CONTROLLABILITY and observability are the central

Transcription

1 1 Complexity of Infiml Oservle Superlnguges Tomáš Msopust Astrt The infiml prefix-losed, ontrollle nd oservle superlnguge plys n essentil role in the reltionship etween ontrollility, oservility nd o-oservility the entrl notions of supervisory ontrol theory. Existing lgorithms for its omputtion re exponentil nd it is not known whether polynomil lgorithm exists. In this pper, we study the stte omplexity of this lnguge. Stte omplexity of lnguge is the numer of sttes of the miniml DFA for the lnguge. For lnguge of stte omplexity n, we show tht the upper-ound stte omplexity on the infiml prefix-losed nd oservle superlnguge is 2 n + 1 nd tht this ound is symptotilly tight. It proves tht there is no lgorithm omputing DFA of the infiml prefix-losed nd oservle superlnguge in polynomil time. Our onstrution further shows tht suh DFA n e omputed in time O(2 n ). The onstrution involves NFAs nd omputtion of the supreml prefix-losed sulnguge. We study the omputtion of the supreml prefix-losed sulnguge nd show tht there is no polynomil-time lgorithm tht omputes n NFA of the supreml prefix-losed sulnguge of lnguge given s n NFA even if the lnguge is unry. Index Terms Disrete event systems; Automt; Prefix-losed lnguge; Oservle lnguge; Complexity. I. INTRODUCTION CONTROLLABILITY nd oservility re the entrl notions of supervisory ontrol theory of disrete event systems in the Rmdge-Wonhm frmework [1] [3]. They form the neessry nd suffiient onditions for the existene of supervisor tht hieves the desired ontrol ehvior of system. In deentrlized supervisory ontrol, where more supervisors ooperte to ontrol the system, every supervisor oserves nd ontrols prt of the system. The oservtion of supervisor is modeled y n oservtion msk or y nturl projetion. Cieslk et l. [1] nd Rudie nd Wonhm [4] hve shown tht ontrollility nd o-oservility re the entrl notions in deentrlized supervisory ontrol. A reltionship etween ontrollility, oservility nd ooservility hs een studied y Kumr nd Shymn [5], who hve shown tht the infiml prefix-losed, ontrollle nd oservle superlnguge plys the essentil role. Another motivtion nd the importne of infiml superlnguges hve een disussed in the fundmentl ook on supervisory ontrol theory [6]. We hve further illustrted its relevne to deentrlized supervisory ontrol with ommunition [7] nd to oordintion ontrol [8]. We refer the reder to these ppers for more detils nd exmples. Infiml superlnguges re of generl interest in supervisory ontrol. There re exmples in modulr nd deentrlized ontrol showing evidene tht supreml sulnguges do not lwys suffie to hieve the est (optiml) solution nd tht Supported y the DFG in Emmy Noether grnt KR 4381/1-1 (DIAMOND). T. Msopust (msopust@mth.s.z) is with CFEAD, TU Dresden, Germny, nd with Institute of Mthemtis, Czeh Ademy of Sienes the optiml solution my e hieved if infiml superlnguges re involved. The exmples show evidene tht the omintion of supreml sulnguges nd infiml superlnguges help hieve optimlity if it is not hievle y supreml sulnguges lone [7], [8]. Therefore our interest in infiml prefix-losed, ontrollle nd oservle superlnguges. Lfortune nd Chen [9] hve shown tht the infiml prefixlosed nd ontrollle superlnguge n e omputed from deterministi finite utomton (DFA) for the lnguge in liner time. Kumr nd Shymn [5] hve further shown tht it is suffiient to onsider the omputtion of the infiml prefixlosed nd oservle superlnguge of lnguge K over Σ wrt the lnguge Σ. Thus, we fous in this pper on the infiml prefix-losed nd oservle superlnguge of K wrt Σ nd study its stte omplexity. Stte omplexity of lnguge is the numer of sttes of the miniml DFA mrking (epting) the lnguge. Sine the miniml DFA is unique (up to isomorphism), stte omplexity is omplexity mesure tht is independent of the representtion nd omputtion of the lnguge. Our ontriution: For lnguge K of stte omplexity n, we show tht the upper-ound on the stte omplexity of the infiml prefix-losed nd oservle superlnguge of K wrt the lnguge Σ is 2 n +1. We further prove tht this ound is symptotilly tight y showing tht the worst-se loweround stte omplexity is t lest 3 4 2n 1 = Ω(2 n ). Sine the stte omplexity is exponentil, so is the time omplexity of ny lgorithm omputing the orresponding miniml DFA. In ddition, our onstrution shows tht DFA representtion of the infiml prefix-losed nd oservle superlnguge of K wrt the lnguge Σ n e omputed in time O(2 n ). Our onstrution involves nondeterministi finite utomt (NFAs) nd is sed on formul equivlent to the formule of Rudie nd Wonhm [10] nd of Kumr nd Shymn [5]. The formule inlude omputtion of the supreml prefix-losed sulnguge. We study the omputtion of the supreml prefixlosed sulnguge nd show tht there is no polynomil-time lgorithm omputing n NFA representtion of the supreml prefix-losed sulnguge of lnguge given s n NFA even if the lnguge is unry. II. PRELIMINARIES We ssume tht the reder is fmilir with supervisory ontrol theory [6] nd utomt theory [11], [12]. For undefined notions, the reder is refer to these referenes. The prefix losure of lnguge L is the set L = {w Σ there is u Σ s.t. wu L}; L is prefix-losed if L = L. The right quotient of lnguge L wrt lnguge M is the set L/M = {w Σ there is x M s.t. wx L}. If M = {} is singleton, we simply write L/ = {w Σ w L}. The empty string is denoted y ε.

2 2 A nondeterministi finite utomton (NFA) is quintuple A = (Q, Σ, δ, Q 0, F ), where Q is finite nonempty set of sttes, Σ is n input lphet, Q 0 Q is set of initil sttes, F Q is set of mrked sttes, nd δ : Q (Σ {ε}) 2 Q is trnsition funtion tht is extended to 2 Q Σ y indution. The lnguge generted y A is the set L(A) = {w Σ δ(q 0, w) } nd the lnguge mrked y A is the set L m (A) = {w Σ δ(q 0, w) F }. The NFA A is n (inomplete) deterministi finite utomton (DFA) if Q 0 1 nd δ(q, ) 1 for every q Q nd Σ. Moreover, DFAs do not dmit ε-trnsitions, tht is, δ is prtil trnsition funtion from Q Σ to Q. For every NFA A there exists DFA B suh tht L m (B) = L m (A) nd L(B) = L(A). The DFA B is onstruted y the stndrd suset onstrution [12] nd is lled the suset utomton of A. Speifilly, for A = (Q, Σ, δ, Q 0, F ), B = (2 Q, Σ, δ, Q 0, F ), where δ : 2 Q Σ 2 Q is defined s δ (X, ) = δ(x, ) nd F = {R Q R F }. Let Σ nd e lphets. An (oservtion) msk is mp P : Σ {ε} tht is extended to Σ so tht P (ε) = ε nd P (s) = P (s)p () for s Σ nd Σ. If L is regulr lnguge, then P (L) = w L P (w) is regulr [13]. A msk P is (nturl) projetion if Σ nd P () =, for, nd P () = ε otherwise. The inverse imge of msk P, denoted y P 1 : 2 2 Σ, is defined s P 1 (L) = {w Σ P (w) L}. Regulr lnguges re losed under the inverse imge of msk [13]. In the rest, the term lnguge stnds for regulr lnguge. III. KNOWN AND PRELIMINARY RESULTS Let inf CO(K, L(G), Σ u, P ) denote the infiml superlnguge of K tht is prefix-losed, ontrollle nd oservle wrt L(G), unontrollle events Σ u, nd msk P. Similrly we use inf C(K, L(G), Σ u ) to denote the infiml prefix-losed nd ontrollle superlnguge nd inf O(K, L(G), P ) to denote the infiml prefix-losed nd oservle superlnguge. Kumr nd Shymn [5] hve proved tht the omputtion of the infiml prefix-losed, ontrollle nd oservle superlnguge of K wrt L(G) depends on the omputtion wrt Σ, nmely inf CO(K, L(G), Σ u, P ) = inf CO(K, Σ, Σ u, P ) L(G). It thus suffies to onsider the omputtion wrt the lnguge Σ. They further proved tht inf CO(K, Σ, Σ u, P ) = inf O(inf C(K, Σ, Σ u ), Σ, P ). Lfortune nd Chen [9] hve shown tht inf C(K, Σ, Σ u ) = KΣ u, whih n e omputed from DFA for K in liner time. The omputtion of the infiml prefix-losed nd ontrollle superlnguge is thus esy nd we fous in the rest on the omputtion of the infiml prefix-losed nd oservle superlnguge. Rudie nd Wonhm [10] showed tht inf O(K, L(G), P ) = L(G) \ (Σ + \ P 1 ( P (K)))Σ, where P is projetion nd P projets ll ut the lst event, indutively defined y P (ε) = ε nd P (s) = P (s). They lso proved tht for K, P 1 [ P (K) = P 1 (P (K K)) Σ ] {ε}. (1) The eqution remins vlid for msks nd Kumr nd Shymn [5] extended it nd simplified to the form inf O(K, L(G), P ) = sup [ P 1 P (K)] L(G) (2) where sup (H) stnds for the supreml prefix-losed