Chapter2Section3_F11

Chapter2Section3_F11 - 91.304 Foundations of (Theoretical)...

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91.304 Foundations of (Theoretical) Computer Science Chapter 2 Lecture Notes (Section 2.3: Non-Context-Free Languages) David Martin m@cs uml edu dm@cs.uml.edu With some modifications by Prof. Karen Daniels, Fall 2011 This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by- 1 sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
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Picture so far ALL B = { 0 n 1 n | n 0 } CFL EG RPP * 01)* FIN REG { 0101, ε } 0 (101) Each point is a language in this Venn diagram Does this exist? 2
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Strategy for finding a non-CFL ± Just as we produced non-regular languages with the assistance of RPP, e'll produce non ontext ee we ll produce non-context-free languages with the assistance of the ontext- ee pumping property context free pumping property ² First we show that CFL CFPP nd then show that a particular language ² And then show that a particular language L is not in CFPP ² Hence L can not be in CFL either 3
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The Context-Free Pumping Property, CFPP Definition L is a member of CFPP if ± There exists p 0 such that ² For every s L satisfying |s| p, ± There exist u,v , x,y,z Σ * such that = v yz 1. s= uv xyz 2. | v y|>0 3. | v xy| p bold, red text shows differences from RPP 4. For all i 0, u v i x y i z L 4
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The non -CFPP Rephrasing L is not in CFPP if ± For every p 0 ² There exists some s L satisfying |s| p such that or every v yz * atisfying 1- : ± For every u,v, x,y,z Σ satisfying 1 3: 1. s= uv xyz, 2. | v y|>0, and 3. | v xy| p ± There exists some i 0 for which v i i 5 u v x y z L
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Game theory formulation he direct (non ontradiction) proof ± The direct (non-contradiction) proof of non-context-freeness can be rmulated as a two- layer game formulated as a two player game ² You are the player who wants to establish that L is not CF-pumpable ² Your opponent wants to make it difficult for you to succeed oth of you have to play by the rules ² Both of you have to play by the rules ² Same setup as with regular pumping (RPP) 6
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Game theory continued The game has just four steps. 1. Your opponent picks p 0 2. You pick s L such that |s| p 3. Your opponent chooses u,v,x,y,z Σ * such that s=uvxyz, |vy|>0, and |vxy| p 4. You produce some i 0 such that uv i xy i zL 7
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Game theory continued you are able to succeed through step 4 ± If you are able to succeed through step 4,
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This note was uploaded on 02/13/2012 for the course CS 91.304 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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Chapter2Section3_F11 - 91.304 Foundations of (Theoretical)...

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