Journl of utomt, Lnguges n Combintorics u (v) w, x{y c OttovonGuerickeUniversitt Mgeburg Tight lower boun for the stte complexity of shue of regulr lnguges Cezr C^mpenu, Ki Slom Computing n Informtion Science Deprtment, Queen' s University, Kingston, Ontrio K7L 3N6, Cn emil: fcezr,kslomg@cs.queensu.c n Sheng Yu Computer Science Deprtment, University of Western Ontrio, Lonon, Ontrio N6 5B7, Cn emil: syu@cs.uwo.c BSTRCT The upper boun for the stte complexity of the shue of two regulr lnguges is mn. We prove tht this boun cn be reche for some (not necessrily complete) eterministic nite utomt with, respectively, m n n sttes. Our construction uses n lphbet of size 5. Keywors: Finite utomt, Shue, Stte Complexity. Introuction By the stte complexity of n opertion on regulr lnguges we men function tht ssocites to the sizes of the DF's representing the operns of the opertion the miniml number of sttes of DF representing the resulting lnguge in the worst cse. Tight lower bouns for the stte complexity of mny bsic opertions on nite n innite regulr lnguges re obtine in [, 3, 4] n in the references liste there. Here we consier the opertion of shue. If L n L re ccepte, respectively, by n mstte DF n n nstte DF it is esy to see tht L L is lwys ccepte by DF hving t most mn sttes. We consier DF's tht re not necessrily complete. We show tht this upper boun is tight, tht is, we construct n exmple for which the lower boun for the number of sttes of DF ccepting L L reches the vlue mn. The DF's use in our construction re incomplete (tht is, we llow unene trnsitions). The construction gives lower boun lso for the shue of complete mstte DF n complete nstte DF but it oes not rech the corresponing
C. C^mpenu, K. Slom, S. Yu upper boun of mn. So our result is tight only for incomplete DF's. For restricte vlues of m n n we present exmples of DF's tht re complete with respect to some sublphbet of the combine lphbet n the stte complexity of their shue reches the upper boun mn. It cn be note tht shue is \icult" opertion even in the cse of regulr lnguges. For instnce, the ecibility of the following question remins open []. For given regulr lnguge L etermine whether or not L hs nontrivil shue ecomposition L = L L where neither one of the lnguges L, L is the singleton lnguge consisting of the empty wor.. Results Here lwys enotes nite lphbet. The shue of wors w ; w is the set w w = fu v u v u m v m j m IN; u i ; v i ; i = ; : : : ; m; w = u u m ; w = v v m g: For exmple, b c = fbc; cb; cbg n b bc = fbbc; bcb; bbc; bcb; bcbg. The shue opertion is extene in the nturl wy for lnguges over the lphbet. For ll unexpline notions concerning nite utomt we refer the reer to [4]. noneterministic nite utomton (NF) is enote s vetuple = (Q; ; ; q ; F ) () where Q is the nite set of sttes, is the nite input lphbet, : Q! Q is the stte trnsition reltion, q Q is the initil stte n F Q is the set of nl sttes. In the stnr wy, the stte trnsition reltion is extene to function ^ : Q! Q. We enote lso ^ simply by. The lnguge ccepte by is L() = fw j (q ; w) \ F 6= ;g: vetuple = (Q; ; ; q ; F ) s in () is si to be eterministic nite utomton (DF) if for ech q Q n z, (q; z) is either singleton set or the empty set (in which cse we sy tht the trnsition is not ene). DF is si to be complete if (q; z) is lwys ene. In our terminology, DF mens eterministic utomton tht my be incomplete unless it is explicitly stte to be complete. For ny DF there is n equivlent complete DF with t most one itionl stte. Both the NF's n (complete) DF's ccept exctly the regulr lnguges. Our min result is the following. Theorem If L is ccepte by n mstte DF n L is ccepte by n nstte DF (m; n ), then the miniml DF for L L nees mn sttes in the worst cse.
