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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern