On Verification of D-Detectability for Discrete Event Systems

On V erificatio n of D-Detecta b ility f or Discrete Ev ent Sy stems ⋆ Jiří Balun a and T omáš Masopu st a , b a Department of Computer Science, F aculty of Science, P alacky Univ ersity , 17. li stopadu 12, 771 46 Olomouc, Czechia b Institute of Mathematics of the Czec h Academy of Sciences Abstract Detectability has been introduced as a g eneralization of state-estimation properties of discrete e v ent sys tems studied in the literature. It asks whether the current and sub sequent states of a sys tem can be determined based on observ ations. Since, i n some applications, to e xactl y determine t he current and subseq uent states may be too stri ct, a relaxed notion of D-detectability has been introduced, distinguishing onl y certain pairs of states rather than all states. Four variants of D-detectability ha v e been defined: strong (per i odic) D-detectability and we ak (periodic) D-detectability . Deciding w eak (periodic) D-detectability is PS p ace -complete, while deciding strong (periodic) detectab ility or strong D -detectability is polyno mial (and we sho w that it is actually NL -complete). Ho we ver , to the bes t of our know ledg e, it is an op en problem whether there exis ts a polynom ial-time algorithm deciding strong periodic D-detectability . W e solv e this problem by sho wing that deciding strong per iodic D-detectability is a PSp a ce -complete problem, and hence there is no pol ynomial-time algorithm unless PS p a ce = P . W e further sho w that there is no po lynomial-time algorithm deciding strong periodic D-detec tability ev en f or sys tems with a single observ able e v ent, unless P = NP . Finally , we propose a class of sys tems fo r which the problem is tractable. Key wor ds: Discrete ev ent systems , finite automata, state estimation, detectability , v erifi cati on, comple xity 1 Introduction Detectability of discrete ev en t sy stems (DESs) mo deled by finite automata has been introduced by Shu et al. [22] as a generalization of other notions studied in the literature, in- cluding stability of Ozveren and Willsky [18] a n d ob ser v- ability o f Caines et al. [5] or of Ramadge [19]. An evidence that many p ractical problems can be formulated as the de- tectability pro b lem for DESs ha s be e n provided by Sh u and Lin [20]. Fur ther mo re, Lin [13] has shown that detectability is closely related to other im por tant pro per ties, such as ob - ser vability , d iagnosability , and opa city . Detectability is a state-es timation proper ty asking whether the cur rent and subsequent states of a DES can be deter - mined after a finite n umber of obser vations. Sh u et al. [ 22] hav e defined f our var iants of detectability: strong (p e r iodic) detectability and weak (per iodic) detectability . In their w o r k, they first studied detectability f or d eter ministic DESs, wh ich are DESs mod eled by deter ministic finite au tomata with a set of initial states. The mo tivation f or a set of initial s tates ⋆ An e xtended abstract of this w ork was presented at the W ork - shop on Discrete Ev ent Systems W ODES 2020 [3]. Cor responding author: T . Masopus t, tel. +4202220907 85, fax +4205412186 57. Email addr esses: jiri.balun01@upol. cz (Jiří Balun), masopust@m ath.cas.cz (T om áš Ma sopust). rather than a single initial state results fr om th e ob ser vation that it is often unknown wh ich state the sys tem is initially in. They proposed an exponential algor ithm f or deciding de- tectability o f a deter ministic DES based on the c o mputation of an obser ver . Sh or tly after, Shu an d Lin [20] extended the problem to nond eter ministic DESs ( DESs modeled by general no ndeter ministic finite autom ata) and d esigned an algor ithm d eciding strong ( per iodic) detectab ility of non de- ter ministic DESs in polynomial time. Deciding s tron g ( p e- r iodic) d etectability was later sh own NL-comp lete [14], that is, the prob lem is efficiently solvable on a parallel compu ter . The comple xity of d eciding weak (per iodic) detectab ility has been in ves tigated o nly recently . Zhang [26] h as shown th at deciding weak (p er iodic) detectability is PSp ace -complete and that it r e mains PSp ace -hard ev en for d e ter ministic DESs with all ev ents obser vable. Masopust [1 4] fur ther s treng th- ened these results by proving the same c o mplexity for struc- turally “simp lest ” deadlock -free DESs tha t are modeled by deter ministic finite autom ata withou t n on-tr ivial cycles. Since the r equirement in the d efinition of d etectability to e xactly deter min e the cur ren t and subsequent states after a finite number of ob ser vations may b e too str ict in som e ap- plications, Shu and Lin [ 20] relaxed the notion o f d e te c t- ability to a so-called D-detectability proper ty . The id ea b e - hind th e relaxation is to d istinguish on ly cer ta in p a ir s of Prep rint submitted to May 19, 2020 states