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Lecture 22 Pumping Lemma for Context Free Languages

Last Time: Useless symbols: generating symbols, useful symbolsAlgorithm for generating and reachable symbolsRemoval of useless symbolsRemoval of epsilon productions; Removal of unit productionsChomsky normal formNew: Chomsky normal formChomsky HierarchyPumping Lemma for Context Free Lang

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Lecture 22 Pumping Lemma for Context Free Languages

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    1. Lecture 22 Pumping Lemma for Context Free Languages Topics: Normal forms Pumping Lemma for CFLs Closure properties

    2. Last Time: Useless symbols: generating symbols, useful symbols Algorithm for generating and reachable symbols Removal of useless symbols Removal of epsilon productions; Removal of unit productions Chomsky normal form New: Chomsky normal form Chomsky Hierarchy Pumping Lemma for Context Free Languages

    3. Useless symbols: generating symbols, useful symbols Algorithm for generating and reachable symbols Removal of useless symbols Removal of epsilon productions; Removal of unit productions Chomsky normal form

    4. Chomsky Normal Form A CFG (Context Free Grammar) is in Chomsky Normal form if productions are one of the following two forms: A ? BC A ? a References http://www.chomsky.info/

    5. Conversion to Chomsky Normal Form Remove: e-productions, unit productions A ? BCDE A ? abc In general For each terminal a create a new non-terminal Na with Na ? a added as a production A ? B1B2Bk create a new non-terminals C1C2Ck and replace the production with A ? B1C1 and Ci ? Bi+1Ci+1 for i=1,k-3 Ck-2 ? Bk-1Bk

    6. Example

    9. Regular Grammars A CFG is regular if all productions are of the form: A ? a or A ? aB Note sentential forms in a derivation based on a regular grammar have a unique form! What is it ? Grammar ? NFA construction Create a state for each nonterminal. A ? aB means d(A, a) = B and A ? a means d(A, a) = Qfinal and

    10. Example

    11. Chomsky Hierarchy http://en.wikipedia.org/wiki/Chomsky_hierarchy

    12. Chomsky Hierarchy Venn Diagram

    13. Backus Naur Form (BNF) Backus Naur Form N ::= a | | (just a CFG) http://en.wikipedia.org/wiki/Backus-Naur_form John Backus Fortran compiler http://en.wikipedia.org/wiki/John_Backus Peter Naur http://en.wikipedia.org/wiki/Peter_Naur

    14. Greibach Normal Form Each production RHS starts with a terminal A ? aa or S? e http://en.wikipedia.org/wiki/Greibach_normal_form

    15. Showing Languages are not CFLs Recursive productions A ? a A | b B ? B a | b D ? aDb | d A ?* a A

    16. Pumping Lemma for CFLs Let L be a CFL. Then there exists a constant n such that if z is a string in L of length at least n, then we can write z = uvwxy such that |vwx| =< n |vx| > 0 uviwxi y is in L for all i >= 0.

    17. Idea behind proof Assume CNF (or do for L(G)-{e}) Consider Parse Tree Sufficiently long string z, means the parse tree must be sufficiently big.

    18. Similarities to Pumping Lemma for Regular Languages Given an arbitrary n. Carefully choose z in L (depending on n) with |z| >= n. Then for any partition z = uvwxy that satisfies |vx| > 0 |vwx| <= n We must be able to pump, i.e. uviwxiy is in L for all i >= 0

    19. Example L = {anbncn | n > 0} Given L as above, suppose we chose n for the Pumping Lemma (for CFLs). Choose z = Consider arbitrary partition of z = uvwxy satisfying | vwx| =< n |vx| > 0 Then show

    20. Example

    36. Homework 7.1.4 7.1.3 7.1.6

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