Closure properties of regular languages

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£' ¢© ¤£#¨£¨¨ £¢!"!"% " ¥%" %$ &¤#¨!"!§¡¦¥¥¤¥£(¡¡¡RegularityBeanda DFA withwithtraversingin£:states,in £in £ in £removablerepeatable“Pumping”02140 Languages and Parsing, MF, Fall 2004 – p.3/8Non-regularityAssume,¢-state DFA:traversingin %:INin $were regular, being accepted by anzeroesafter removalinrepeatableremovable onesin Shows that any DFA recognizing all strings infurther strings outside ¥.INrecognizescan’t be recognized by finite automata!02140 Languages and Parsing, MF, Fall 2004 – p.4/8
++*¤§)¢!*¦)£+*)¤£#¥¢©£©¤¨£¨£+¨#§¨!*¦)¤¥¤©£¥£¨£¤--¤¨+¢!* ¨,* ¨,*¤)*)*)¨¨£The pumping lemma for regular languagesThm: For each regular languagethat each ¥with1.,2. ¢,3.for eachIN.there is a constantcan be split intoIN suchsuch thatN.B.: Doesn’t imply that all regular languages are pumpable!All infinite ones are, however.Main use of pumping lemma is to prove non-regularity of some¥!02140 Languages and Parsing, MF, Fall 2004 – p.5/8Non-regularity — proof recipeIn order to showto be non-regular, proceed as follows:1. Take IN arbitrarily (i.e., no constraints to be imposed fromyour side);2. Provide construction rules — obviously dependent onthe a word ¥with ¢.3. Take an arbitrary split of(a), and(b) ¢.intosatisfyingI.e., (a) and (b) are the only properties enforced.4. Provide a construction rule forIN such that,¤¢— for5. Prove that your constructions satisfy all their claimed properties( ¥, ¢, ¥), unless obvious.,¤02140 Languages and Parsing, MF, Fall 2004 – p.6/8¥.
Page 1: The Pumping Lemma& Closure Properti

Closure properties of regular languages

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