rather than all states of the system. F our varian ts of D-detectability hav e been defined: strong (per iodic) D-de- tectability and weak (per iod ic) D-detectab ility . The notion o f (D-)detectab ility has been extended in many directions. T o mentio n a fe w , Shu and Lin [ 21] extended strong (D-)d e te c tability to delay ed (D-)d etectability , moti- vated by discrete ev ent systems with d e lay s, an d de signed a polynomial-time algor ith m to check strong (D-)detectability f or d elay ed DESs. Zhan g and Giu a [2 7] h av e recently im- prov ed th e algor ithm for checking strong delay ed (D-)detect- ability . Th ey fur ther introduce d sev eral other notions of de - tectability , see Zhan g et al. [28] f or more details. Alv es an d Basilio [1] studied (D-)detectability f or d iscrete ev ent sys- tems w ith m u lti-channel commun ication netw ork s. Yin and Laf or tun e [25] e xamined the verification o f weak and strong detectability proper ties f or mod ular DESs, and show ed th at chec king both is PSp a ce -h a rd. The ex act com ple xities of these tw o problem s hav e recently been resolv ed by Masopu st and Y in [17]. They are, respectiv ely , PSp a ce -comp lete and ExpSp a ce -co m plete. W e ref er the reader to Hadjicostis [8] f or the latest dev elopm e nt of state-estimation pr oper ties. Since detectability is a special case o f D-detec tability , de- ciding D-detectability is at least as ha r d as deciding detect- ability . An immed iate consequence is th at the co mple xity of deciding wh ether a DES satisfies weak (per io dic) D-detec t- ability is PSp a ce -com plete. The case of strong D-detectability is similar to that of strong detectability . For strong (pe r io dic) detectability , Shu an d Lin [20] designed a detector that can decid e, in polynomial time, whether a DE S satisfies strong (per iod ic) detectability . They hav e f ur ther shown that their detector is also suitable f or decidin g strong D-d etectability . Con sequently , the com- ple xity of v er ifying whethe r a DES satisfies strong D-detect- ability is p olynomial; see Zhang and Giua [27] f or details on the algo r ithmic complexity . W e fur ther im p rov e th is re- sult by showing tha t decid in g whether a DES satisfies strong D-detectability is NL -co mplete (Theor em 1). Since NL is the class of pro blems th at can b e efficien tly parallelized, see Arora and Barak [2] for details, we obtain th at th e verifica- tion of strong D-detectability can be e fficiently verified on a parallel computer. Ho we v er, d eciding strong p eriodic D-detectability is mo re inv olv ed. Altho ugh the d etector -based technique p rovides a polynomial-time a lg or ithm to d ecide strong per io dic detect- ability , Sh u a nd Lin [ 20] g iv e an example that th is algor ithm does no t work for checking strong p er iodic D-detectability . They leav e the question of the existence of a polynomial- time algor ithm d eciding strong p er iodic D-detectab ility o f a DES op en. T o the be st of o ur know ledge, this question has n ot yet been answ ered in the literatur e. W e answer this question by showing that ther e do e s n o t exis t any algor ithm that would decide, in polynomial time, whether a DE S sat- isfies strong per io d ic D-detectability (Theo rem 3), un less P = PSp a ce . The question wheth er P = PSp a ce is a longstand- ing op en pro blem of compu ter science asking whe th er ev er y problem solv able in po lynomial space can also be solv ed in polynomial time. It is generally believ ed that it is not the case. In par ticu lar , Theorem 3 sh o ws th at the strong per i- odic D - detectability problem is PSp ace -complete. For mu- lated differently , th e result sa ys th at the technique based on the co mputation of the ob ser ver is in p r inciple optimal. No- tice that since NL is a s tr ict sub c lass of PSp ace , strong pe - r iodic D -detec tability is significan tly mo re complex than its non-p er iodic cou nter par t—strong D-detectability . W e fur ther show that strong p eriodic D-detectab ility is more complex than strong D-detectability ev en for sys tems h aving only a single o b ser vable ev ent. Namel y , we show th at strong per iodic D-detecta b ility canno t be v er ified in p olynomial time ev en for DESs th at hav e o nly a sin gle ob ser vable ev en t (Theor e m 4), unless P = NP . Finall y , we specify a class of systems for which decid - ing strong periodic D-detectability is in polynomial time. Namel y , we co nsider the class of sys tems mo deled by NF As where all cycles are in the f or m of self-loo ps and where there is n o nond eter ministic choice between a step changin g the state an d a step n o t changing the state under the same obser vation. These r e strictions are pur ely structur al, an d th e models a re called rpo DES , see Section 5 f or details. Our contr ibution s, compar ed with kno wn results, are sum- mar ized in T ab les 1 an d 2. 