Tight lower boun for the stte complexity of shue of regulr lnguges 3 Proof. Choose = f b; c; g. Let L i = L( i ), i = ;, where = (f; ; : : : ; m g; ; ; ; fg) n = (f; ; : : : ; n g; ; ; ; fg). The trnsition reltions i, i = ; re ene below. ll trnsitions not liste below re unene. For we ene: (i; ) = i + mo m, i m ; (i; c) = i +, i < m ; (i; ) = i, i m ; (i; f) = i, i m. For we ene: (i; b) = i + mo n, i n ; (i; c) = i, i n ; (i; ) = i +, i < n ; (i; f) = i, i n. The utomt i, i = ; re epicte in Figure. 3 c ; m f c c; f c; f 3 c; f b; c; n f 6 b Figure : Exmple of DF for which the shue complexity reches mn The lnguge L L is ccepte by the NF = (f; ; : : : ; m g f; ; : : : ; n g; ; ; (; ); f(; )g) where the trnsition reltion is ene by setting ((i; j); x) = f( (i; x); j); (i; (j; x)g; i m ; j n ; x : bove if (i; x) or/n (j; x) is unene then ((i; j); x) is singleton set or the empty set, respectively. The NF is epicte in Figure.
4 C. C^mpenu, K. Slom, S. Yu (; ) c (; ) (; ) (; ) c (; ) (; ) c (; ) c (; ) c (; ) c (; n ) c (; n ) b b b c (; n ) ; c ; c ; c ; c ; c; f ; c ; c ; c ; c; f ; c ; c ; c ; c; f ; c ; c ; c (m ; ) c (m ; ) (m ; ) b (m ; n ) Figure : The NF for the result of the shue between L( ) n L( ) of Figure Denote m = f; ; : : : ; m g n n = f; ; : : : ; n g. Let B = ( mn ; ; '; f(; )g; F B ) be the DF obtine from the NF using the stnr subset construction. Here F B = fx m n j (; ) Xg. For X m n n y we ene '(X; y) = f(x; y) j x Xg. In orer to complete the proof it is sucient to show tht: (i) None of the sttes of B, tht is, subsets of m n re equivlent (tht is, rech
Tight lower boun for the stte complexity of shue of regulr lnguges 5 nl stte for exctly the sme inputs). (ii) ll subsets of m n re rechble from the initil stte f(; )g. Note tht lthough ; is rechble from the initil stte, it nee not be use since we llow incomplete utomt, n hence the boun is mn. To see (i), let X; Y m n where X 6= Y. Then there exists (i; j) X Y (or in Y X), i m, j n. Now '(X; mi b nj ) F B () becuse tkes (i; j) to (; ) with input mi b nj n, '(Y; mi b nj ) 6 F B (3) becuse with input mi b nj oes not tke ny element (i ; j ) 6= (i; j) to (; ). (In the cse (i; j) Y X the reltions () n (3) re simply interchnge.) For (ii) rst we observe tht '(f(; )g; c m n ) = m n: >From the set mn we cn rech ny of its subsets by \eliminting" one t time ny sequence of elements. This relies on the following observtion. Let X m n n (i; j) X, i m, j n. Then '(X; mi b nj f i b j ) = X f(i; j)g: (4) Note tht for ny Y m n, '(Y; f) = Y f(; )g. The wor mi b nj shifts the originl set X so tht the element (i; j) becomes (; ) n the trnsition by f \elimintes" exctly this element. ll the other elements re unchnge by trnsition with f. In the NF, the wor mi b nj i b j eterministiclly trnsltes ny stte bck into itself. Thus the only eect on X of the trnsitions ccoring to the input mi b nj f i b j is tht the element (i; j) is eliminte n (4) hols. It is esy to see tht the shue of, respectively, n mstte n n nstte DF lnguge cn lwys be ccepte by (not necessrily complete) DF hving t most mn sttes. Theorem implies tht this upper boun cn be reche. The DF's n constructe for the proof of Theorem re not complete. We cn for i \e stte" ummy i, i = ;, such tht ll previously unene trnsitions hve trget ummy i. Then the sttes of B re subsets of (m [ fummy g) (n [ fummy g). For the moie utomton B we ene the following nottion. For X (m [ fummy g) (n [ fummy g) enote the \live prt" of X s live(x) = X \ (m n): The set live(x) consists of exctly the elements of X tht cn rech the nl stte on some input. The construction implies the following. (i) ssume tht X; Y (m [ fummy g) (n [ fummy g). If live(x) 6= live(y ), then the sttes X n Y re not equivlent.
