Optimal Constructions for Active Diagnosis

Transcription

1 Optiml Construtions or Ative Dignosis Sten Hr 1, Serge Hddd 1, Trek Melliti 2, nd Sten Shwoon 1 1 LSV (CNRS & ENS Chn) & INRIA, Frne 2 IBISC (Univ Evry Vl-Essonne), Frne Astrt The tsk o dignosis onsists in deteting, without miguity, ourrene o ults in prtilly oserved system. Depending on the degree o oservility, disrete event system my e dignosle or not. Ative dignosis ims t ontrolling the system in order to mke it dignosle. Solutions hve lredy een proposed or the tive dignosis prolem, ut their omplexity remins to e improved. We solve here the tive dignosility deision prolem nd the tive dignoser synthesis prolem, proving tht (1) our proedures re optiml w.r.t. to omputtionl omplexity, nd (2) the memory required or the tive dignoser produed y the synthesis is miniml. We then ous on the dely etween the ourrene o ult nd its detetion y the dignoser. We onstrut memory-optiml dignoser whose dely is t most twie the miniml dely, wheres the memory required or dignoser with optiml dely my e highly greter ACM Sujet Clssiition F.4.3 Forml Lnguges Keywords nd phrses Dignosis, Control theory, Automt theory, Gmes Digitl Ojet Identiier /LIPIs.xxx.yyy.p 1 Introdution In monitoring disrete event systems, one o the entrl tsks is tht o dignosis: Given inite leled trnsition system A (lso lled plnt ) whose events re prtilly oservle, our tsk is to deide - sed on the strem o oservtion lels - whether or not prtiulr unoservle events, lled ults, hve ourred. More preisely, the system is onsidered k-dignosle i t most k events ter the ourrene o ult, the oservtion is suiient to detet tht ourrene with ertinty, i.e. ll possile system runs omptile with the prtil oservtion olleted so r re ulty. The system A is dignosle i there exists k 1 suh tht A is k-dignosle. As the system my e insuiiently oservle, or the oservtion not disriminting enough, dignosility veriition hs reeived onsiderle ttention sine the seminl pper y Smpth et l [13]; see lso [4, 3]. Those works onstrut dedited deterministi version o the originl plnt, so-lled dignoser; the sene o indeterminte yles in this uxiliry utomton is equivlent to dignosility. On the other hnd, one system hs een shown to e undignosle - in sense tht we will ormlize lter - severl tions n ollow, suh s omplete redesign o the system, or dding urther sensors to enhne oservility. Smpth et l [12] hve initited dierent pproh, tht o tive dignosis: i the given plnt A is not dignosle, synthesize prtil-oservtion ontroller C tht ores A to sty within dignosle suset o its ehviors (or, equivlently, suh tht the ontrolled plnt A C is dignosle). The pir onsisting o the ontroller nd the dignoser is lled n tive dignoser. Lter, Chnthery nd Penolé [5] hve proposed plnning-sed pproh vi twin plnt onstrution. This work hs een supported y projet ImpRo ANR-2010-BLAN-0317 nd the Europen Union Seventh Frmework Progrmme [FP7/ ] under grnt greement HYCON2 NOE. Sten Hr, Serge Hddd, Trek Melliti, nd Sten Shwoon; liensed under Cretive Commons Liense CC-BY Conerene title on whih this volume is sed on. Editors: Billy Editor, Bill Editors; pp Leiniz Interntionl Proeedings in Inormtis Shloss Dgstuhl Leiniz-Zentrum ür Inormtik, Dgstuhl Pulishing, Germny

2 2 Optiml Construtions or Ative Dignosis Our ontriutions We ollow the pproh o Smpth et l [12], ut vi dierent method sed on utomt nd gme theory. This llows us to improve the onstrution o dignosers nd moreover estlish omplexity results, whih were not treted in previous work: 1. We uild deterministi Bühi utomton tht epts the sulnguge o ininite unmiguous oservle sequenes, i.e. those tht re either (i) triggered y set o orret runs or (ii) triggered y set o ulty runs. Its size is upper-ounded y 2 O(n), where n is the numer o sttes, whih is etter thn ll previous onstrutions. In ddition we show the optimlity o our onstrution proving tht there is mily o systems or whih ny orresponding deterministi Bühi utomton must hve size in 2 Ω(n). 2. We then design Bühi gme, where winning strtegy yields n tive dignoser or the system, nd vie vers. We thus solve the tive dignosis prolem y deiding whether there exists winning strtegy, nd the synthesis prolem y giving n tive dignoser ssoited with positionl strtegy. The size o the tive dignoser is singly exponentil w.r.t. the size o the system, while tht o [12] is douly exponentil. We lso prove tht the deision proedure is EXPTIME-omplete nd tht the synthesis proedure is optiml w.r.t. the numer o sttes o the tive dignoser (still in 2 O(n) ). 