2 Preliminaries and Definitions F or a set A , | A | denotes the cardin ality of A and 2 A its pow er set. An a lp habet Σ is a finite non empty set o f ev ents. A strin g o ver Σ is a sequence of ev ents of Σ . Let Σ ∗ denote the set of all finite strin gs ov er Σ ; the empty str ing is deno ted by ε . F o r a str ing u ∈ Σ ∗ , | u | den otes its length. As usu al, the notation Σ + stands f or Σ ∗ \ { ε } . A nondeterministic finite a utomaton (NF A) o ver an alphab et Σ is a structu r e A = ( Q , Σ , δ , I , F ) , where Q is a finite set of states, I ⊆ Q is a set of initial states, F ⊆ Q is a set of marked states, and δ : Q × Σ → 2 Q is a tra n sition function that can be extended to th e dom ain 2 Q × Σ ∗ by induction. The lang u ag e recognized by A is the set L (A ) = { w ∈ Σ ∗ | δ ( I , w ) ∩ F , ∅ } . Equivalently , the transition f u nction δ is a relation δ ⊆ Q × Σ × Q , wh ere, f or instance, δ ( q , a ) = { s , t } denotes the tw o transition s ( q , a , s ) a n d ( q , a , t ) . The NF A A is deterministic (DF A) if it has a un ique initial state, i.e., | I | = 1, and no nondeter ministic transitions, i.e., | δ ( q , a ) | ≤ 1 f or ev er y q ∈ Q and a ∈ Σ . Th e DF A A is total if in e very state, a transition under ev er y e vent is defined, i.e., | δ ( q , a ) | = 1 f or ev er y q ∈ Q an d a ∈ Σ . F o r DF As, w e identify singleto ns with their elem ents and simply wr ite p instead of { p } . Spec ificall y , we wr ite δ ( q , a ) = p in stead of δ ( q , a ) = { p } . A discr et e ev ent sy stem ( D E S) is an NF A G with all states marked. Hen c e we simply wr ite G = ( Q , Σ , δ , I ) leaving 2 DESs r poDES s detectability D-detectability detectability D-detectability kno wn kno wn ne w kno wn kno wn ne w strong NL -c [14] in P [20] NL -c (Thm 1) NL -c [14] in P [20] NL -c (Thm 1 & Cor 9) we ak PSp a ce -c [26] PSp a ce -c — PSp a ce -c [14 ] PSp a ce -c — strong per iodic NL -c [14 ] ? PSp a ce -c (T hm 3) NL -c [14] ? NL -c (Thm 8) we ak periodic PSp a ce -c [26] PSp a ce -c — PSp a ce -c [14 ] PSp a ce -c — T able 1 Summary of kn o wn and ne w results f or DESs and r poDESs; ? means that the problem was open; results easil y der iva ble from e xisting results are also placed among kno wn r esults unary DESs unary rpoDESs detectability D-detectability detectability D-detectability strong NL -c [14] NL -c (Thm 1) NL -c (Cor 9) NL -c (Cor 9) we ak NL -c (Thm 6) NL -c (Thm 6) NL -c (Thm 6) NL -c (Thm 6) strong periodic NL -c [14] NP -c (T hm 4) NL -c (Thm 8) NL -c (T hm 8) we ak periodic NL -c (Thm 6) NP -c (Thm 6) NL -c (T hm 6) NL -c (Thm 6) T able 2 Summary of kno wn and new results f or DESs and r poDESs with a single observable e vent out the set of marked states. Additionally , th e alphab et Σ is par titioned into the set Σ o of observable events and th e set Σ u o = Σ \ Σ o of unob servable events . S tate-estimation proper ties are based on the obser vation of ev ents. The o bser vation is descr ibed by projections. The pro- jection P : Σ ∗ → Σ ∗ o is a mor ph ism define d by P ( a ) = ε f or a ∈ Σ \ Σ o , an d P ( a ) = a for a ∈ Σ o . The action of P on a s tr ing w = a 1 a 2 · · · a n , where a i ∈ Σ f or 1 ≤ i ≤ n , is to erase all e v ents fro m w that do not b elong to Σ o ; in par tic- ular , P ( a 1 a 2 · · · a n ) = P ( a 1 ) P ( a 2 ) · · · P ( a n ) . The de finition can readily be extended to infin ite str ings and langu ages. Shu and Lin [2 0] make the follo wing tw o reasonab le as- sumptions on the DES G = ( Q , Σ , δ, I ) that we ad opt: (1) G is deadlo ck free – it m e a ns that for ev er y state o f the sy stem, at least one ev ent can occur; f or m ally , for ev er y q ∈ Q , there exis ts σ ∈ Σ such th at δ ( q , σ ) , ∅ . (2) No lo op in G consists solely of un obser vable ev ents – f or ev er y q ∈ Q and ev er y w ∈ Σ + u o , q < δ ( q , w ) . W e point out that to v e r ify wh ether a sys tem satisfies these tw o proper ties is very easy . The violatio n of any of the p rop- er ties is of ten co n sidered a modelin g er ror. Moreov er, omit- ting the condition s does not change our results. The set of infinite sequences of ev en ts (or trajectories) gen- erated by the DES G is deno ted by L ω ( G ) . Given a set Q ′ ⊆ Q , the set o f all po ssible states after ob ser ving a strin g t ∈ Σ ∗ o is den o ted by R ( Q ′ , t ) = Ø w ∈ Σ ∗ , P ( w ) = t δ ( Q ′ , w ) . F or w ∈ L ω ( G ) , we denote the set of its pr efixes by P r ( w ) . 