sulnguge of lnguge H. Note tht it immeditely implies tht inf O(K, L(G), P ) = inf O(K, Σ, P ) L(G). The formule onsist of opertions studied in the literture nd their worst-se stte omplexities give rough estimte on the stte omplexity of the lnguge inf O(K, Σ, P ). By Yu et l. [14], the ound is no more thn 2 Σ (4n2 +8n+1), where n is the stte omplexity of K. Nmely, Yu et l. [14] show tht K needs no more thn 4n + 8 sttes nd K K no more thn (4n + 8)n sttes. Then P 1 P (K K) needs t most 2 (4n+8)n sttes. (If P is nturl projetion, the ound is lower [15], [16].) The intersetion with Σ then needs no more thn 2 (4n+8)n 2 sttes nd the union over ll events in Σ no more thn (2 (4n+8)n 2) Σ sttes. The supreml prefix-losed sulnguge of DFA n e omputed in liner time nd does not inrese the stte omplexity; it requires to remove ll non-mrked sttes nd orresponding trnsitions. Results of Yu et l. [14] hold for ny lnguge nd the reder my notie tht the lnguges of the formule re of speil forms. The worst-se stte omplexity of Yu et l. [14] is thus mostly not tight for them. For instne, it n e shown tht the tight stte omplexity on K K is 2n rther thn (4n + 8)n, whih dereses the upper ound to 2 Σ 2n. We now show tht the upper ound on the stte omplexity of the lnguge inf O(K, Σ, P ) is no more thn 2 n + 1. To this im, we express the formul for inf O(K, Σ, P ) in n equivlent form using the opertion of right quotient. This expression is sed on the following reltion etween the msk, intersetion nd right quotient opertions. Lemm 1: Let P e msk from Σ to. For prefix-losed lnguge K over Σ nd n event Σ, it holds tht P 1 (P (K K)) Σ = (P 1 P (K/)). Proof: The lim holds for K =. Assume tht K. Let x P 1 (P (K K)) Σ. Then P (x) P (K K) nd there exists y K K suh tht P (x) = P (y). Sine y K, we hve tht y K/, hene x P 1 (P (y)) (P 1 P (K/)). On the other hnd, let x (P 1 P (K/)). Then x P 1 P (K/) nd there is y K/ with P (x) = P (y). Sine y K/, y K. Beuse K is prefix-losed, y K, whih implies tht y K K. Thus, P (x) P (K K), tht is, x P 1 (P (K K)) Σ. The ssumption tht the lnguge is prefix-losed is essentil. The lemm does not hold for non-prefix-losed lnguges even if P is the identity msk. In this se, Lemm 1 redues to K K = (K/). If K = {} is non-prefix-losed, then K K =, wheres (K/) = {}. We n now express the formul of Kumr nd Shymn [5] in n equivlent form using the opertion of right quotient. Theorem 2: Let K e nonempty lnguge over Σ, nd let P e msk from Σ to. Then inf O(K, Σ, P ) = sup ( (P 1 P (K/)) {ε}). Proof: By (1), (2), nd Lemm 1, inf O(K, Σ, P ) = sup ( P 1 P (K)) = sup ( [P 1 (P (K K)) Σ ] {ε}) = sup ( (P 1 P (K/)) {ε}), respetively. We further modify the formul y moving the union opertion deeper into the formul. It is then pplied to struturlly simpler suformul, whih is useful for our gol.

