lec0417 - Recursive Definitions for Languages Not only can...

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Recursive Definitions for Languages Not only can we use regular expressions to define a language, but we can also define languages recursively. Here is an example of such a definition: 1) λ L 2) if w L, then awb L 3) A string w L only if it can be obtained from the basis(1) by finite number of applications of the recursive step. From this definition, it’s fairly easy to see that L = {a n b n | n N}. However, just because we can “see it” doesn’t constitute a proof. We will use induction to prove this result for L, and then look at a few other similar examples. First, we will show that all strings of the form a n b n L. We will use induction on n. Base case n=0 : a 0 b 0 = λ∈ L. Inductive Hypothesis : Assume for an arbitrary n=k that a k b k L. Inductive Step : Under this assumption, we must prove for n=k+1 that a k+1 b k+1 L. a k+1 b k+1 = a(a k b k )b. (Let w = a k b k .) = awb. By our inductive hypothesis, w L. Furthermore, step 2 tells us that if w L, then awb w L as well. Thus, we have deduced that a k+1 b k+1 n L, as desired.
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It remains to be shown that all strings in L fit this form. Assume to the contrary that there exists a string in L NOT of the form a n b n . We know that all strings in L (except for the empty string) start with a and end with b. (Why?) Let w be the smallest string in L that is NOT of the form a
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lec0417 - Recursive Definitions for Languages Not only can...

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