On Equations for Regular

We consider systems of equations of the form Xi=,IJ,a l &~LJ& i=l,...,n where A is the underlying alphabet, the Xi are variables, the Pi.0 are boolean functions in the variables J&B and each & is either the empty word or the empty set. The symbols and u denote concatenation and union of languages over A. We show that any such system has a unique solution which, moreover, is regular. These equations correspond to a type of automaton, called boolean automaton, which is a generalization of a nondeterministic automaton. The equations are then used to determine the language accepted by a sequential network; they are obtainable directly from the network. 1. Notation Let A be a finite alphabet and A* the free monoid generated by A. An element of A is called a letter and an element of A* is called a word over A. The unit element of the monoid A* is the empty word A. The length 1 w 1 of a word w over A is the number of letters in w ; note that IA I= 0. The concatenation (product in the monoid A*) of two words u and v is denoted by u l v. The reverse wp of a word w over A is defined recursively: hP = A, and (vu)’ = ad’ for a E A, v E A*. A subset of A* is called a language over A. The empty language is denoted by 0, and I is a shorthand for the language A*. The concatenation of two languages L and L,’ is L l L’ = (u l v 1 u EL, v EL’}. If L is a language, then L* = UnacLn, where Lo = {A ). The left quotient w\L of a language L over A with respect to a word w over A is the language {x I wx E L}; similarly for the right quotient, L/w = {x I xw E L}. The reverse L.* of a language L is the language {wP I IV E L}. * This research was supported by the Natural Sciences and Engineering Research Council, Canada under grant No. A-1617. ** Present address: Department of Computer Science, University of Kentucky, Lexington, KY 405069 U.S.A. 19 20 J.A. Brzozowski, E. Leiss The set P(A*) of all languages over A together with the set operations union (u), intersection ( n ), and complement (‘) forms a boolean algebra, in which 0 and I act as zero and one, respectively. We also consider the finite boolean algebra L, of ‘language, functions f:X:i,P(A*)+ P(A*), i.e. the functions which can be expressed in terms of unions, intersections, and complements of the variables. (Note that Xi”,lS denotes the Cartesian product of it copies of S.) The constant functions 0 and I act as zero and one, respectively. Another finite boolean algebra which will be used is the set B, of boolean functions f : X:=1(0, l}+ (0, 1) in the variables xl, . . . , xn, x being {xl, . . . , xJ, together with the operations OR ( v ), AND ( A ), and complement (‘). The constant functions 0 and 1 act as zero and one, respectively. Clearly, & and L, are isomorphic as boolean algebras. The following correspondence will be used: