Closure properties of regular languages

The Pumping Lemma& Closure PropertiesMartin FränzleInformatics and Mathematical ModellingThe Technical University of Denmark02140 Languages and Parsing, MF, Fall 2004 – p.1/8What you’ll learn1. Repeatable sub-strings:What’s “regular” about regular languages2. The pumping lemma:Shows that repeatable sub-strings are ubiquitious in infiniteregular languages3. How to prove languages being non-regular:An application of the pumping lemma4. Some closure properties of the class of regular languages02140 Languages and Parsing, MF, Fall 2004 – p.2/8

£' ¢© ¤£#¨£¨¨ £¢!"!"% " ¥%" %$ &¤#¨!"!§¡¦¥¥¤¥£(¡¡¡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¥.

36 45§ 4*¢¦)©2$§£% /-¦0/¨.$"¤"¦1$§ ++££§ £)£ ¨ )¨¨¨ *§ ¦*¨.¨ ¦¤$¢¢¤¤££5 6 43* § 4©) ¦¨+¨+¥¥¨¨ExamplesProve the following languages to be non-regular:For anyIN, take%, then pump.For anyIN, takein the splitis prime 7 " 87 ¨for a prime£. Then pump tois not prime.such that+.02140 Languages and Parsing, MF, Fall 2004 – p.7/8Closure properties of regular languagesThe regular languages are closed underunion — i.e., the union of two regular languages is regular,intersection — i.e., the intersection of two reg. lang.s is regular,complement — i.e., the complement of a reg. lang. is regular,difference — i.e., the difference of two reg. lang.s is regular,reversal — i.e., the language containing the reversals of thewords of a reg. lang. is regular,closure (Kleene star) — i.e., the closure of a reg. lang. is regular,concatenation — i.e., the catenation of two reg. lang.s is regular,homomorphism — i.e., the image of a reg. lang. under ahomomorphism is regular,inverse homomorphism — i.e., the pre-image of a reg. lang.under a homomorphism is regular.02140 Languages and Parsing, MF, Fall 2004 – p.8/8