Lecture 22 Pumping Lemma for Context Free Languages CSCE 355 Foundations of Computation Lecture 22 Pumping Lemma for Context Free Languages Topics: Normal forms Pumping Lemma for CFLs Closure properties November 19, 2008

Last Time: New: 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 Hierarchy Pumping Lemma for Context Free Languages

Algorithm for generating and reachable symbols 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

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/

Conversion to Chomsky Normal Form Remove: ε-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  B1B2…Bk create a new non-terminals C1C2…Ck and replace the production with A  B1C1 and Ci  Bi+1Ci+1 for i=1,…k-3 Ck-2  Bk-1Bk

Example

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 δ(A, a) = B and A  a means δ(A, a) = Qfinal and

Example

Chomsky Hierarchy http://en.wikipedia.org/wiki/Chomsky_hierarchy Grammar Languages Automaton Production rules (constraints) Type-0 Recursively enumerable Turing machine α  β no restrictions Type-1 Context-sensitive Linear-bounded non-deterministic Turing machine αAβ  αγβ Type-2 Context-free Non-deterministic pushdown automaton A  α Type-3 Regular DFA A  a or A  aB

Chomsky Hierarchy Venn Diagram

Backus Naur Form (BNF) Backus Naur Form John Backus Peter Naur N ::= α | … | β (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

Greibach Normal Form Each production RHS starts with a terminal A  aα or S ε http://en.wikipedia.org/wiki/Greibach_normal_form

Showing Languages are not CFLs Recursive productions A  a A | b B  B a | b D  aDb | d A * α A β

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.

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

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

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 …

Example

Homework 7.1.4 7.1.3 7.1.6