Unrestricted State Complexity Of Binary Operations On Regular And Ideal Languages
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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.
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Janusz Brzozowski, Corwin Sinnamon (2017). Unrestricted State Complexity Of Binary Operations On Regular And Ideal Languages. UWSpace. http://hdl.handle.net/10012/12498
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Brzozowski, Janusz (World Scientific Publishing, 2013-09-01)Sequences (L-n vertical bar n >= k), called streams, of regular languages L-n are considered, where k is some small positive integer, n is the state complexity of L-n, and the languages in a stream differ only in the ...
Brzozowski, Janusz; Sinnamon, Corwin (Springer, 2017-06-27)A language L over an alphabet Σ is prefix-convex if, for any words x,y,z∈Σ∗, whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages, which were ...
Brzozowski, Janusz; Sinnamon, Corwin (Springer, 2017-03-06)A language L over an alphabet Σ is suffix-convex if, for any words x,y,z∈Σ∗, whenever z and xyz are in L, then so is yz. Suffix-convex languages include three special cases: left-ideal, suffix-closed, and suffix-free ...