Pushdown Automaton: CFL and the pumping lemma

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Proof - value of p • Let |V| be the number of variables in G. • We set p = b |V|+2 . (h>log b |p|). • Then for any string of length p the parse tree requires height at least |V|+2 (Note, b>1 since there are no unit rules). • Given a string wL, s.t. |w| p , since G has only |V| variables, at least one of the variables repeats (height |V|+2 |V|+1 variables + terminal). • W.l.o.g. assume this variable is R 40
Proof – condition 1 • To prove condition 1 (|vy|>0) we have to show it is impossible that both v and y are . • We use a grammar without unit rules. • But the only way to have v=y=, is to have a rule R R, which is a unit rule. Contradiction. • So condition 1 is satisfied. S R R u v x y z 41
Page 1 and 2: Unit 9 More Pushdown Automata Conte
Page 3 and 4: Example: L = {a i b i c j d j |i,j0
Page 5 and 6: CFG=PDA Theorem: A language is cont
Page 7 and 8: Different derivations for the same
Page 9 and 10: control E E E E E E input: stac
Page 11 and 12: E E E E E E control 5 E E 5
Page 13 and 14: From CFG to PDA Informally: 1. Plac
Page 15 and 16: From CFG to PDA • We define as f
Page 17 and 18: From PDA to CFG context-free gramma
Page 19 and 20: From PDA to CFG • $ is always pop
Page 21 and 22: From PDA to CFG • For any word w
Page 23 and 24: From PDA to CFG Option 1: The stack
Page 25 and 26: From PDA to CFG Let P=(Q, , , , q 0
Page 27 and 28: Example: 0,A ,$ #, q s q 0 1,A q 1
Page 29 and 30: CFG=PDA • We have shown that a la
Page 31 and 32: Non context-free Languages • Cons
Page 33 and 34: The Pumping Lemma - background •
Page 35 and 36: The Pumping Lemma - background •
Page 37 and 38: The Pumping Length • That means t
Page 39: Proof - value of p • First we fin
Page 43 and 44: Proof - condition 3 S S R R R x u v
Page 45 and 46: Usage of the lemma - Example • We
Page 47 and 48: L n n n p p p { a b c | n 0}. Let
Page 49 and 50: L={a i b i c i |i0} Case 2: v or y
Page 51 and 52: Intersection 1 CFL are not closed u
Page 53: Complement Theorem: CFL are not clo

Pushdown Automaton: CFL and the pumping lemma

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