3. We then study the dely etween ult nd its detetion y n tive dignoser. We irst Ω(n log(n)) present mily o systems or whih miniml-dely dignoser must hve 2 sttes. However, reining our erlier onstrution yields n tive dignoser with size 2 O(n), whose dely is t most twie the miniml possile dely. In ddition, we sketh the onstrution o miniml-dely tive dignoser with t most 2 O(n2) sttes. Orgniztion Setion 2 rells notions relted to dignosis nd tive dignosis. In Setion 3, we estlish the lower ounds relted to the omputtionl omplexity, the memory requirements nd the index. Setion 4.1 presents the onstrution o the deterministi Bühi utomton. Then in Setion 4.2, we solve the deision nd the synthesis prolems or tive dignosis. Ater tht, Setion 4.3 reines the synthesis prolem w.r.t. the dely. Setion 5 gives some perspetives o this work. A long version with ll proos is ville [7]. 2 The tive dignosis prolem Leled trnsition systems When deling with disrete event systems (DES) dignosis, systems re oten modeled using leled trnsition systems (LTS). So we deine LTS, their properties nd lnguges. Deinition 1. A leled trnsition system is tuple A = Q, q 0, Σ, T where: Q is set o sttes with q 0 Q the initil stte; Σ is inite set o events; T Q Σ Q is the set o trnsitions. We note q q or (q,, q ) T ; this trnsition is then sid to e enled in q. A run over the word σ = Σ ω i+1 is sequene o sttes (q i ) i 0 suh tht q i qi+1 σ or ll i 0, nd we write q 0 = i suh run exists. A inite run over w Σ is deined nlogously, nd we write q = w q i suh run ends t stte q. A stte q is rehle i w there exists run q 0 = q or some w. Deinition 2 (Lnguges o n LTS). Let A = Q, q 0, Σ, T e n LTS. The inite lnguge L (A) Σ o A nd the ininite lnguge L ω (A) Σ ω o A re deined y: L (A) = { w Σ q : q 0 w = q } L ω (A) = { σ Σ ω q 0 σ = }

3 S.Hr, S. Hddd, T. Melliti, nd S. Shwoon 3 An LTS A is live i or ny rehle stte there exists trnsition enled in tht stte. An LTS A is deterministi i or every pir q Q, Σ there is t most one q suh tht q q. For deterministi utomton we write T (q, ) = q i q q. Oservtions In order to ormlize prolems relted to dignosis, we prtition Σ into two disjoint sets Σ o nd Σ uo, the sets o oservle nd o unoservle events, respetively. Moreover, we distinguish speil ult event Σ uo. Let σ e inite word; its length is denoted σ. For Σ Σ, deine P Σ (σ) indutively y: P Σ (ε) = ε; or Σ, P Σ (σ) = P Σ (σ); nd P Σ (σ) = P Σ (σ) or / Σ. Write σ Σ or P Σ (σ), nd or Σ, write σ or σ {}. When σ is n ininite word, its projetion is the limit o the projetions o its inite preixes. This projetion n e either inite or ininite. As usul the projetion is extended to lnguges. P Σo will e more simply denoted y P. An LTS A is onvergent i L ω (A) Σ Σ ω uo = (i.e. no ininite sequene o unoservle events rom ny, rehle stte). When A is onvergent, then or ll σ L ω (A), one hs P(σ) Σ ω o. We shll ssume tht the system under dignosis is live nd onvergent. q 0 q 1 q 2 Exmple 3. Figure 1 shows live nd onvergent LTS with Σ o = {,, } nd Σ uo = {}. Dignosility Figure 1 An LTS. A inite (resp. ininite) sequene σ is orret i it elongs to (Σ \ {}) (resp. (Σ \ {}) ω ). Otherwise σ is lled ulty. An oservtion sequene my e the projetion o oth orret nd ulty sequene, hene miguous. Deinition 4 (miguous nd surely ulty sequene). Let A e n LTS, σ 1, σ 2 L ω (A) e two sequenes nd σ Σ ω o e n oservle sequene suh tht: (1) P(σ 1 ) = P(σ 2 ) = σ, (2) σ 1 is orret nd (3) σ 2 is ulty. Then σ is lled miguous nd the pir (σ 1, σ 2 ) is witness or the miguity o σ. Amiguous inite oservle sequenes re deined nlogously. A sequene σ P(L (A)) is surely ulty i P 1 (σ ) L (A) Σ Σ. Deinition 5 (Dignosility). Let k N. An LTS A is k-dignosle i: σ = σ σ L (A) σ Σo k P(σ) is surely ulty sequene, Furthermore, A is dignosle i there exists k suh tht A is k-dignosle. Our deinition o dignosility is slight vrition o the one given in [13]. Indeed the numer k ove is relted to oservle events while in ormer works, it is relted to ny kind o events. However or inite-stte onvergent systems (whih re the ones ddressed y oth works) the deinitions o dignosility oinide. Exmple 6. The LTS o Figure 1 is not dignosle sine the orret ininite tre ω nd the ulty ininite tre ω hve the sme projetion. Ative dignosility We suppose tht Σ o is prtitioned into susets Σ Σ o o ontrollle nd Σ u = Σ o \Σ o unontrollle tions. Intuitively, ontroller my orid suset o the ontrollle tions sed on the oservtions mde so r, therey restriting the ehviour o A.

