the Prove following languages are contextfree by designing contextfree grammars which generate them: L1 = {an bm : n m, n, m 0}
1 n m n m
Hint: L = {a b : n < m} {a b : n > m} L2 = { c u c v c : |u|=|v|, u, v {a, b}* }
Announcements: Assignment #4 is posted: Due Wed. July 9 Tutorial today: contextfree grammars, discussion of questions from the midterm (in particular, 1(a), 1(c) and 3(b) students tended to provide solutions to these more complex than necessary). No tutorial or class Tuesday July 1 (Reading break/Canada Day). Academic drop deadline for courses starting week of May 5: June 26.
Contextfree grammars and regular languages. More examples of contextfree languages. All regular languages are contextfree and a subclass of contextfree languages (those with regular contextfree grammars) are regular.
Which rules below are not in the correct form to correspond to a regular contextfree grammar? 1. S A B 2. S B b 3. S a A 4. S aaaA 5. B b 6. B [0 or more terminals] then [at most one nonterminal] 1. S A S B 2. Aa 3. A 4. A aaa bb
Theorem: If L is regular, then L is contextfree. Proof: A contextfree grammar can be constructed from a DFA for L. Definition: A regular contextfree grammar is a contextfree grammar where each has rule its righthand side equal to an element of * ({} ( V ) )
[0 or more terminals] then [at most one nonterminal]
Our proof constructs a regular contextfree grammar.
Theorem: If L has a regular contextfree grammar, then L is a regular language. Proof: A NDFA can be constructed that accepts the language that the regular contextfree grammar generates. Note: this is in the text as Exercise 3.1.10.
Given the regular contextfree grammar G=(V, , R, S) construct a NDFA M = (K, , , s, F) where K= (V ) { f }, s= S, F = { f } For each rule T u R with u *, RV, add a transition (T, u, R) to . For each rule Tu with u in *, add a transition (T, u, f) to .
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