Intersection of Regular Languages and State Complexit y Jean-Camille Birge t Department of Computer Science and Engineerin g University of Nebraska, Lincoln, NE 68588-011 5 birget c@, fergvax .unl .ed u In [1] B . Ravikumar considers the following problem . Given n languages L n (1) , L n (2) , . . ., Ln (n) , each of which is recognized by a DFA with n states : what is the number of states of the minimum DFA fo r the intersection nLn (i) in the worst case? He proves a lower bound of n! ; but, as Leiss [2] points out , this does not completely solve the problem, since the best known upper bound is n n . Below we sho w that a slight extension of Ravikumar's proof yields a lower bound of n n ; so the upper bound is reachable . For unexplained notation, see [1], [2], and [3] . Let N be the set of natural integers . For n E N le t B n be the set of all binary relations on the set {1, ¢ ¢ ¢, n} ; let F, 1 be the set of all (total) functions on {1, ¢ . ¢ , n} . For G, G' E B n we denote the composition of these relations by G ¢G' . Just like Ravikumar, we pick the following languages : Ln(i) = (G 1 G2 ¢ . ¢G,n E E B n* / / (i,i) E G 1 ¢G2 ¢ ¢G,,, m E N} . Then Ln = flLn ( i ) is the languag e E L n = {G 1 G 2 ¢ ¢ ¢ Gil Claim, B n* G 1 ¢G 2 ¢ . ¢ ¢ ¢ G,n is a reflexive relation, m N} . Any DFA recognizing L n = flL n (i) needs > n n states . Proof. Let L n be recognized by a DFA (Q, B n , 8, q 0 , F) . We shall show that for all pairs of functions f, g E F n with f ~ g, we have 8(g 0 ,f) ~ 6(g 0 ,g) ; this implies that IQI > n n . Suppose, by contradiction, tha t 6(g 0 ,f) = 6(g 0 ,g) . Then if the next input letter is f- 1 E B n we have 6(g 0 ,ff- 1 ) = 6(g0 ,gf- 1 ) ; so the string s ff- 1 and gf- 1 should be either both accepted or both rejected . But f ¢f- 1 is a reflexive relation, wherea s g ¢f- 1 is not reflexive when f # g . ((k)g)f- 1 (Indeed : Let k E {1, ¢ . ¢, n } be such that (k)f ~ (k)g . Then k = (k)g ¢f- 1 , by the definition of f 1 . Thus (k,k) g ¢f- 1 . Note : To be consistent with the usua l definition of composition of relations, we assume that g ¢f- 1 means that g is applied first, and then f- 1 ; accordingly we write (x)f for the image of x under f. ) Remark 1 . One could have used the smaller alphabet F žuF n- 1 instead of B n (where Fn - 1 consist s of all the inverses of functions) . Furthermore, since F n is generated under composition by just thre e functions (e .g ., the cycle (1, 2, ¢ ¢ n), the transposition (1, 2), and the collapse (1, 1, 3, 4, ", n)) one coul d replace F nuFn - 1 by just four relations . So the above result holds for a four-letter alphabet. Remark 2 . From the above result (and from the upper bound) it follows immediately that any DF A recognizing 10kLn(i) has > n k states, for any k n . This leaves the problem open of finding worst cas e lower bounds for the number of states of a DFA recognizing the intersection of k languages (each of whic h has a DFA with n states), when k > n . The lower bound n k cannot hold in general (at least if the alphabet i s kept fixed) since, for a given n, there are only boundedly many regular languages . Reference s [ 1 ] B . Ravikumar, "Some applications of a technique of Sakoda and Sipser", SIGACT News, 21 .4 (1990) 73-77 . [2] E . Leiss, "Some comments on a recent note of Ravikumar", SIGACT News, 22 .1 (1991) 64 . [3] J . Hoperoft, J . Ullman, "lntrod. to Automata Theory, Languages, and Computation", Addison-Wesley (1979) .

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