Notes-SecE - COT 4210 Section E Spring 2001 Pumping Lemma...

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COT 4210 Section E Spring 2001 Pumping Lemma for Regular Languages We have completed our study of closure properties of Regular Languages and various formal systems for defining or describing them. The question we wish to consider next is, how can non-Regular languages be identified? The first such tool is the pumping lemma for Regular languages . It defines a key property possessed by all Regular languages, specifically all infinite Regular languages (for it is only the infinite languages that can be non-Regular). As a logical assertion, it states that, “L is regular ] L has property P”. Negating this implication we have, “L does not have property P L is not regular.” It is the contra-positive form that is useful in proving languages to be non-Regular. That is, typical proofs using the Pumping Lemma assert that a given language, L, is Regular, and then proceed to obtain a contradiction by showing L does not have property, P. We’ll explain what property “P” is after the statement of our next theorem. Theorem 9 (Pumping Lemma for Regular Languages(PLR)). Let L be a Regular language over . There exits a positive integer α, depending only on L, such that for every x L with |x| α , then there exists a decomposition of x in the form uvw , where |uv| α and |v| 1, such that for every value of k 0, uv k w T L. The “property P” of Regular languages described in the PLR is: there is a fixed number α associated with each regular language, L, such that all strings, x, belonging to L and having length at least α , define an infinite subset of L, denoted L x , defined by L x = { uv k w | k 0, where x = uvw for some strings u,v,w satisfying |uv| α and |v| 1 } The figure below illustrates this concept. Feb. 22, 2001 Page 48
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COT 4210 Section E Spring 2001 NOTE: You may be wondering about finite languages. Since the PLR applies to all Regular languages, how does the result hold for finite Regular languages? The answer is that the value of α for finite languages, L, is always greater than the length of the longest string in L; all members of a finite language are therefore considered to be “short” strings. Before proving the PLR we apply it to an example. We had stated earlier that certain languages were known to be non-Regular without giving any proof. One of these is L = {a n b n | n 0 }. Using the PLR we now supply the proof. Corollary PLR-1 : L = { a n b n | n 0 } is non-Regular. Proof. The proof will proceed by contradiction. Suppose L is Regular, then the conclusion of the PLR must hold for L. In particular, consider x = a α b α h L (a long member of L), where α is the parameter defined for L by the PLR. To obtain a contradiction we must show that for each decomposition of the selected long string x in the form uvw, there exists a k [ 0, such that uv k w T L. It is really easier to think of applying the contra-positive of the PLR,
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Notes-SecE - COT 4210 Section E Spring 2001 Pumping Lemma...

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