3 3 Lemm 3: Let K Σ e lnguge nd P : Σ {ε} e msk. Let Σ = { Σ} e opy of Σ disjoint from oth Σ nd. Let h: Σ Σ Σ {ε} e msk defined y h() = P (), for Σ, nd h( ) =, for Σ. Let g : Σ Σ e msk defined y g( ) =, for Σ. Then ( ( ) ) (P 1 P (K/)) = g h 1 h (K/) Σ Σ. Proof: By the properties of msks, we hve tht g(h 1 (h( (K/) )) Σ Σ ) = g([ h 1 (h((k/) ))] Σ Σ ) = g([ h 1 (h(k/)h( ))] Σ Σ ) = g([ h 1 (P (K/) )] Σ Σ ) = g([ h 1 (P (K/))h 1 ( )] Σ Σ ) = g([ P 1 (P (K/)) P 1 (ε)] Σ Σ ) = g( [P 1 (P (K/)) P 1 (ε) Σ Σ ]) = g( P 1 (P (K/)) ) = g(p 1 (P (K/)) ) = (P 1 P (K/)). This ompletes the proof. As orollry of Theorem 2 nd Lemm 3, we otin the following formul, whih we use to show the symptotilly tight ound on the stte omplexity of inf O(K, Σ, P ). Corollry 4: Under the ssumptions of Lemm 3, if K, inf O(K, Σ, P ) = [ ( ( ) ) ] sup g h 1 h (K/) Σ Σ {ε}. IV. DETERMINISTIC STATE COMPLEXITY We now use Corollry 4 to show tht 2 n + 1 is n upperound on the stte omplexity of the lnguge inf O(K, Σ, P ) nd tht the ound is symptotilly tight. Corollry 4 suggests n lgorithm (Algorithm 1) to ompute the lnguge inf O(K, Σ, P ). We now disuss stte omplexities of its steps. Consequently we otin its time omplexity. Lemm 5 (Yu et l. [14]): Let A e DFA over Σ with n sttes, nd let Σ. Then the miniml DFA for L m (A)/ hs t most n sttes. The ound is tight. Algorithm 1 Computtion of inf O(K, Σ, P ) Input: DFA for K over Σ nd msk P Output: DFA for the lnguge inf O(K, Σ, P ) 1: if K = then return the DFA for K 2: else 3: Compute DFA for K 4: Compute DFA for (K/) 5: Compute n NFA for g(h 1 h( (K/) ) Σ Σ ) 6: Determinize the NFA 7: Compute the union with {ε} 8: Compute the supreml prefix-losed sulnguge , Fig. 1. Automt A (left) nd B (right) for (L m(a)/) The onstrution is s follows. Let A = (Q, Σ, δ A, q 0, F A ) e DFA. Construt the DFA A = (Q, Σ, δ A, q 0, F A ), where F A = {q Q δ A (q, ) F A }. Then L m (A ) = L m (A)/. We now study the size of the miniml DFA for the lnguge omputed in Step 4 of the lgorithm. Lemm 6: Let A e DFA over Σ with n sttes. Then the miniml DFA for (L m (A)/) hs t most n+1 sttes. The ound is tight even for prefix-losed lnguges. Proof: Let A = (Q, Σ, δ A, q 0, F A ) e DFA with n sttes Q = {0, 1,..., n 1}. For every Σ, we onstrut the set F = {q Q δ A (q, ) F A } of ll sttes of A from whih n -trnsition rehes mrked stte. We onstrut the DFA B = (Q {n}, Σ, δ B, 0, {n}) from A y dding new stte, n, whih is the only mrked stte, nd y defining the trnsitions δ B (q, ) = δ A (q, ), for 0 q n 1 nd Σ, nd δ B (f, ) = n, for every f F. The onstrution is illustrted in Fig. 1. The orresponding sets re F = {0, 1}, F = {0} nd F =. We lim tht B mrks the lnguge (L m (A)/). If string is mrked y B, it is of the form w, for some Σ, whih mens tht δ B (0, w) F. By the onstrution of F, w L m (A)/, hene w (L m (A)/). On the other hnd, if w (L m (A)/), then w L m (A)/, hene δ B (0, w) = f, for some f F, whih implies tht δ B (0, w ) = δ B (f, ) = n, hene it is mrked y B. To show tht the ound is tight, we onsider the DFA A depited in Fig. 2 (solid rrows) with sttes {0,..., n 1}, where stte 0 is initil nd ll sttes re mrked. The DFA is miniml; two sttes re distinguishle y string in. The DFA B for (L m (A)/) (L m (A)/) is depited in Fig. 2 (ll rrows), where the sttes re {0,..., n} with n eing the only mrked stte. There is n -trnsition from stte i to stte n for every 0 i n 1, nd -trnsition from stte j to stte n for every 0 j n 2. The DFA B is miniml; sttes {0,..., n 1} re distinguishle y the sme rgument s for A nd n is not equivlent with ny other stte sine it is the only mrked stte. We now use the previous results to otin our upper-ound on the stte omplexity of the lnguge inf O(K, Σ, P ). Theorem 7 (Upper ound): Let K over Σ e nonempty lnguge mrked y DFA with n sttes. Then the miniml,,, 0 1 n 2 n 1 n Fig. 2. Automt A (solid rrows) nd B (ll rrows)

4 4 DFA for inf O(K, Σ, P ) hs no more thn 2 n + 1 sttes. Proof: Let P : Σ {ε}. By Corollry 4, we hve tht inf O(K, Σ, P ) = sup [g(h 1 h( (K/) ) Σ Σ ) {ε}]. From Lemm 6, we hve tht the miniml DFA mrking the lnguge (K/) hs t most n+1 sttes, only one of whih is mrked. We denote this stte y f. Notie tht, y the onstrution, there is no trnsition from stte f. We represent the lnguge g(h 1 h( (K/) ) Σ Σ ) s n NFA s follows. The lnguge h( (K/) ) is omputed y repling every x-trnsition, x Σ, with the h(x)-trnsition. The lnguge h 1 h( (K/) ) is then omputed y repling every y-trnsition, y, y n x- trnsition for every x Σ suh tht h(x) = y. In ddition, for every x Σ suh tht h(x) = ε, we dd selfloop under x to every stte of the NFA. To ompute n NFA for h 1 h( (K/) ) Σ Σ then mens to remove ll trnsitions from stte f. This n e done during the omputtion of n NFA for h 1 h( (K/) ) so tht no self-loop is dded to stte f. The omputtion of n NFA for the msk g is similr to tht of h. The resulting NFA hs t most n + 1 sttes. Thus, DFA equivlent to the NFA, onstruted y the stndrd suset onstrution, hs t most 2 n+1 rehle sttes. However, sine every mrked stte of the suset utomton must ontin f, nd there re t most 2 n susets ontining f, there re t most 2 n mrked sttes in the omputed DFA. To ompute the union with {ε}, the DFA my require one more (initil nd mrked) stte. Thus, the resulting DFA hs t most 2 n sttes, where t most 2 n + 1 sttes re mrked. Sine inf O(K, Σ, P ) is prefix-losed, its miniml DFA must hve ll sttes mrked. There re t most 2 n + 1 mrked sttes in the ove onstruted utomton, therefore the miniml DFA for inf O(K, Σ, P ) n hve t most so mny sttes. Consequently, the time omplexity of Algorithm 1 is O(2 n ). Indeed, let n e the stte omplexity of K. Step 3 requires time O(n). To ompute Step 4, we dd new stte, f, nd sn the utomton in liner time using, e.g., the redthfirst serh (BFS) lgorithm [17]. For every stte q nd its out-going trnsition under x, if δ(q, x) is mrked, we dd n x -trnsition from q to f. This n e done in time O(1 + n + 2n Σ ) = O(n Σ ), sine there re n sttes, one dded new stte, nd t most n Σ trnsitions tht my e duplited to f. Step 5 n e omputed in time O(n Σ ) s follows. The pplition of h n e done in time O(n Σ ) y the BFS lgorithm. The pplition of h 1 n e done in time O(n Σ + n Σ \ ) = O(n Σ ), where the seond prt orresponds to dding self-loops under unoservle events. As explined ove, the intersetion with Σ Σ is done so tht no trnsitions re dded to f during the omputtion of h 1. Step 6 n e omputed in time O(2 n Σ ), sine, y the proof of Theorem 7, the DFA hs t most 2 n sttes nd Σ trnsitions in every stte. Step 7 n e omputed in time O( Σ ) s follows: let q 0 e the initil stte of the DFA, nd let q i e new mrked stte. We hnge the DFA so tht q i is the only initil stte, i.e., q 0 is not initil nymore, nd for every x Σ, we define δ(q i, x) = δ(q 0, x). Finlly, Step 8 n e omputed in liner n 2 n 1 Fig. 3. The miniml DFA A n for K n time wrt the size of the input DFA y removing ll nonmrked sttes nd the orresponding trnsitions. The overll time omplexity is O( Σ 2 n ). Considering the size of the lphet s onstnt results in the limed omplexity O(2 n ). We now disuss the lower-ound stte omplexity nd show tht it is Ω(2 n ). It holds even for projetions. Theorem 8 (Lower ound): Let P : {, } {} e projetion. For every n 2, there exists miniml DFA with n sttes mrking lnguge K n {, }, suh tht the stte omplexity of inf O(K n, Σ, P ) is t lest 3 4 2n 1. Proof: Let K n e the lnguge mrked y the DFA A n depited in Fig. 3. It hs n sttes {0, 1,..., n 1}, where stte 0 is the sole initil nd mrked stte. For 0 i n 1, δ(i, ) = (i + 1 mod n). For 1 i n 3, δ(i, ) = i + 1, δ(n 2, ) = 0, nd, for i {0, n 1}, δ(i, ) = i. Finlly, there is single -trnsition δ(n 1, ) = 0. An NFA B n for the lnguge g(h 1 h( (K n /) ) Σ Σ ) is uild from A n ording to the ove onstrutions in the following steps nd the result is depited in Fig. 4: 1) We ompute K n y mrking ll sttes of A n. 2) To ompute (K n /), we dd new stte, n. From every stte of A n, trnsitions under nd go to stte n, nd trnsition under goes from stte n 1 to stte n. The only mrked stte is stte n. 3) The lnguge h( (K/) ) is omputed y repling the -trnsition y n ε-trnsition. 