4 4 Optiml Construtions or Ative Dignosis Deinition 7 (Controller). Let A e n LTS. A ontroller or A is mpping ont : P(L (A)) 2 Σ suh tht or ll σ, Σ u Σ uo ont(σ). The ontrolled LTS A ont = Q ont, q 0ont, Σ, T ont is deined y: Q ont is the smllest suset o Σ o Q suh tht 1. (ε, q 0 ) Q ont ; 2. (σ, q) Q ont ont(σ) q q implies (P(σ), q ) Q ont. q 0ont = (ε, q 0 ) ((σ, q),, (σ, q )) T ont i q q ont(σ) σ = P(σ) In the dignosis rmework, the gol o our ontroller is to mke the system dignosle, nd to perom dignosis. However, one requires tht the ontrol nnot introdue dedloks. Deinition 8 (Pilot nd Ative Dignoser). Let A e n LTS. We ll h = ont, dig pilot or A i ont is ontroller nd dig is mpping rom P(L (A ont )) to {, }. Moreover, h is lled n tive dignoser i: 1. A ont is live; 2. P(L ω (A ont )) does not ontin ny miguous sequene; 3. dig(σ) = i nd only i σ is surely ulty sequene or σ P(L (A ont )). For k 1, we sy tht h is k-tive dignoser, i or ll σ = σ σ L (A ont ) with σ Σo k, dig(p(σ)) =, i.e. every ult is dignosed ter t most k oservtions. The miniml k suh tht h is k-tive dignoser is lled the dely o h. We ll A (k-)tively dignosle i (k-)tive dignoser exists, nd the miniml suh k the index o A. Exmple 9. In the LTS o Figure 1, ssume tht Σ = {, }. Let h n = ont n, dig, with n 1, e the pilot deined y: ont n (σ n ) = {,, } or σ Σ nd ont n (σ) = Σ otherwise; dig(σ) = i σ Σ oσ o. Then h n is n tive dignoser with dely n + 2. Notie tht n tive dignoser does not neessrily hve inite dely. For instne, in Figure 1, there is n tive dignoser tht dmits the sequene 2 3 nd is not n k-tive dignoser or ny k. However, we will see tht i A is tively dignosle, there does exist k-tive dignoser (or some k). We ome k to this point in Setion 4.3. We re now in position to ormlly stte the relevnt prolems or tive dignosis. Let A e live nd onvergent LTS with initely mny sttes. We re interested in: the tive dignosis deision prolem, i.e. deide whether A is tively dignosle; the synthesis prolem, i.e. deide whether A is tively dignosle nd in the positive se uild n tive dignoser. the miniml-dely synthesis prolem, i.e. deide whether A is tively dignosle nd in the positive se uild n tive dignoser with miniml dely. We introdue the notion o stte-sed pilot s inite representtion o n tive dignoser. Deinition 10 (stte-sed pilot). A stte-sed pilot C = B, ont C, dig C onsists o deterministi LTS B = Q, q0, Σ o, T nd lellings ont C, dig C : Q 2 Σ {, }, suh tht or ll q Q, Σ u Σ uo ont C (q). The pilot h C = ont, dig ssoited with C is given y ont(σ) = ont C (q) nd dig(σ) = dig C (q) or ll σ P(L (A)), where q is the unique stte suh tht q0 σ = q. Exmple 11. Figure 2 shows stte-sed pilot or the LTS o Figure 1. Oserve tht there is n outgoing trnsition rom the rightmost stte (to ulil the lnguge inlusion requirement) ut is disled in this stte (in order to implement the tive dignoser h 1 ).

