# Lec 09 Pumping Lemma.ppt - Theory of Automata Pumping Lemma...

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Theory of Automata2ContentsDefinition of non-Regular LanguagesTheorem 13Proof & ExampleTheorem 14Proof & Example
Theory of Automata3IntroductionBy using FAs and regular expressions, we have been able to define manylanguages.Although these languages have many different structures, theytake only a few basic forms:Languages with required substringsLanguages that forbid some substringsLanguages that begin or end with certain substringsLanguages with certain even (or odd) properties, and so on.We now turn our attention to some new forms, such as the languagePALINDROME, or the language PRIME of all words ap, where p is a primenumber.We shall see that neither of these is a regular language. We can describe themin English, but they can not be defined by an FA. We need to build morepowerful machines to define them.
Theory of Automata4Definition of Non-Regular LanguagesDefinition:A language that cannot be defined by aregular expression is called a non-regularlanguage.Notes:By Kleene’s theorem, a non-regular language can alsonotbe accepted by any FA or TG.All languages are either regular or non-regular; noneare both.
Theory of Automata5Case StudyConsider the languageL = {Λ; ab; aabb; aaabbb; aaaabbbb; aaaaabbbbb; …}The language L can also be written asL = {anbnfor n = 0; 1; 2; 3; …}or for shortL = {anbn}Note that although L is a subset of many regularlanguages, such as a*b*; the language defined by ab alsoincludes such strings as aab and bb that are not in L.
Theory of Automata6ExampleWe shall show that the language L = {anbn} is non-regular.Suppose on the contrary that L were regular. Then there must exist some FA thataccepts L.Just for the sake of argument, let us assume that this FA has 95 states.We know that this FA must accept the word a96b96.The first 96 letter a’s of this input string trace a path through this machine.The path cannot visit a new state when each input letter is read, because thereare only 95 states. Therefore, at some point the path returns to a state that it hasalready visited.In other words, the path must contain acircuitin it. (A circuit is a loop that can bemade of several edges).
Theory of Automata7So, the path first wanders up to the circuit and then starts to looparound the circuit (maybe many times) until a b is read from the input.

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