# Pump - Harvard CS 121 and CSCI E-207 Lecture 8: Non-Regular...

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Unformatted text preview: Harvard CS 121 and CSCI E-207 Lecture 8: Non-Regular Languages Harry Lewis September 29, 2009 Reading: Sipser, 1.4. Harvard CS 121 &amp; CSCI E-207 September 29, 2009 Cardinality of Languages An alphabet is finite by definition Proposition: * is countably infinite So every language is either finite or countably infinite P ( * ) is uncountable, being the set of subsets of a countably infinite set. i.e. There are uncountably many languages over any alphabet Q: Even if | | = 1 ? 1 Harvard CS 121 &amp; CSCI E-207 September 29, 2009 Existence of Non-regular Languages Theorem: For every alphabet , there exists a non-regular language over . Proof: There are only countably many regular expressions over . There are only countably many regular languages over . There are uncountably many languages over . Thus at least one language must be non-regular. In fact, almost all languages must be non-regular. Q: Could we do this proof using DFAs instead? Q: Can we get our hands on an explicit non-regular language? 2 Harvard CS 121 &amp; CSCI E-207 September 29, 2009 Goal: Explicit Non-Regular Languages It appears that a language such as L = { x * : | x | = 2 n for some n } = { a, b, aa, ab, ba, bb, aaaa, . . . , bbbb, aaaaaaaa, . . . } cant be regular because the gaps in the set of possible lengths become arbitrarily large, and no DFA could keep track of them....
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Pump - Harvard CS 121 and CSCI E-207 Lecture 8: Non-Regular...

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