2.1 A Brief Complexity Review W e n o w b r iefly r evie w the basics of com ple xity theor y needed to un derstand the r esults. A decision problem is a y es-no question. A decision problem is decid able if there e xists a n algor ithm that can sol ve the problem. Co m ple xity theor y classifies de cidable pro blems to classes accord in g to the time or space an algo r ithm needs to sol ve the p roblem. The comp le xity classes we consider in th is pa per are NL , P , NP , and PSp a ce . Th ey denote the classes of problems that are solv able by a non deter ministic logarithmic - space, dete r min- istic pol ynomial- time, nonde ter ministic pol yno mial-time, and deter m in istic polynomial-sp a ce alg or ithm, respectivel y . The h ierarchy of classes is NL ⊆ P ⊆ NP ⊆ PSp a ce . Which of the inclusion s are str ict is a long standing ope n pro blem in compu ter science. The widely accepte d con jecture is that all inclusions are strict. How ev er, so far o n ly th e inclusion NL ⊆ PSp ace is known to be s tr ict. A decision problem is NL -comp lete (resp . NP -com p lete, PSp a ce -co m plete) if it belongs to NL (resp. NP , PSp ace ) and ev er y problem fro m NL (resp. NP , PSp ace ) can b e redu ced to it by a deter min- istic logar ithmic-spac e (resp . p o lynomial-time) algor ithm. 3 Definitions of the D-Detect ability Problems Shu and Lin [20] d efined D-detectab ility as a generalizatio n of detecta b ility by making the states that nee d to be distin- guished e xplicit. Let G = ( Q , Σ , δ, I ) be a DES, and let T s p e c ⊆ Q × Q be a relation on the set o f states of G . Th e relation T s p e c specifies 3 pairs of states tha t must b e d istinguished, an d is therefore called a specifica tion . The idea behin d the definition of D-de- tectability is to ensure that the p airs of states fro m T s p e c are distinguished after a finite number o f ob ser vations. W e now recall the definitions of the four variants of D-detectability . A DES G = ( Q , Σ , δ, I ) is str ong ly D-detectab le with resp ect to projection P : Σ ∗ → Σ ∗ o and a specificatio n T s p e c if, for all tr ajectories of the sys tem, th e pairs of states o f T s p e c can be distinguished in ev er y step of the sys tem after a finite number of ob ser vations. T h is is form a ll y d efined as follo ws: ( ∃ n ∈ N )( ∀ s ∈ L ω ( G ))( ∀ t ∈ Pr ( s )) | P ( t ) | > n ⇒ ( R ( I , P ( t )) × R ( I , P ( t )) ) ∩ T s p e c = ∅ . A DES G = ( Q , Σ , δ, I ) is w eakly D-detectable with respect to projection P : Σ ∗ → Σ ∗ o and a specificatio n T s p e c if, for some trajector ies of the sys tem, th e pa irs of s tates of T s p e c can be distinguished in ev er y step of the syst em after a finite number of ob ser vations. T h is is form a ll y d efined as follo ws: ( ∃ n ∈ N )( ∃ s ∈ L ω ( G ))( ∀ t ∈ Pr ( s )) | P ( t ) | > n ⇒ ( R ( I , P ( t )) × R ( I , P ( t )) ) ∩ T s p e c = ∅ . A DES G = ( Q , Σ , δ, I ) is str o ng ly periodica lly D -detectab le with respect to projection P : Σ ∗ → Σ ∗ o and a specification T s p e c if the pa irs of states of T s p e c can be per io dically d is- tinguished f or all trajectories of the system. Formally , ( ∃ n ∈ N )( ∀ s ∈ L ω ( G ))( ∀ t ∈ Pr ( s ))( ∃ t ′ ∈ Σ ∗ ) t t ′ ∈ P r ( s ) ∧ | P ( t ′ ) | < n ∧ ( R ( I , P ( t t ′ )) × R ( I , P ( t t ′ ))) ∩ T s p e c = ∅ . A DES G = ( Q , Σ , δ, I ) is weakly periodic a lly D-detectable with respect to projection P : Σ ∗ → Σ ∗ o and a specification T s p e c if the pa irs of states of T s p e c can be per io dically d is- tinguished f or some trajectories of the sys tem. Form ally , ( ∃ n ∈ N )( ∃ s ∈ L ω ( G ))( ∀ t ∈ Pr ( s ))( ∃ t ′ ∈ Σ ∗ ) t t ′ ∈ P r ( s ) ∧ | P ( t ′ ) | < n ∧ ( R ( I , P ( t t ′ )) × R ( I , P ( t t ′ ))) ∩ T s p e c = ∅ . 4 Res ults W e now d iscuss th e complexity o f dec id ing whether a DES satisfies D-detectability . As alrea dy poin ted o u t in the in- troductio n, the complexity of chec king wheth er a DES sat- isfies weak (per iodic ) D-detectability f ollow s directl y from the comp le xity of checking weak (per io dic) d etectability . In- deed, a polynomial space is sufficien t f or an alg or ithm based on th e in spection of states in th e ob ser v er and w ork s for all the D-detectability varian ts. Theref ore, d eciding weak (p e- r iodic) D-d e tectability is in PSp ace . On the othe r hand , de- tectability is a special case o f D-d etectability for T s p e c = Q × Q \ {( q , q ) | q ∈ Q } . Th erefore, d eciding weak (pe- r iodic) D-detectability is at least as har d as decidin g weak (per iodic) detectability . Since the latter is PSp ace -hard, so is the f or mer. 4.1 V erific a tion o f Str ong D-Detectability Shu an d Lin [2 0] designe d an algor ithm tha t verifies strong (per iodic) detectability in p olynomial time. T heir alg or ithm is based on the constru ction of a finite automaton called a detect or . Intuitiv ely , giv en a DES G , their detector G d et is constr u cted from G so that (i) the set of initial states of G d et is the set of all states o f G r e a chable from th e initial states of G un der str ings consisting o nly o f unobser vable ev ents, (ii) all the oth er states of G d et are one - or two-element sub sets of the set of states of G , an d (iii) the transition relation of G d et is con structed in th e similar way as that of the ob ser v er, but if the reached state X in th e obser ver consists of m ore than tw o states, then the detec tor G d et has sev er a l tr a nsitions each leadin g to a tw o-elemen