4) To ompute h 1 h( (K/) ) Σ Σ, self-loop under is dded to every stte of A n. Note tht it is not dded to stte n, sine it would e eliminted y the intersetion with Σ Σ. Thus, this n e done in liner time without omputing the intersetion. 5) Finlly, to pply g mens to renme ll trnsitions under, nd, whih ll go to stte n. We show tht the miniml DFA equivlent to the NFA B n hs t lest 3 4 2n 1 rehle mrked sttes. Using the stndrd suset onstrution, we first show tht ll sttes of the, 0 n, 1 2 n 2 n 1 ε, Fig. 4. An NFA B n mrking lnguge g(h 1 h ( (Kn/) ) Σ Σ )

5 5 suset utomton orresponding to the NFA B n re pirwise distinguishle. Indeed, B n mrks ε only from stte n nd i only from stte n 1 i, for 0 i n 1. Therefore, the sttes of the suset utomton re pirwise distinguishle. To prove the theorem, we show tht the suset utomton hs 2 n n 2 1 mrked sttes tht re ll rehle vi other mrked sttes. Stte {0} is initil, ut not mrked; we resolve this issue lter. We now prove, y indution on the size of the suset, tht every suset of {0, 1,..., n 1, n} ontining 0 nd n is rehle in the suset utomton from stte {0} y nonempty string over {}. Sine there is n -trnsition nd -trnsition from every stte 0 through n 1 to n, ll susets rehle y suh string must ontin stte n, i.e., they re mrked in the suset utomton. Stte {0, n} is rehle from stte {0} y. Stte {n 2, n} is rehle from {0} y n 2. Stte {0, n 2, n} is rehle from stte {n 2, n} y 2 n 3. Stte {0, n 2, n} goes to stte {0, 1, n 1, n} y, nd then y string in to sttes {0, i, n 1, n} with 1 i n 2. Stte {0, n 2, n 1, n} goes to stte {0, n 1, n} y, nd then to stte {0, 1, n} y. By string in, stte {0, 1, n} goes to sttes {0, i, n} with 1 i n 2. Thus, eh suset of size two or three ontining 0 nd n is rehle. Now, let X = {0, i 1, i 2,..., i t, n} e set of size t + 2, where 2 t n 1 nd 1 i 1 < i 2 < < i t n 1. We onsider two ses: 1) If i t = n 1, then X is rehle from stte {0, i 2 i 1,..., i t 1 i 1, n 2, n} y i1 1, nd the ltter set of size t + 1 is rehle y the indution hypothesis. 2) If i t < n 1, then X is rehle from stte {0, i 2 i 1,..., i t i 1, n 1, n} y i1 1, nd the ltter set of size t + 2 ontins stte n 1, nd is rehle y 1). This proves rehility of ll susets of {0, 1,..., n} ontining 0 nd n. There re 2 n 1 suh susets. Next, if X = {i 1, i 2,..., i t } is non-empty suset of the set {1, 2,..., n 2}, then the set X {n} is rehle from the set {0, i 2 i 1, i 3 i 1,..., i t i 1, n} ontining 0 nd n y i1. Thus, for every X {1, 2,..., n 2}, stte X {n} is rehle in the suset utomton. These sets do not ontin 0, hene they re different from the rehle sttes onsidered ove. There re 2 n 2 1 suh susets. Finlly, we ompute the union with the lnguge {ε}. To do this, we rete new initil nd epting stte, I, (stte {0} is not initil nymore) with trnsitions defined extly s for stte {0}, tht is, δ(i, x) = δ({0}, x), for every x {, }. This hs resolved the prolem with the non-mrked initil stte, sine stte I is mrked nd hs the sme trnsitions s stte {0}, tht is, ll sttes rehle from stte {0} re lso rehle from stte I. Thus, we hve shown tht the miniml DFA onstruted y the suset onstrution hs t lest 2 n n 2 mrked sttes tht re ll rehle from the initil mrked stte I vi mrked sttes. However, stte I is equivlent to stte {0, n}. Indeed, oth sttes I nd {0, n} go to stte {1, n} under, to stte {0, n} under, nd to stte {0} under. It remins to show tht if the non-mrked sttes re eliminted, the onstruted mrked sttes different from I re still pirwise distinguishle. Let X nd Y e two sets different Fig. 5. A prefix-losed NFA A with L m(a) = {} from I onstruted ove. They oth ontin n nd, without loss of generlity, we my ssume tht there exists i suh tht n 1 i X \ Y. Then the set rehle from X under i ontins n 1, ut the set rehle from Y under i does not. It mens tht i is mrked from X, ut not from Y, whih distinguishes the sttes X nd Y. Therefore, the miniml DFA of the supreml prefix-losed sulnguge hs t lest 2 n