5 S.Hr, S. Hddd, T. Melliti, nd S. Shwoon 5 Σ, Σ, Σ\{}, Figure 2 A stte-sed pilot.,,, l 0 l 1 l 2 l n 1, l n l n+1 d,,, q 0 q 1 q 2 q n 1, q n, d q n+1, r 0 r 1 r 2 r n 1,,,, r n d r n+1 Figure 3 An LTS A n with Σ o = {,,, d}, Σ = {, d} used in Theorem Lower ounds We irst estlish tht the tive dignosis deision prolem is EXPTIME-hrd. The proo [7] relies on redution rom sety gmes with imperet inormtion [1]. Theorem 12 (hrdness). The tive dignosis deision prolem is EXPTIME-hrd. The next theorems ous on the memory required or synthesis prolems relted to tive dignosis. We strt with the lnguge o unmiguous sequenes o n LTS. Deinition 13 (Bühi utomton). A Bühi utomton over Σ is tuple B = B, F, where B = S, s 0, Σ, δ is n LTS suh tht S is inite, nd F S n eptne ondition. A run (q i ) i 0 is epting i q i F or ininitely mny vlues o i. The lnguge L(B) onsists o ll words in L ω (B ) or whih there exists n epting run. A Bühi utomton is lled deterministi (live) i its underlying LTS is. Theorem 14 (lower ound or determiniztion). There exists mily (A n ) n 1 o LTS with the size o A n in O(n) suh tht ny deterministi Bühi utomton reognizing the unmiguous sequenes o A n hs t lest 2 n sttes. The mily o LTS (A n ) n 1 is depited in Figure 3. During the n irst steps ult n our leding to the upper (resp. lower) rnh o the LTS when ollowed y (resp. ). However the orresponding oservle sequene eomes deinitively miguous i n steps lter the LTS perorms d (resp. ). So ny deterministi utomton should led to dierent sttes when reding two dierent words o length n. With n pproprite hoie o ontrollle events, this mily lso provides lower ound or stte-sed tive dignoser.

6 6 Optiml Construtions or Ative Dignosis Theorem 15 (lower ound or pilots). There exists mily (A n ) n 1 o tively dignosle LTS with the size o A n in O(n) suh tht the LTS o ny stte-sed pilot C, where h C is n tive dignoser or A, hs t lest 2 n sttes. We shll now see tht the lower ound is even higher when one tries to minimize the ult-detetion dely. The LTS A n o Figure 4 ontins the oservle (nd unontrollle) oservtion sequene π(1)... π(n), where π is permuttion. Suh sequene is miguous sine ult my hve ourred eore ny oservle event. To remove miguity with miniml dely (i.e. n + 2) n tive dignoser must disllow t time n + i ll events in B exept π(i) tht ores the potentil ulty sequene to reh stte s where only is possile. Thus, ny tive dignoser or A n must rememer the permuttion π. Theorem 16 (miniml-dely dignoser). There exists mily (A n ) n 1 o (n)-tively dignosle LTS (or some untion ) with O(n) sttes suh tht the LTS o ny sttesed pilot C, where h C is n (n)-tive dignoser or A, hs t lest n! sttes. Note tht in Figure 4 the lphet size depends on n; however Theorem 16 lso holds or ixed-size lphet [7]. While the previous exmples exhiit n index liner w.r.t. the size o the LTS, this index my e exponentil in the worst se (nd no more s shown in the Setion 4). An exmple or this is shown in [7]. Theorem 17 (lower ound or index). There exists mily (A n ) n 1 o tively dignosle LTS with O(n) sttes suh tht the index o A n is t lest 2 n. 4 Size-Optiml Controller 4.1 Chrteriztion o unmiguous sequenes In this setion, we hrterize the ininite unmiguous sequenes in n eiient wy. Fix inite-stte live, onvergent LTS A = Q, q 0, Σ, T or the rest o the setion. We uild Bühi utomton B = (B, F ) tht epts the unmiguous oservtion sequenes. Sine B is the se o the tive dignoser onstruted in Setion 4.2, we wnt B to e deterministi. A potentil proedure or otining deterministi utomton epting unmiguous sequenes is s ollows: First, uild non-deterministi Bühi utomton tht epts oservle sequenes explinle y oth orret nd ulty sequene. This leds to s 1 2 n A, B \ { 1 } q 1 q 2 A, B \ { 2 } q n A, B \ { n } 1 2 n A = { 1,..., n } B = { 1,..., n } p 1 p 2 p n r A A A A B Figure 4 An LTS A n with Σ o = A B {}, Σ = B whose miniml-dely tive dignoser requires t lest O(n!) sttes.

