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Pushdown Automata – Simplifying the Grammar

Pushdown Automata – Simplifying the Grammar. Lecture 20 Section 2.2 Fri, Oct 5, 2007. The Grammar Rules. We now have the grammar rules A pp  A pp A pp A pq  A pp A pq A pq  A pq A qq A pr  A pp A pr A pr  A pq A qr A pr  A pr A rr A qq  A qq A qq. The Grammar Rules.

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Pushdown Automata – Simplifying the Grammar

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  1. Pushdown Automata – Simplifying the Grammar Lecture 20 Section 2.2 Fri, Oct 5, 2007

  2. The Grammar Rules • We now have the grammar rules • AppAppApp • ApqAppApq • ApqApqAqq • AprAppApr • AprApqAqr • AprAprArr • AqqAqqAqq

  3. The Grammar Rules • AqrAqqAqr • AqrAqrArr • ArrArrArr • AprAqq • AqqaAqqb • App • Aqq • Arr

  4. The Grammar • The start symbol is Apr.

  5. Simplifying the Grammar • We may simplify these grammar rules considerably. • Eliminate any rule involving a variable that never appears on the left. • Eliminate any rule involving a variable, other than the start symbol, that never appears on the right.

  6. Simplifying the Grammar • Eliminate any rule involving a variable that is purely recursive. • Eliminate any unnecessary unit rules. • Use common sense to eliminate any other rules that are useless.

  7. Simplifying the Grammar • The grammar rules are reduced to • AppAppApp |  • AprAppApr | AprArr | Aqq • AqqAqqAqq | aAqqb |  • ArrArrArr | 

  8. Simplifying the Grammar • Then we notice that the rules • AppAppApp |  • ArrArrArr |  are useless since the only other choice for App and Arr is . • We should eliminate them and replace App and Arr with  everywhere they occur.

  9. Simplifying the Grammar • Now the grammar rules are reduced to • AprApr | Apr | Aqq • AqqAqqAqq | aAqqb | 

  10. Simplifying the Grammar • Now it is obvious that the rule • AprApr is pointless, so eliminate it. • We now have • AprAqq • AqqAqqAqq | aAqqb | 

  11. Simplifying the Grammar • Now it is obvious that the unit rule • AprAqq serves no purpose. • Eliminate it, making Aqq the new start symbol.

  12. Simplifying the Grammar • We now have • AqqAqqAqq | aAqqb |  or, more simply, • SSS | aSb | 

  13. a,; a b, a;  , ;  , ; $ , $;  p q r s Exercise • Find a context-free grammar for the language of the PDA

  14. Exercise • Find a context-free grammar for the language of the PDA (,; ( ),(;  a, ;  , ; $ , $;  p q r s +, ; 

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