t sub set of X , see Shu an d Lin [20] for details. Since th e states of the d etector a re on e- or tw o-elemen t subsets, their num ber is p olynomial. Shu and Lin [20] sho wed that a DES G satisfies strong (pe- r iodic) d etectability if and only if any state reachable fr o m any loop in G d et consists solely ( per iodically) of d istinguish- able states. They fu r ther prov ed that their algor ithm, respec- tiv ely the detector, w ork s for checking whe ther a DES sat- isfies strong D-detectability . T his in par ticular implies th at the co mplexity o f v er ifying wheth e r a DES satisfies strong D-detectability is polynomial. Z hang and Giu a [27] recently improv ed the algor ithmic complexity of this p roblem. W e no w discuss the compu tational comple xity of deciding strong D-de tec tability and show th a t it is a n NL - complete problem . Consequently , sinc e NL is the class of problems that can be efficiently parallelized, see Arora and Barak [2] f or d etails, ou r result show s th a t the question whethe r a DES satisfies strong D-d etectability can be efficiently verified o n a p arallel compu te r. Theorem 1. Deciding whether a DE S is strong ly D-detect- able is an NL -complete p r oblem. Proof. W e pr o ve memb ership o f th e prob lem in NL by giving a nondeter min istic logar ithmic-spac e algor ith m th at chec ks whether th e cond itio n do es not hold . Since NL is closed under co mplemen t, see Immer m an [10] or Szelepc- sényi [24] for deta ils, it show s that there is a no n deter minis- tic logarith mic-space algor ithm chec king wh e ther the co n- dition is satisfied. T o check that th e proper ty is no t satisfied, ou r NL algor ithm guesses tw o states of G d et , sa y x and y , wh ere y contains 4 s p r t x G a a a a a a ? Figure 1. The DES A constructed from G in the N L -hardness proof of Theorem 1 1 2 3 a b a a Figure 2. The DES G from Example 2 indistinguishable states, and v er ifies that ( i) y is rea chable from x , ( ii) x is r eachable from the initial state of G d et , and (iii) x is in a cycle, i. e ., x is reachab le fr o m x by a p a th having at least one transition. Notice that ou r algor ithm do es not constr u ct the detector G d et . It only stores a constant nu mber of states of G d et and compu tes the required tr a n sitions o f G d et on d emand. Therefore, our a lg or ithm do es not need more than a logarithm ic space. For more details how to check reachability in NL , we r e fer th e r eader to Masopust [15]. T o show NL -hard ness, we r educe the D A G non- r eachability problem , see Cho and Huyn h [ 6] for d etails: Giv en a directed acy clic graph G = ( V , E ) and tw o n odes s , t ∈ V , it asks whether t is not re a chable from s . From G , we con struct a DES A = ( V ∪ { x } , { a } , δ, s ) , wh ere x < V is a new state and a is an obser vable ev ent. For e very edge ( p , r ) ∈ E , we add the transition ( p , a , r ) to δ , and f or ev er y p ∈ V \ { t } , we add the transition ( p , a , x ) to δ . Moreov er , we add th e self- loop transitions ( x , a , x ) and ( t , a , t ) to δ . The c onstru ction is depicted in Fig. 1. Notice that A is deadlo ck -f r ee and has no unobser vable ev ents. Let the specificatio n T s p e c be d efined as T s p e c = { ( t , x ) } . W e no w sho w that t is not reachable fr om s in the g raph G if and only if the DES A is strongly D-de- tectable. If t is no t reachable fr om s in G , then, f or ev er y k ≥ | V | , δ ( s , a k ) = { x } . Therefore, A is strongly D-detect- able. If t is rea chable from s in G , then, for ev er y k ≥ | V | , δ ( s , a k ) = { t , x } . T herefore, A is not strongly D- detectable, because ( t , x ) ∈ T s p e c .  4.2 V erification of Str ong P eriod ic D - Detectability Although the d e tector -based technique leads to a p olynomi- al-time algor ithm decid ing wh ether a DE S satisfies strong per iodic detectability , Shu and Lin [20] hav e shown that th is algor ithm d oes no t work f or checking strong pe r io dic D-de- tectability . T o giv e the reader an idea of the d e tector -based polynomial-time algor ithm and of the p roblem why it d oes not w ork for checking strong p er iodic D-detectability , we slightly elabor ate the ex ample of Shu an d Lin [20]. Example 2. Let G = ( { 1 , 2 , 3 } , { a , b } , δ, { 1 , 2 , 3 } ) b e th e DES depicted in Fig. 2, where both ev en ts are ob ser vable. { 1 , 2 , 3 } { 1 , 3 } { 1 , 2 } { 2 , 3 } { 2 } { 3 } a a a a a a b b b a a Figure 3. Detector G d et constructed from the DES G of Example 2 { 1 , 2 , 3 } { 2 } { 3 } a b a a Figure 4. The observ er of the DES G from Example 2 Let the specification T s p e c = { ( 1 , 3 ) } . The detector G d et is depicted in Fig. 3. In G d et , we can immediately see th at G is not strong ly D-detectable, since there is an infinite path tha t goes per io dically thro ugh state { 1 , 3 } , v iolating thus s tron g D-detectability . From th e same infinite p ath going per iodi- cally