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Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death

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Parse Tree Describes derivation of a string from a CFG Examples: S→aSaa|B B→BB|b|ε S S S B BB aa aa b b S a for each a L S (S) | SS | S* | S+S S S S SS (a+b)*

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Pumping Lemma for Context Free Languages If L is a CFL then there is some number p such that for any string s in L where |s|>p we can divide s into 5 substrings s=uvxyz in a way to satisfy all of the following: 1.For each i≥0, 2.|vy|>0 3.|vxy|≤p

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Pumping Lemma for Context Free Languages 1.For each i≥0, 2.|vy|>0 3.|vxy|≤p Proof: Consider the CFG for L. The grammar has a finite number of production rules. If a string is long enough, some variable must have been reached twice. S R R uvxyz S R u x z S uz R vy R vxy R

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Pumping Lemma for Context Free Languages 1.For each i≥0, 2.|vy|>0 3.|vxy|≤p Example: Consider: If any of a,b,c are not part of vy then we definitely fail rule 1. Because of rule 3 if v contains the letter ‘a’ then y cannot contain the letter ‘b’

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Pumping Lemma for Context Free Languages 1.For each i≥0, 2.|vy|>0 3.|vxy|≤p Example:. Consider: If neither a,c or b,d are not part of vy then we definitely fail rule 1. Because of rule 3 if v contains the letter ‘a’ then y cannot contain the letter ‘c’ If v contains the letter ‘b’ then y cannot contain the letter ‘d’

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Pumping Lemma for Context Free Languages 1.For each i≥0, 2.|vy|>0 3.|vxy|≤p Example: Consider: If I modify the first set of ‘a’s then I must also modify the second set. If v contains one of the first set of ‘a’s then y cannot contain one of the second set due to rule 3. The case for the ‘b’s is similar.

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Pumping Lemma for Context Free Languages 1.For each i≥0, 2.|vy|>0 3.|vxy|≤p Example: Prove its not context free for extra credit! Triple extra credit if you show the following first: If |∑|=1 then for any language, if L is context free then L is also regular

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Pumping Lemma for Context Free Languages Does EqLenCat Preserve Regularity? No, Let A=a*, B=b*, Does EqLenCat Preserve Context Freeness? No, Let,. Then If A,B are regular, is EqLenCat Context Free? Yes, by construction of PDA. Idea: Build a PDA which simulate the FSA for A, where it pushes a character onto the stack for each input. It then non- deterministically switches to the FSA for B, where it pops from the stack for each input. Accept if in accept state of B with empty stack

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Pumping Lemma for Context Free Languages If A,B are regular, is EqLenCat Context Free? Modified Concatenation Construction: a a a stack AB ε ε Push ‘a’ on each transition Pop ‘a’ on each transition

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Pumping Lemma for Context Free Languages Does YesNo Preserve Regularity? Yes, Does YesNo Preserve Context Freeness? No. Idea: show that for an arbitrary context free language L, YesNo(L) is context free iff its complement is context free

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Pumping Lemma for Context Free Languages Does YesNo Preserve Context Freeness? Consider a CFL language L whose complement is not context free. We will assume that L contains ε. We will also assume that ‘$’ is not a character in Σ. Let Since So if YesNo preserves context-freeness then L complement must be context free

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Here Be an extra slide

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Pumping Lemma for Context Free Languages 1.For each i≥0, 2.|vy|>0 3.|vxy|≤p Proof: Consider the CFG for L. The grammar has a finite number of production rules. If a string is long enough, some rule must have been applied twice. S R R uvxyz S R u x z S R R uvxyz R vy

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