7 S.Hr, S. Hddd, T. Melliti, nd S. Shwoon 7 qudrti low up w.r.t. the size o A. Then, determinize it y the Sr proedure [11], yielding deterministi Rin utomton, nd omplement it so it epts the unmiguous sequenes. However, we now provide the onstrution o simpler nd smller deterministi Bühi utomton. More preisely, the utomton tht we uild hs the ollowing properties: B is deterministi; B reds the oservle sequenes o A, i.e. L ω (B ) = P(L ω (A)); B epts extly the unmiguous oservtion sequenes. We irst give some intuition out the wy B works. Its sttes re triples U, V, W, where U, V, W Q. The sttes in U represent sttes rehle y non-ulty tres in A, wheres V W re sttes rehle y ommitting ult. Let σ = Σ ω o e n oservtion sequene. An miguous preix o σ will led to stte in whih oth U nd V W re non-empty, nd i σ is miguous, then its run will eventully remin in suh sttes orever. Unortuntely, the reverse implition is not true, s the exmple rom Figure 1 shows: every inite preix o the sequene ω is miguous, ut ω is not. In order to distinguish miguous sequenes rom those tht merely hve ininitely mny miguous preixes, V nd W ssume dierent untions: W represents wthlist, initilly empty. Suppose tht the oservtion 1... j, or some j, orresponds to some ulty exeution. Then we put the stte rehle y tht ulty exeution into W nd tre its suessor sttes there while mking urther oservtions. I W never eomes empty, then indeed there exists ulty element o P(σ) in L ω (A). On the other hnd, i some oservtion j, or j > j, is impossile in ll sttes o W, then we n onlude tht no ult hs ourred eore j. In the mentime, V serves s witing room : it stores sttes tht n e rehed y ulty sequenes where the ult hs ourred etween oservtions j nd j. When W eomes empty, those sttes re shited rom V to W to orm the new wthlist. Let S S, Σ o, nd L Σ uo e lnguge o unoservle tions. We denote δ L (S, ) := { q Q q S, w L : q w = q }, nd introdue the revitions δ n or L = (Σ uo \ {}) (non-ulty exeutions), δ or L = Σ uoσ uo (ulty exeutions), nd δ or L = Σ uo (ritrry exeutions). We n now stte the orml onstrution o B = S, s 0, Σ o, δ, F s ollows: S = 2 Q 2 Q 2 Q \ {,, } nd s 0 = {q 0 },, ; F = {, S 1, S 2, S 1, S 2, S 1, S 2 Q }; or s = U, V, W S nd Σ o suh tht δ (U V W, ), let := δ (U, ) δ (V, ); then { δn (U, ),, i W = ; δ(s, ) = δ n (U, ), \ δ (W, ), δ (W, ) otherwise. Oserve tht disregrding the eptne ondition, the sequenes red y B extly orrespond to oservle sequenes o A, i.e. P(L ω (A)). Theorem 18. A sequene o oservtions σ Σ ω o is epted y B i it is unmiguous. Exmple 19. Figure 5 shows the result o the onstrution on the system rom Figure 1. Sine ll non-empty sets re singletons we hve represented them y their item. Notie tht ny sequene ending in ω is miguous in Figure 1 nd hene not epted in Figure 5. On the other hnd, e.g., sequene ω is epted: while every preix i, or i 1, is miguous, we lwys know ter i+1 oservtion tht no ult hs ourred eore the i-th oservtion.

8 8 Optiml Construtions or Ative Dignosis 1,, 1,, 3 1, 3, 1,, 2 1, 2,,, 3 Figure 5 Bühi utomton resulting rom Figure 1; epting sttes hve doule rmes. We riely disuss the reltionship o our determiniztion onstrution with other stndrd onstrutions in dignosis nd utomt theory. In [16], dignosility o n LTS A is deided y uilding two utomt: one is modiition o A tht epts the projetions ll non-ulty sequenes, the other epts the projetions o ll ulty sequenes, rememering whether ult hs ourred in the urrent stte. The ross produt o these two is non-deterministi Bühi utomton o size 2n 2 (or Q = n) tht epts ll miguous sequenes. A diret determiniztion [11] o tht ross produt would yield Rin utomton o size 2 O(n2 log n). However, given tht the ross produt is wek in the sense tht ll its strongly onneted omponents re either ully epting or ully non-epting, one ould pply the rekpoint onstrution o Miyno nd Hyshi [9] to otin deterministi Bühi utomton o its omplement lnguge, o size 3 2n2. Our onstrution, while similr in spirit to tht o [9], is more eiient thn tht: or rehle Bühi stte U, V, W S, ny LTS stte q Q my or my not pper in U, nd it my pper in t most one o V or W, ut not in oth. Thus, the numer o rehle sttes in B is ounded y 2 n 3 n = 6 n = 2 O(n). Theorem 14 shows tht n exponentil lowup in n is unvoidle in generl, i.e. our onstrution is optiml up to onstnt tor in the exponent. 