n 2 1 sttes, whih ompletes the proof. Comining the upper nd lower ounds of Theorems 7 nd 8 gives the following orollry. Corollry 9: Let K over Σ e lnguge with stte omplexity n, nd let P e msk. Then the worst-se stte omplexity of the lnguge inf O(K, Σ, P ) is Θ(2 n ). We lso hve the following onsequene on the time omplexity of Algorithm 1. Corollry 10: The time omplexity of Algorithm 1 is Θ(2 n ), where n is the stte omplexity of the input lnguge. V. NONDETERMINISTIC STATE COMPLEXITY Algorithm 1 represents the lnguge s n NFA nd it is determinized efore omputing the opertion sup ( ). The lgorithm omputing sup ( ) on DFA uts off ll non-mrked sttes nd the orresponding trnsitions, whih requires liner time wrt the size of the input DFA. However, s shown ove, this DFA my e exponentilly lrger thn the DFA for K. Another possiility is to exeute sup ( ) diretly on n NFA. We now disuss this possiility nd show tht, in generl, there is no polynomil-time lgorithm tht, given n NFA A, would ompute n NFA mrking the lnguge sup (L m (A)). We first provide rief insight into the differene etween the omputtion of sup ( ) for DFAs nd NFAs. Indeed, if ll sttes of n NFA re mrked, then its lnguge is prefix-losed. However, ompred to DFAs, the prolem with NFAs is tht hving non-mrked stte does not yet men tht the lnguge is not prefix-losed, f. Fig. 5 for n exmple. It n e shown tht, given n NFA, it is PSPACE-omplete to deide whether its mrked lnguge is prefix-losed [18]. Theorem 11: The prolem whether the mrked lnguge of n NFA is prefix-losed is PSPACE-omplete. We now show tht there is no polynomil-time lgorithm omputing n NFA representtion of sup (L m (A)) in generl. Theorem 12: Let A e n NFA. There is no polynomil-time lgorithm omputing n NFA for the lnguge sup (L m (A)). The lim holds even for unry lnguges. Proof: We prove the theorem y onstruting, for ny n 1, n NFA A n with polynomilly mny sttes in n suh tht ny NFA for sup (L m (A n )) hs t lest exponentilly mny sttes in n. Clerly, suh n NFA nnot e omputed in polynomil time wrt the size of A n. To onstrut the NFAs A n, we first onstrut uxiliry DFAs B n, for every n 0. The DFA B 0 = (X 0, {},

6 Fig. 6. The NFA A 2 ; nondeterministi union of DFAs B 0, B 1, nd B 2 γ 0, X i,0, X m,0 ), where X 0 = X i,0 = X m,0 = {0 0 } nd γ 0 (0 0, ) is undefined. For n 1, let p n denote the nth prime numer. We define the DFA B n = (X n, {}, γ n, X i,n, X m,n ), where the stte set is X n = {0 n, 1 n,..., (p n 1) n }, the set of initil sttes is X i,n = {0 n }, the set of mrked sttes is X m,n = X n \ {0 n }, nd the trnsition funtion is γ n (i n, ) = (i + 1 mod p n ) n, for ll i n X n. Then L m (B 0 ) = {ε} nd L m (B n ) = \ ( pn ), f. Fig. 6 for utomt B 0, B 1, nd B 2. We ssume tht the stte sets X i nd X j re disjoint for ny i j. For n 1, we uild the NFA A n = (Q n, {}, δ n, Q i,n, F n ) s nondeterministi union of the DFAs B 0, B 1,..., B n. The NFA A 2 is depited in Fig. 6. Formlly, Q n = n k=0 X k, δ n (i k, ) = γ k (i k, ), Q i,n = n k=0 X i,k, nd F n = n k=0 X m,k. The numer of sttes of A n whih hs een estimted y Bh nd Shllit [19] to e n 2 ln n = O(n 2 ln n). The mrked lnguge of A n is L m (A n ) = \ ( pn# ) +, where p n # = Π n i=1 p i. Indeed, for m 1, string m is mrked y A n if nd only if there is p i {p 1,..., p n } suh tht m mod p i 0. Thus, the shortest string tht is not mrked y A n is of length p n #. Therefore, the supreml prefix-losed sulnguge of L m (A n ) is the finite lnguge { pn# 1 }. We now show, using the fooling set tehnique [20], tht ny NFA mrking this lnguge requires t lest p n # sttes. Ft 13 (Fooling set tehnique): Let L Σ e lnguge, nd let S = {(x i, y i ) 1 i k} e set of pirs suh tht (i) x i y i L for 1 i k, nd (ii) if i j, then x i y j / L or x j y i / L, for 1 i, j k. Then ny NFA mrking the lnguge L hs t lest k sttes. Set S is lled fooling set for L. Let S = {( i, pn# i 1 ) 0 i p n # 1}. Then i+pn# i 1 = pn# 1 elongs to the lnguge { pn# 1 }. Thus, S stisfies item (i) of the fooling set tehnique. To show tht it