is context sensistive if and only if it is accepted by a Linear-Bounded automaton Theorem: There is a language which is context-sensitive but not decidable Observation:

12. Fall 2006Costas Busch - RPI12 Non Turing-Acceptable Turing-Acceptable decidable Context-sensitive Context-free Regular The Chomsky Hierarchy