4.2 Synthesizing the ontroller We simultneously solve the deision nd synthesis prolems. As eore, we ix n LTS A = Q, q 0, Σ, T. We shll try to onstrut stte-sed pilot C suh tht h C is n tive dignoser or A. The onstrution sueeds i A is tively dignosle. Aording to Deinition 8, the min hllenges in uilding n tive dignoser re to ensure tht (i) the ontrolled system remins live, (ii) the ontroller exludes the miguous sequenes, nd (iii) dignosis inormtion is provided. For this, we introdue Bühi gmes. Deinition 20 (gme). A gme G (etween two plyers lled Control nd Environment) is tuple V C, V E, E, v 0, V F, where V C, V E re the verties owned y Control nd Environment, respetively; V G denotes ll verties, nd v 0 V C is n initil vertex. E V G V G re direted edges suh tht or ll v V C there exists some (v, w) E, nd V F V G is winning ondition. A ply is untion ρ: N V G suh tht ρ(0) = v 0 nd ρ i, ρ i+1 E or ll i 0; we ll ρ k := ρ(0) ρ(k), or some k 0, prtil ply i ρ(k) V C, nd set stte(ρ k ) := ρ(k). We write Ply (G) or the set o prtil plys o G. A ply ρ is lled winning (or Control) i ρ(i) V F or ininitely mny i. Deinition 21 (strtegy). Let G = V C, V E, E, v 0, V F e gme. A strtegy (or Control) is untion θ : Ply (G) V G suh tht stte(ξ), θ(ξ) E or ll ξ Ply (G). A ply ρ dheres to θ i ρ(i) V C implies ρ(i + 1) = θ(ρ i ) or ll i 0. A strtegy is lled winning i every ply ρ tht dheres to θ is winning. A positionl strtegy is untion

9 S.Hr, S. Hddd, T. Melliti, nd S. Shwoon 9 1,, 1,,, {, } 1,,, {,, } 1,,, 1,, 3 1,,, 1,,, {, } 1,,, 1,, 2 Figure 6 Exerpt o the Bühi gme or Exmple 1. θ : V C V G suh tht v, θ (v) E or ll v V C ; we ll θ winning i the strtegy θ with θ(ξ) = θ (stte(ξ)) is winning. In the gme tht we hve deined, ply n only e stuk in stte o Environment. Thus we do not onsider inite mximl plys or deining the winning strtegies o Control. Let B = B, F, with B = S, s 0, Σ o, δ, e the deterministi Bühi utomton onstruted rom A in Setion 4.1. We shll tke B s the LTS omponent o C. To determine ont C, we onstrut Bühi gme sed on B. The ojetive o Control is to otin n epting run y suitly restriting the possile tions, nd ny winning strtegy will e suitle ndidte or ont C. Intuitively, round o the gme is plyed s ollows: 1. Control restrits the set o possile tions to Σ. 2. Environment hooses n tion Σ to determine the next stte o B. The hoies o Control re sujet to some restritions. Indeed, eh stte s = U, V, W represents Control s knowledge out the urrent potentil sttes o A. To ensure tht the ontrolled system remins live, Σ must not use dedloks in ny stte rehle y unoservle events rom U V W. Also, Control nnot prevent the unontrollle events. So we deine the dmissile sets nd the gme s ollows. Deinition 22 (dmissile tion set). Let s = U, V, W e stte o B. We ll Σ Σ o dmissile or s i (i) Σ u Σ nd (ii) or ll sttes q o A with q = w q or some q U V W nd w Σ uo, there exists Σ nd q Q with q q. The dmissile sets or s re denoted dm(s). Deinition 23 (ontroller-synthesis gme). Let B = S, s 0, Σ o, δ, F e Bühi utomton. We denote G(B) the gme V C, V E, E, s 0, F, where V C = S, V E = (S 2 Σo ) (S Σ o ), nd E = E 1 E 2 E 3, where E 1 = { s, s, Σ s S, Σ dm(s) }; E 2 = { s, Σ, s, s S, Σ }; E 3 = { s,, s δ(s, ) = s }. The set E 3 is only introdued to reord the sequene o oservle tions tht our during ply. Furthermore Environment n e stuk in vertex o E 3 mening tht the tion hosen y Environment does not orrespond to possile ehvior o the system. Exmple 24. Figure 6 depits n exerpt o the gme or Exmple 1. In the initil stte, there re three possile dmissile sets, ll inluding, the unontrollle oservle tion. {} is not n dmissile set s it loks the system. I Environment hooses tion, it immeditely loses sine is not possile initilly even ter ult.