6 C. C^mpenu, K. Slom, S. Yu (ii) For ny Z m n there exists stte X rechble from the initil stte f(; )g such tht live(x) = Z. The bove observtions give the following lower boun for the stte complexity of the shue of two complete DF's. Note tht lso in the cse where both the originl DF's n the DF for the shue lnguge re require to be complete, the (trivil) upper boun is gin mn since the empty set cnnot pper in the resulting subset construction. Corollry If L is ccepte by complete mstte DF n L is ccepte by complete nstte DF (m; n ), then the miniml complete DF for L L nees t lest (m)(n) sttes in the worst cse. s we cn see, for the worst cse exmple there is no trnsition with letter b in the rst utomton n no trnsition with letter in the secon one, n there is no obvious wy to moify the construction so tht these trnsitions coul be ene without chnging the stte complexity of the resulting shue. lso, we nee for both DF's the unene trnsition in stte with letter f, so the utomt re incomplete, even if we consier restricte lphbet. For the following prticulr cses we hve the worst cse of complexity mn in cse of semicomplete DF, i.e., both utomt re complete with respect to subset of the originl lphbet, so the theoreticl bouns re reche. For m 3 n n = the exmple is given in Figure 3. ; c ; c ; c ; c 3 m Figure 3: The DF for m 3 n n =, the shue complexity is m. For m 4 n n = the exmple is given in Figure 4. For m = 3 n n = the exmple is given in Figure 5.
c ; e c; ; e ; e c; ; e b; c 6 b; c c; ; e Tight lower boun for the stte complexity of shue of regulr lnguges 7 3 m ; e e ; c Figure 4: The DF for m 4 n n = the shue complexity is m. ; c; e b; e e 6 Figure 5: The DF for m = 3 n n = the shue complexity is m = 6 = 63. We lso hve similr exmples for (m; n) f(; ); (; ); (; )g. However, for ll of the exmples, the rst utomton n the secon utomton o not hve the sme unerlying lphbet, tht is, they re only semicomplete. To preserve completeness n hve the sme lphbet we hve to ummy stte, so the new upper boun will be (m)(n), which is one stte less thn the boun (m)(n) from Corollry. In the cse of complete DF, when trnsforming the shue NF to DF we cnnot rech the stte corresponing to the empty set. 3. Open questions The lnguge L L in the proof of Theorem is over 5letter lphbet. It remins n open question wht is the optiml lower boun for lphbet sizes, 3 n 4. For unry lphbets, the shue opertion simply reuces to the opertion of ctention. In the unry cse the boun for ctention is known to be mn [4]. lso, it remins n open question whether we cn improve the lower boun from Corollry for the stte complexity of the shue of two complete DF's.
8 C. C^mpenu, K. Slom, S. Yu References [] C. C^mpenu, K. Culik II, K. Slom, S. Yu. Stte complexity of bsic opertions on nite lnguges, Proceeings of the Fourth Interntionl Workshop on Implementing utomt, WI'99, (July 999, Potsm, Germny), Springer Verlg, to pper. [] C. C^mpenu, K. Slom, S. Vgvolgyi. Shue quotient n ecompositions, Proceeings of the Fifth Interntionl Conference on Developments in Lnguge Theory, DLT, (July, Vienn, ustri), to pper. [3] C. C^mpenu, K. Slom, S. Yu. Stte complexity of regulr lnguges: Finite versus innite. In: \Finite vs Innite: Contributions to n Eternl Dilemm", C. S. Clue, G. Pun (Es.), SpringerVerlg,, pp. 53{73. [4] S. Yu. Regulr lnguges. In: Hnbook of Forml Lnguges, Vol. I, G. Rozenberg n. Slom, es., SpringerVerlg, pp. 4{, 997.