throu gh the states { 1 , 2 } and { 1 , 3 } , the reade r could get an impression that G is strongly p e r iodically D-detectable. Ho we v er, this is not the case as can be seen from th e obser ver depicted in Fig. 4. There, there is an infinite tr ajectory a ω in state { 1 , 2 , 3 } that violates strong per iodic D-detecta b ility . ⋄ Shu and Lin [20] hav e left the question whether ther e exists a p o lynomial-time algor ithm de c iding strong per io dic D-de- tectability of a DES open . T o the best of our know led ge, this question has n ot yet b e en answered in the literature . W e answer this question in the sequel. W e distinguish tw o cases based on the numb er of obser vable ev ents in the sys tem: (i) The general case where the sys tem has two or more ob- ser vable ev ents; (ii) A special case where the sy stem has only a single ob- ser vable ev ent. The case of two or more ob servable events As pointed ou t abov e, the pr o blem whether a D E S satisfies strong o r weak (per iod ic) D-d e te c tability is in PSp ace . In this section, we sho w that decid ing strong p er iodic D-detect- ability is PSp ace -hard, and henc e PSp a ce -com plete. Conse- quently , th ere is no algorith m solving this p roblem in po l y- nomial time, unless P = PSp ace . Theorem 3. Deciding whether a DES is str o ng ly p eriodi- cally D-detectable is a PSp ace -complete p r oblem. Th e prob- lem is PSp ace -har d e ven if the DES has only tw o obser vable and no unob ser vable events. 5 Proof. Membe r ship in PSp ace follo ws f rom the in spection of states of the obser ver that are built on d emand [20,26,15]. T o show PSp a ce -h ardness, w e re duce the intersection em pti- ness pro blem. The pro blem is PSp ace -complete [7] an d asks, giv en a sequen ce A 1 , . . . , A n of total DF As o ver a commo n alphabet Σ w ith | Σ | ≥ 2, wh ether the lan guage ∩ n i = 1 L ( A i ) is empty? Without loss of generality , we may assume that Σ = { 0 , 1 } . From A 1 , . . . , A n , we constru ct a DES G that is strongly p er iodically D-detectable if and on ly if th e inter - section of the languages of A 1 , . . . , A n is empty . The m ain id ea of our proof is to co nstruc t G as a non- deter ministic u n ion o f the au tomata A 1 , . . . , A n together with n + 1 new states such that all and only these states are reachable at the same time if and only if the intersection is nonemp ty . In the case the inter section is empty , o nly a str ict subset of the new states can be reached at the same time. After reaching the new states, th e compu tation remains in the new states. The new states (up to o n e special state) form a cy cle, and hen ce, d u r ing any f ur ther com putation, the cur - rent states are perio dically r otated. This allo ws u s to make one of the new states per iodically in distinguishable from the special state, an d one p er iodically distinguishable. F or ma ll y , let A 1 , . . . , A n be to tal DF As o ver a comm on alphabet Σ , and let A i = ( Q i , Σ , δ i , q 0 , i , F i ) . Without loss of generality , we may assume th at the states of the DF As are p a ir wise disjoint. W e constr uct a DES G as a no n de- ter ministic union of the auto mata A i , i.e., G con tains all states and transitions of ev er y A i , and we add n + 1 new states q − , q + 1 , . . . , q + n and sev eral new transition s u nder a new ev ent a < Σ as dep ic ted in Fig. 5. Namel y , f o r i = 1 , . . . , n , we add the transition ( q , a , q − ) for ev er y no n -marked st ate q ∈ Q i \ F i , and the transition ( q , a , q + i ) f or ev e r y marked state q ∈ F i . Further more, we add the self-loop ( q − , σ , q − ) f or ev er y σ ∈ Σ ′ = Σ ∪ { a } . Finally , we create a cy- cle on the states Q + = { q + 1 , . . . , q + n } by add ing, for ev er y σ ∈ Σ ′ , the transitions ( q + i , σ , q + i + 1 ) , f or 1 ≤ i < n , and th e transition ( q + n , σ , q + 1 ) . T he set of initial states o f G is th e set I = { q − , q 0 , 1 , . . . , q 0 , n } of initial states o f the a utomata A i plus the new ly ad ded state q − . The alph abet of G is Σ ′ = Σ ∪ { a } = { 0 , 1 , a } , all ev ents of which a r e obser vable. T o show that the p roblem is PSp a ce -har d for | Σ ′ | = 2, we modify G by enco ding the ev en ts of Σ ′ in binar y as follo ws . Let b be a new ev ent, and let f : Σ ∗ → { a , b } ∗ be a mor ph ism defined by f ( 0 ) = ba and f ( 1 ) = b b . No w , in G , we replace each transition t = ( p , 0 , q ) with tw o transitions ( p , b , p t ) and ( p t , a , q ) , whe r e p t is a n ew state. Similarl y , we r e p lace each transition r = ( p , 1 , q ) with tw o transition s ( p , b , p r ) and ( p r , b , q ) , wh ere p r is a new state; see Fig. 7 for an illustration ho w to rep lace the transitions of the DF As o f Fig. 6. Notice that this replacem ent requires to add a new state f or e a ch transition o f G , which ca n indeed be done in polynomial time. T h is results in a DES G ′ with the alpha b et Σ ′ = { a , b } , wher e both a and b are obser vable. q + 1 q + 2 q + n A 1 A 2 A n . . . . . . q − a a a Σ ′ Σ ′ Σ ′ a a a Σ ′ Σ ′ Figure 5. Cons truction of the DE S G from the PSpa ce -hardness part of the proof of T heorem 3 W e define the specificatio n T s p e c = { ( q − , q + 1 ) } , an d sho w that G ′ is strongly per iodically D- detectab le if an d on ly if the intersection ∩ n i = 1 L ( A i ) is empty . Assume that the intersection is empty . A trajectory that nev er reaches the states of Q + cannot violate strong p e r iodic D- de- tectability , because it can n ot e nter state q + 1 from the speci- fication. T herefore, assume that G ′ ev entu all y en te r s a state of Q + . When G ′ enters a state o f Q + , it leav es all states out of Q + ∪ { q − } . Th us, let s ∈ L ω ( G ′ ) be an arbitrar y trajec- tor y that enters Q + . Then, s = s 1 s 2 where G ′ generates s 1 in states ou tside Q + , and s 2 is the p ar t after G ′ first en ters Q + . In this case, s 1 = f ( w ) ∈ { a , b } ∗ , for some w ∈ { 0 , 1 } ∗ , and s 2 ∈ a { a , b } ω . Then, after generating the first ev ent of s 2 , the obser ver of G ′ is in a set of states co n sisting of q − and a str ict subset of Q + ; indeed, G ′ cannot tr ansit to all s tates of Q + at th e sam e tim e, since the assumptio n that the inter - section is em pty imp lies that, for ev er y w ∈ Σ ∗ , there e xists i ∈ { 1 , . . . , n } suc h that w < L ( A i ) . Let p i ∈ Q + = { q + 1 , . . . , q + n } denote the state of Q + with the minimal ind ex, in which G ′ cannot be when th e i th ev ent of s 2 is generated. By constru ction, the cycle on Q + ensures th at p i per iodically alter nates amon g q + 1 and som e other states of Q + when generating s 2 . Theref ore, in the infinite sequence p 1 , p 2 , . . . , th ere are infinitely many j such that p j = q + 1 , and hence q − and q + 1 are per iodically distinguished, which sho ws that G is strongly per iod ically D-detectab le. On th e o ther han d, assume that the intersection is n o nempty , and let w ∈ ∩ n i = 1 L ( A i ) . Then, after generating the str ing f ( w ) a , the obser ver o f G ′ reaches the state { q − } ∪ Q + . No w , ev er y transition keeps G ′ in all states o f { q − } ∪ Q + , and hen ce it results in a self-loop in the obser ver of G ′ . How ev er, this self-loop v iolates strong per iod ic D - detectability , because it contains b oth states q − and q + 1 . Therefore, any tr ajectory s ∈ L ω ( G ′ ) with f ( w ) a as its pr efix leads to a set of states where the states of T s p e c can n ev er be distinguished, and hence G ′ is not strongly per iodically D-detectable.  6 a b c d A 1 A 2 0 , 1 0 , 1 0 , 1 0 , 1 Figure 6. The DF As A 1 and A 2 o v er Σ = { 0 , 1 } T w o illustrativ e examples o f our constr uction without the bi- nar y encod ing can be f ound in the co n f erenc e version [3]. Here we illustrate the binar y encoding on on e o f the ex am- ples. Let A 1 and A 2 be total DF As ov er the a lp habet { 0 , 1 } depicted in Fig. 6; L ( A 1 ) consists of str ings of od d length, and L (A 2 ) of str ings o f ev en le n gth. Our co nstruc tion r esults in a DES G ′ depicted in Fig. 7. Since L (A 1 ) ∩ L ( A 2 ) = ∅ , G ′ is strongly per io d ically D-d etectable, which is evident from the obser ver o f G 1 depicted in Fig. 8, where on ev e r y trajectory , q − and q + 1 can be per io dically distinguished . a e f h g b q + 1 c i j ℓ k d q + 2 A 1 A 2 q − a a b b b a b b b a b b b a b b b a a , b a , b a a a , b Figure 7. The DES G ′ with Σ ′ = { a , b } , where 0 is encoded as ba and 1 as bb The case of a singl e obser vable event In the previous sub section, we hav e shown that d eciding strong p er iodic D-detectability is PSp ace -complete for DESs with at leas t tw o obser vable ev ents. W e now show th at the problem is still more d ifficult than its no n-per iod ic co unter - par t ev en for DESs having only a single obser vable ev e nt. { a , d , q − } X Y { b , c , q − } { q − , q + 1 } { q − , q + 2 } a a a , b a , b b a , b b a , b Figure 8. The observ er of G ′ ; states marked by double circles contain indistinguishable states of G ′ ; here X = { e , f , k , ℓ, q − } and Y = { g , h , i , j , q − } Theorem 4. Deciding str on g p e riodic D-detectab ility for DESs with a single ob ser vable event is NP -co mplete. Proof. Consider a DES with a single o bser vable ev en t { a } . If the DES ha s un obser vable ev ents, we can eliminate th em as follo ws. First, we re place each unob ser vable tra n sition, i.e., a transition of the f or m ( p , u , q ) with u being an uno b- ser vable ev ent, by an ε -transition ( p , ε, q ) . Th en, we use the standard technique to eliminate ε - transitions [9]. This elimi- nation results in a DES and can be d one in polynomial time . Therefore, without loss of generality , we may assume tha t the DES is of the form G = ( Q , { a } , δ , I ) . A proof that w e can d e cide st ron g p er iodic D-detectability of G in no