Unrestricted State Complexity Of Binary Operations On Regular And Ideal Languages
Abstract
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 twosided 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); twosided 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, rightideal, leftideal, and twosidedideal languages to include the unrestricted cases.
Collections
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
Other formats
Related items
Showing items related by title, author, creator and subject.

Complexity of Proper PrefixConvex Regular Languages
Brzozowski, Janusz; Sinnamon, Corwin (Springer, 20170627)A language L over an alphabet Σ is prefixconvex if, for any words x,y,z∈Σ∗, whenever x and xyz are in L, then so is xy. Prefixconvex languages include rightideal, prefixclosed, and prefixfree languages, which were ... 
Complexity of RightIdeal, PrefixClosed, and PrefixFree Regular Languages
Brzozowski, Janusz; Sinnamon, Corwin (Institute of Informatics: University of Szeged, 2017)A language L over an alphabet E is prefixconvex if, for any words x, y, z is an element of Sigma*, whenever x and xyz are in L, then so is xy. Prefixconvex languages include rightideal, prefixclosed, and prefixfree ... 
Complexity of LeftIdeal, SuffixClosed and SuffixFree Regular Languages
Brzozowski, Janusz; Sinnamon, Corwin (Springer, 20170306)A language L over an alphabet Σ is suffixconvex if, for any words x,y,z∈Σ∗, whenever z and xyz are in L, then so is yz. Suffixconvex languages include three special cases: leftideal, suffixclosed, and suffixfree ...