l7-handout

# l7-handout - Lecture 7 Closure properties Outline Closure...

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Lecture 7: Closure properties David Dill Department of Computer Science 1 Outline Closure properties of regular languages 2 Closure Properties A ”closure property” is a theorem that shows that certain operations preserve some property. People often say that a set is closed under certain operations. For example, the integers are closed under addition, subtraction, and multiplication, but not division. So a closure property is that addition of integers gives an integer. Closure properties are generally of interest to mathematics. They provide powerful ways to prove that certain sets have a property, and, sometimes more importantly, that the sets do not have the property. That’s why we’re interested in closure properties of regular languages. We can use all the different representations we have (reg exprs, DFAs, NFAs e-NFAs) to prove closure properties. 3 Regular Operators We can use all the different representations we have (reg exprs, DFAs, NFEs e-NFAs) to prove closure properties. Theorem Regular sets are closed under the operations: union, concatenation, Kleene closure. Proof: Suppose L 1 and L 2 are regular languages. Then there exist regular expressions R 1 and R 2 such that L ( R 1 ) = L 1 and L ( R 2 ) = L 2 . But then L ( R 1 + R 2 ) = L 1 L 2 , L ( R 1 R 2 ) = L 1 L 2 , and L ( R * 1 ) = L * 1 . So each of these languages is regular as well. 4

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Boolean Operations Def. The complement of a language (relative to an alphabet Σ ) L = Σ * - L . The Boolean operations are union, intersection, and complement. Theorem: The regular languages are closed under Boolean operations. Proof: We’ve already got union. Complement: Build the DFA D . Define D C as the same DFA, but complement the set of final states ( F C = Q - F ). It’s easy to see that this DFA accepts the complement of the language.
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