10 10 Optiml Construtions or Ative Dignosis We n now ddress the deision nd synthesis prolems. The next theorem is sed on the ollowing: (1) Bühi gmes n e solved in polynomil time, (2) positionl winning strtegy n lwys e hosen or Control i it wins nd (3) there is tight orrespondene etween winning strtegies nd tive dignosers. Theorem 25. Let A e n LTS with n sttes nd m ontrollle tions. The tive dignosis deision nd synthesis prolems or A n e solved in 2 O(n+m) time. Moreover, i A is tively dignosle, then one n synthesize stte-sed pilot C with t most 6 n sttes suh tht h C is n tive dignoser or A. We riely disuss the reltionship o our onstrution with tht o [12]. There, n tive dignoser is uilt on the sis o powerset onstrution tht is similr to ours ut without splitting the possily ulty sttes into wthlist W nd witing room V. However, they then e the orementioned prolem o distinguishing sequenes with ininitely mny miguous preixes (like ω in Exmple 1) rom truly miguous sequenes (like ω ), whih they resolve y exmining eh yle o the utomton. Sine the numer o sttes in tht utomton is 3 n, 1 nd there n e exponentilly mny yles, this proedure is douly exponentil in n. Our onstrution is only singly exponentil in n. Using Theorems 12 nd 25, we get the ollowing orollry. Corollry 26. The tive dignosis deision prolem is EXPTIME-omplete. 4.3 Index nd witing time We ssume tht A is tively dignosle nd develop the onstrution o n tive dignoser with dely lose to the index o A, nd omputtionl omplexity still in 2 O(n). For simpliity, we denote the gme G(B) y G. Let G e ny gme. Given strtegy θ or G, we denote y Ply ω θ (G ) the set o plys tht dhere to strtegy θ, nd y R(θ) the suset o sttes o S tht re visited y ply o Ply ω θ (G ). We re now in the position to introdue the min onept o this setion, the witing time o strtegy: the mximl numer o sttes visited without enountering n epting stte. Deinition 27 (witing time). Let θ e strtegy or G. Then the witing time K(θ) is deined s sup( {k i k j ρ(k) S} i, j ρ Ply ω θ (G) F {ρ(k)} i k j = ) with the onvention sup( ) = 0. Oserve tht K(θ) my e ininite or non-positionl winning strtegy. However, it is inite nd stritly smller thn S (sine there is t lest one epting stte) or winning positionl strtegy. In t, or θ winning positionl strtegy, K(θ) n e omputed in liner time (with pproprite dt strutures) w.r.t. the size o G. In order to present it nd or susequent use, we introdue the ollowing nottion. Let s e stte o the Bühi utomton, Out(s) := { Σ o δ(s, ) is deined }. First one omputes, y inresing vlues, the miniml solution o the ollowing eqution system: { 0 i s R(θ) F ; V θ (s) = 1 + mx(v θ (δ(s, )) Σ Out(s) s.t. (s, Σ ) = θ(s)) i s R(θ) \ F. Then K(θ) = mx(v θ (s) s R(θ)). Denote y D(θ) the dely o the tive dignoser relted to strtegy θ. Lemm 28 shows tht K(θ) provides useul inormtion out D(θ). 1 This is the result when only one ult type is onsidered; [12] tully provides or severl ult types, whih we omit here or ske o simpliity.

11 S.Hr, S. Hddd, T. Melliti, nd S. Shwoon 11 Lemm 28. Let θ e strtegy or gme G with inite witing time. Then: 1 + K(θ) D(θ) 1 + 2K(θ) Intuitively, the upper ound is potentilly due to ult stying in the witing room o B or t most K(θ) steps, then in the wthlist or t most K(θ) + 1 steps. The lower ound is due to the t tht long surun with non-empty wthlist, possile ult ould hve ourred eore this surun. Deine K A = min(k(θ)), where θ rnges over the winning strtegies or G. Sine positionl suh strtegy exists, we know tht K A is inite nd elongs to 2 O(n). Let us note D A = min(d(θ)) the index o A. The ollowing orollry provides tight rme or D A nd shows tht the index is in 2 O(n). Corollry 29. Let A e tively dignosle. Then: 1 + K A D A 1 + 2K A Let us ompute n tive dignoser or, equivlently, strtegy θ tht hieves K(θ) = K A. To this im, we introdue mily o gmes {G i } i N deined s ollows. The set o verties o G i re: V Gi = {v j v V G 0 j i} {lost} where the suset o verties owned y Control re {v j v V C 0 j i} {lost}, the initil vertex is s 0 0, nd the set o epting sttes re {s 0 s F }. Its set o edges E = E 1 E 2 E 3 is deined y: or ll j i, v j, w j elongs to E 1 i v, w elongs to E 1 ; or ll j i, v j, w j elongs to E 2 i v, w elongs to E 2 ; or ll j i, s, j, s 0 E 3 i s,, s E 3 nd s F ; or ll j < i, s, j, s j+1 E 3 i s,, s E 3 nd s / F ; s, i, lost elongs to E 3 i s,, s elongs to E 3 nd s / F ; lost, lost elongs to E 3 nd there is no other edge. Gme G i hs the ollowing properties: n ininite ply either ends up in lost or visits the epting sttes ininitely oten, with t most i visits o the set {v j v S \ F, 0 j i} etween two visits o epting sttes. The ollowing lemm reltes strtegies in G nd G i. Bsed on it n eiient omputtion o n optiml strtegy w.r.t. K(θ) n e perormed. Lemm 30. There is winning strtegy θ in G with K(θ) i i there is winning strtegy θ i in G i. Moreover, in the positive se, θ n e hosen to e positionl. Theorem 31. I A is tively dignosle, there exists positionl strtegy θ tht ulills K(θ) = K A. Moreover, suh strtegy n e omputed in 2 O(n). This onstrution represents resonle trdeo, sine, due to Theorem 16, n tive dignoser tht relizes dely equl to the index o A my need to e muh lrger, i.e. 2 Ω(n log(n)). We sketh the onstrution o ontroller with miniml dely one one knows tht the system is tively dignosle. One itertively uilds sety gme G i prmetrized y inresing vlues o i. A ontroller stte o this gme is deined y (U, d) where U is the set o sttes rehed y orret sequene while d ssoites with every stte s rehed y ulty sequene durtion d(s) i + 1 sine the ourrene o the erliest ult tht would led to s. As in the previous gmes the ontroller selets suset o oservle tions letting the environment selet n tion mong them. The im o the ontroller is to void sttes with some d(s) = i + 1. The irst i or whih G i hs winning strtegy is the index nd the winning strtegy yields n tive dignoser with miniml dely. Oserve tht sine the index is ounded y 2 O(n), in the worst se the inl gme hs 2 O(n2) sttes.

