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Closure properties of RE, Rec

[Section 10.1]. Closure properties of RE, Rec. Recall that a language L is - recursively enumerable (RE) if there exists a TM for L, recursive (Rec) if there exists a TM for L which halts on every input (i.e. also on strings not from L).

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Closure properties of RE, Rec

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  1. [Section 10.1] Closure properties of RE, Rec • Recall that a language L is • - recursively enumerable (RE) if there exists a TM for L, • recursive (Rec) if there exists a TM for L which halts on every input (i.e. also on strings not from L). • Is the class RE closed under union ? And intersection ? • Is the class Rec closed under union ? And intersection ? • What can we say about complement ?

  2. [Section 10.1] Closure properties of RE, Rec Lemma : RE languages are closed under union. Lemma : RE languages are closed under intersection.

  3. [Section 10.1] Closure properties of RE, Rec Lemma : Recursive languages are closed under union. Lemma : Recursive languages are closed under intersection.

  4. [Section 10.1] Closure properties of RE, Rec Lemma : Recursive languages are closed under complement. Lemma : RE languages are not closed under complement.

  5. [Section 10.1] Closure properties of RE, Rec Thm : L and L’ are RE iff L is recursive.

  6. [Section 10.4] The Chomsky Hierarchy • Noam Chomsky studied grammars as potential models for natural languages. He classified grammars according to these four types: • Type 0 Grammars: Unrestricted Grammars (generate RE languages) • Type 1 Grammars: Context-sensitive (monotone) Grammars (generate context-sensitive languages) • Type 2 Grammars: Context-free Grammars (generate context-free languages) • Type 3 Grammars: Regular Grammars (generate regular languages)

  7. [Section 10.3] Unrestricted Grammars (Type 0) • Def: An unrestricted grammar is a 4-tuple G=(V,,S,P) where • V is a finite set of variables •  is a finite set of terminal symbols • S 2 is the start symbol • P is a finite set of productions of the form  where 2 (V [)+ and 2 (V [)* • (V and  are assumed to be disjoint)

  8. [Section 10.3] Unrestricted Grammars (Type 0) Example: Give an unrestricted grammar for { akbkck | k ¸ 0 }

  9. [Section 10.3] Unrestricted Grammars (Type 0) Example: Give an unrestricted grammar for { aj | j = 2k, k ¸ 0 }

  10. [Section 10.3] Context-sensitive Gram. (Type 1) Def: A type 0 grammar G=(V,,S,P) is context-sensitive if for every production rule  in P, ||·||. Which of our examples of type 0 grammars are context-sensitive ?

  11. [Section 10.3] Context-sensitive Gram. (Type 1) Def: A type 0 grammar G=(V,,S,P) is context-sensitive if for every production rule  in P, ||·||. Lemma: Every context-free language which does not contain  is context-sensitive.

  12. [Section 10.3] Context-sensitive Gram. (Type 1) Def: A type 0 grammar G=(V,,S,P) is context-sensitive if for every production rule  in P, ||·||. Lemma: Every context-free language which does not contain  is context-sensitive. Def: A linear-bounded automaton A is a TM which never rewrites a blank to a non-blank symbol. Lemma: A language L is context-sensitive iff there exists a linear-bounded automaton accepting L.

  13. [Section 6.3] Regular Grammars (Type 3) Def: A type 0 grammar G=(V,,S,P) is regular if every production rule in P is of the form A B or A , where A,B 2 V and 2. Lemma: A language L is regular iff there exists a regular grammar for L-{}.

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