lso stisfies item (ii), let ( i, pn# i 1 ) nd ( j, pn# j 1 ) e two elements of S. Without loss of generlity, we ssume tht i < j. Then j + p n # i 1 > p n # 1, whih implies tht j pn# i 1 does not elong to { pn# 1 }, i.e., it proves tht S stisfies item (ii). Thus, S is fooling set for the lnguge { pn# 1 } of size p n #. Therefore, ny NFA mrking the lnguge { pn# 1 } hs t lest p n # sttes. Sine p n # = e (1+o(1))n log n [21] is exponentil wrt n, hene not polynomil wrt the size of A n, there is no lgorithm tht would ompute n NFA for the lnguge { pn# 1 } in polynomil time. VI. CONCLUSION is 1 + n i=1 p i, A onsequene of the exponentil stte omplexity is tht ny lgorithm omputing DFA for inf O(K, Σ, P ) requires, in the worst se, exponentil time (nd exponentil spe to store it). Algorithm 1 further shows tht the exponentil time is suffiient. The lgorithm is thus optiml in the sense tht there is no symptotilly more effiient lgorithm. Conerning the NFA representtion, we showed tht even for unry lnguges, the lgorithm would need more thn polynomil time to ompute the result nd more thn polynomil spe to store it. This is in ontrst to heking whether the lnguge of n NFA is prefix losed, whih n e done in polynomil spe nd it is not known whether it n e done in polynomil time. ACKNOWLEDGMENT The uthor grtefully knowledges very useful suggestions nd omments of the nonymous referees. The upper ound on the stte omplexity of the opertion K K is due to n nonymous referee. REFERENCES [1] R. Cieslk, C. Deslux, A. S. Fwz, nd P. Vriy, Supervisory ontrol of disrete-event proesses with prtil oservtions, IEEE Trns. Automt. Control, vol. 33, pp , [2] F. Lin nd W. M. Wonhm, On oservility of disrete-event systems, Inform. Si., vol. 44, no. 3, pp , [3] P. J. Rmdge nd W. M. Wonhm, The ontrol of disrete event systems, Pro. of the IEEE, vol. 77, pp , [4] K. Rudie nd W. M. Wonhm, Think glolly, t lolly: Deentrlized supervisory ontrol, IEEE Trns. Automt. Control, vol. 37, no. 11, pp , [5] R. Kumr nd M. A. Shymn, Formule relting ontrollility, oservility, nd o-oservility, Automti, vol. 34, no. 2, pp , [6] C. G. Cssndrs nd S. Lfortune, Introdution to disrete event systems, 2nd ed. Springer, [7] J. Komend nd T. Msopust, Computtion of ontrollle nd ooservle sulnguges in deentrlized supervisory ontrol vi ommunition, 2016, sumitted, [8] J. Komend, T. Msopust, nd J. H. vn Shuppen, On distriuted omputtion of supervisors in modulr supervisory ontrol, in Int. Conferene on Complex Systems Engineering (ICCSE), 2015, pp [9] S. Lfortune nd E. Chen, The infiml losed ontrollle superlnguge nd its pplition in supervisory ontrol, IEEE Trns. Automt. Control, vol. 35, no. 4, pp , [10] K. Rudie nd W. M. Wonhm, The infiml prefix-losed nd oservle superlngunge of given lnguge, Systems Control Lett., vol. 15, no. 5, pp , [11] J. E. Hoproft nd J. D. Ullmn, Introdution to Automt Theory, Lnguges nd Computtion. Addison-Wesley, [12] M. Sipser, Introdution to the theory of omputtion, 2nd ed. Thompson Course Tehnology, [13] S. Ginsurg, Algeri nd Automt-theoreti Properties of Forml Lnguges. Amsterdm: North-Hollnd, [14] S. Yu, Q. Zhung, nd K. Slom, The stte omplexities of some si opertions on regulr lnguges, Theoret. Comput. Si., vol. 125, no. 2, pp , [15] G. Jirásková nd T. Msopust, On struturl property in the stte omplexity of projeted regulr lnguges, Theoret. Comput. Si., vol. 449, pp , [16] K. Wong, On the omplexity of projetions of disrete-event systems, in Pro. of WODES, Cgliri, Itly, 1998, pp [17] T. Cormen, C. Leiserson, R. Rivest, nd C. Stein, Introdution to Algorithms, 3rd ed. MIT Press, [18] T. Msopust, Complexity of verifying nonlokingness in modulr supervisory ontrol, 2016, sumitted. Preprint ville online t [19] E. Bh nd J. Shllit, Algorithmi Numer Theory, Volume I: Effiient Algorithms. MIT Press, [20] J.-C. Birget, Intersetion nd union of regulr lnguges nd stte omplexity, Inform. Proess. Lett., vol. 43, pp , [21] N. J. A. Slone, The on-line enylopedi of integer sequenes, Sequene A Aessed on Otoer 18, 2016.