A Hierarchy of Formal Languages and Automata

1. A Hierarchy of Formal Languages and Automata • How large is the family of languages accepted by Turing machines?. • Is there any language that cannot be accepted by Turing machines?.

2. Recursively Enumerable Languages A language L is said to be recursively enumerable if there exists a Turing machine that accepts it: q0w |*Mx1qfx2qf F for everyw  L.

3. Recursive Languages A language L on  is said to be recursive if there exists a Turing machine that accepts L and that halts on every w  +.

4. Theorem There exist languages that are not recursively enumerable.

5. Theorem There exist recursively enumerable languages that are not recursive.

6. Unrestricted Grammars G = (V, T, S, P) Productions are of the form: u  v u  (V  T)+ and v  (V  T)*

7. Theorem Any language generated by an unrestricted grammar is recursively enumerable.

8. Theorem For every recursively enumerable language, there exists an unrestrictedgrammar that generates it.

9. Context-Sensitive Grammars G = (V, T, S, P) Productions are of the form: x  y x, y  (V  T)+ and |x|  |y|

10. Example • L = {anbncn | n  1} is context-sensitive. G = ({S, A, B}, {a, b}, S, P) P = { S  abc | aAbc Ab  bA Ac  Bbcc bB  Bb aB  aa | aaA }

11. Linear Bounded Automata End markers [ ] Read-write head Control unit q0

12. Theorem For every context-sensitive L not containing , there exists a LBA M such that L = L(M).

13. Theorem If a language is accepted by some LBA M, then there exists a context-sensitive grammar G such that L(G) = L(M).

14. Context-Sensitive and Recursive Languages • Every context-sensitive language is recursive. • There exists recursive languages that are not context-sensitive.

15. The Chomsky Hierarchy LRE type 0 (TM) type 1 (LBA) type 2 (PDA) type 3 (FA) LCS LCF LREG