Unrestricted State Complexity Of Binary Operations On Regular And Ideal Languages
MetadataShow full item record
We study the state complexity of binary operations on regular languages over diﬀerent alphabets. It is known that if L′m and Ln are languages of state complexities m and n, respectively, and restricted to the same alphabet, the state complexity of any binary boolean operation on L′m and Ln is mn, and that of product (concatenation) is m2n − 2n−1. In contrast to this, we show that if L′m and Ln are over diﬀerent alphabets, the state complexity of union and symmetric diﬀerence is (m + 1)(n + 1), that of diﬀerence is mn + m, that of intersection is mn, and that of product is m2n + 2n−1. We also study unrestricted complexity of binary operations in the classes of regular right, left, and two-sided ideals, and derive tight upper bounds. The bounds for product of the unrestricted cases (with the bounds for the restricted cases in parentheses) are as follows: right ideals m + 2n−2 + 2n−1 + 1 (m + 2n−2); left ideals mn + m + n (m + n − 1); two-sided ideals m+2n (m+n−1). The state complexities of boolean operations on all three types of ideals are the same as those of arbitrary regular languages, whereas that is not the case if the alphabets of the arguments are the same. Finally, we update the known results about most complex regular, right-ideal, left-ideal, and two-sided-ideal languages to include the unrestricted cases.
Cite this version of the work
Janusz Brzozowski, Corwin Sinnamon (2017). Unrestricted State Complexity Of Binary Operations On Regular And Ideal Languages. UWSpace. http://hdl.handle.net/10012/12498
Showing items related by title, author, creator and subject.
Saleh-Anaraki, Payam (University of Waterloo, 2011-01-20)Application and implication of using complex vectors and complex transformations in solutions of Maxwell’s equations is investigated. Complex vectors are used in complex plane waves and help to represent this type of waves ...
Six Empty Shells: Contextualizing the Aspirations of Mexican Modernity Through the Tlatelolco Housing Complex. Abad, Daniel (University of Waterloo, 2019-12-12)During the 1950’s, the Mexican federal government pursued an ambitious plan to modernize the country. With the growing apparatus of government and facing a demographic explosion, the capital, Mexico City, faced a critical ...
Ofili, Patricia (University of Waterloo, 2019-05-21)This dissertation revolves around the complex political circumstances in apartheid South Africa that produced Nelson Mandela the rhetorician, human rights activist, and the longest political prisoner in human history. The ...