12 12 Optiml Construtions or Ative Dignosis 5 Conlusion nd Perspetives We hve developed n tive-dignosis method or inite-stte systems nd shown it to e optiml w.r.t. severl riteri. For instne, our work llows to minimize the dely etween the ourrene o ult nd its detetion. In generl, striving or miniml dely my led to n overly restritive ontroller, e.g., in Figure 1 the ontroller ould ompletely orid the tion. We hve thereore undertken work to llow or prmetrized tive dignosis, whih onstruts the most permissive ontroller tht respets user-speiied dely. This work, not inluded here or spe resons, is ontined in the long version [7]. Future work hs some reserh leds to ddress. First, it remins to determine the preise memory requirements or the miniml-dely dignoser sine we showed tht it lies etween 2 Θ(n log(n)) nd 2 Θ(n2). Seond, the ontrol or tive dignosis ould e reined into se ontrol, i.e. one tht does not enourge the ulty ehviours. Lst we im t ddressing ininite-stte systems or systems with quntittive etures, s or pssive dignosility in pushdown systems [10], Petri nets [2], timed [17, 15] nd proilisti systems [14]. Reerenes 1 D. Berwnger nd L. Doyen. On the power o imperet inormtion. In Pro. FSTTCS, volume 2 o LIPICS, Bnglore, Indi, M.P. Csino, A. Giu, S. Lortune, nd C. Setzu. Dignosility nlysis o unounded Petri nets. In CDC09: 48th IEEE Con. on Deision nd Control, pges , C. G. Cssndrs nd S. Lortune. Introdution to Disrete Event Systems - Seond Edition. Springer, F. Cssez nd S. Tripkis. Fult dignosis with stti nd dynmi oservers. Fundment Inormtie, 88: , E. Chnthery nd Y. Penolé. Monitoring nd tive dignosis or disrete-event systems. In Pro. SeProess 09, A. Cimtti, C. Peheur, nd R. Cvd. Forml veriition o dignosility vi symoli model heking. In Proeedings o IJCAI, pges Morgn Kumnn, Sten Hr, Serge Hddd, Trek Melliti, nd Sten Shwoon. Optiml onstrutions or tive dignosis. Reserh Report LSV-13-12, ENS Chn, Septemer R. Küsters. Memoryless Determiny o Prity Gmes, pges LNCS S. Miyno nd T. Hyshi. Alternting inite utomt on ω-words. Theoretil Computer Siene, 32: , C. Morvn nd S. Pinhint. Dignosility o pushdown systems. In Proeedings o the Hi Veriition Conerene, LNCS 6405, S. Sr. On the omplexity o omeg-utomt. In FOCS, pges IEEE, M. Smpth, S. Lortune, nd D. Teneketzis. Ative dignosis o disrete-event systems. IEEE Trnstions on Automti Control, 43(7): , July M. Smpth, R. Sengupt, S. Lortune, K. Sinnmohideen, nd D. Teneketzis. Dignosility o disrete-event systems. IEEE Trns. Aut. Cont., 40(9): , D. Thorsley nd D. Teneketzis. Dignosility o stohsti disrete-event systems. IEEE Trnstions on Automti Control, 50(4): , S. Xu, S. Jing, nd R. Kumr. Dignosis o dense-time systems using digitl loks. IEEE Trnstions on Automtion Siene nd Engineering, 7(4): , T-S. Yoo nd S. Lortune. Polynomil-time veriition o dignosility o prtilly oserved disrete-event systems. IEEE Trns. Automt. Contr., 47(9): , S. Hshtrudi Zd, R.H. Kwong, nd W.M. Wonhm. Fult dignosis in disrete-event systems: Inorporting timing inormtion. Trns. Aut. Cont., 50(7): , 2005.