ndeter min istic polynomial time uses a so- called fas t matrix multiplication tec hnique . The b asic idea of this technique is to r epresent the transition fu nction δ of G as a binar y matr ix M , wher e M [ i , j ] = 1 if an d on ly if there is a transition from state i to state j in G . Then, f or r ≥ 1, M r represents the reachability in G under the str ing a r . Fur ther - more, using the fact that M 2 = M × M , M 4 = M 2 × M 2 , etc., we can comp ute M r by O ( lo g r ) matr ix multiplication s, each m ultiplication in polynomial time . For mo re details and e xamples on th is technique, we refer to Masop ust [15]. Assume that G has n states. Then, the obser ver of G consists of a sequen ce of k states follo wed by a cycle con sisting of ℓ states, that is, the langu age of G is a k ( a ℓ ) ∗ . Since the number of states o f th e o bser ver of G is at most 2 n , k + ℓ ≤ 2 n . No w , G is strongl y per io dically D-dete c table if and only if there is a state X ⊆ Q in the cycle o f the obser ver of G (we assume tha t the obser ver is constr ucted by the standard sub- set constr uction [9]) that is disjoint from the specification , that is, X ∩ T s p e c = ∅ . Indeed , to chec k wheth er X ∩ T s p e c = ∅ can be done in polynomial time. It rem a ins to show ho w to find X in polynomial time. This means to fin d m ≤ 2 n such that δ ( I , a 2 n + m ) ∩ T s p e c = ∅ . How ev er, an NP algo r ithm can guess m in b inar y and verify the guess in p olynomial time by computin g δ ( I , a 2 n + m ) using the fas t matr ix multiplica tio n, cf. Masopust [15] for mo r e details an d an e xample. T o p rov e NP -h ardness, we use the constr uction of Stoc k - mey er and Mey er [23] enc oding a boo lean f or mula in 3CNF in the form of a u nar y NF A. 1 F or a n illu stration, the rea der may follo w Examp le 5 in parallel with the pr o of. 1 A boolean f ormula is built from propositional v ar iables, oper - ators conjunction, disjunction, and negation, and parentheses. A f ormula i s satisfiable if there is an assignment of 1 ( tru e ) and 0 ( false ) t o its v ar iables making it true . A literal is a variable or its negation. A clause is a disjun ction of literals. A formula is in conjunctiv e normal form (CNF) i f it is a con junction of clauses; e.g., ϕ = ( x ∨ y ∨ z ) ∧ ( ¬ x ∨ y ∨ z ) is a fo rmula i n CNF with two clauses x ∨ y ∨ z and ¬ x ∨ y ∨ z . If ev ery clause has at most three literals, the formula is in 3CNF . G i v en a formula in 3CNF, 3CNF satisfiability asks whether the fo r mula i s satisfiable; e. g., ϕ is sat- isfiable f or ( x , y , z ) = ( 0 , 1 , 0 ) . 3CNF satisfiability is NP -complete. 7 Let ϕ b e a f or mu la in 3 CNF with n variables and m clauses, and let C k be the set of litera ls in the k th clause, k = 1 , . . . , m . The assignment to the variables is represented b y a binar y v ector of leng th n . L et p 1 , . . . , p n be the first n p r ime num- bers. F or a n atural nu mber z cong r uent with 0 or 1 mo d ulo p i , f or ev er y i = 1 , . . . , n , we say th at z satisfies ϕ if the assign- ment ( z mod p 1 , z mod p 2 , . . . , z mod p n ) ∈ { 0 , 1 } n satisfies ϕ . Let A 0 be an NF A rec ognizing the language o f the e x- pression Ð n i = 1 Ð p i − 1 j = 2 0 j · ( 0 p i ) ∗ , that is, L (A 0 ) = { 0 z | ∃ k ≤ n , z . 0 mo d p k and z . 1 mod p k } is the set of all natu ral number s th at d o not en code an assignme nt to the variab les. F or each C k , we constr uct an NF A A k such th at if 0 z ∈ L ( A k ) and z is an assignment, then z does not assign 1 ( true ) to an y literal in C k ; e.g., if C k = { x r , ¬ x s , x t } , 1 ≤ r , s , t ≤ n and r , s , t are d istinct, let z k be the unique in teger such that 0 ≤ z k < p r p s p t , z k ≡ 0 m od p r , z k ≡ 1 mod p s , and z k ≡ 0 mo d p t . Then L (A k ) = 0 z k · ( 0 p r p s p t ) ∗ . No w , ϕ is satisfiable if and on ly if there exis ts z such that z encodes an assignme n t to ϕ and 0 z < L ( A k ) f or all 1 ≤ k ≤ m , which is if an d o nly if L (A 0 ) ∪ Ð m k = 1 L ( A k ) , 0 ∗ . The con structio n o f all the automa ta A 0 , A 1 , . . . , A k can be done in polynomial time [23]. Let A denote th e NF A obta in ed by taking the au tomata A 0 , A 1 , . . . , A k as a single NF A, and let p = Π n i = 1 p i . If z enco des a n assignme n t to ϕ , the n so does z + c p for any natural c : if z ≡ x i mod p i , then z + c p ≡ x i mod p i , f or ev er y 1 ≤ i ≤ n , as well. Thus, if 0 z < L ( A k ) f or all k , then 0 z ( 0 p ) ∗ ∩ L ( A ) = ∅ . Since b oth langu ages a r e infinite, the minimal DF A reco gnizing L ( A ) mu st hav e a nontr ivial cy cle alter nating between marked and non- m arked states, and hence the same hold s for the o b ser v er of A . W e now show that ϕ is satisfiable if and o nly if A is strong ly per iodically D-detectable with respe c t to th e specification T s p e c consisting of all pairs of states, wh ere th e states co me from two differ ent automata A i and A j , i , j , and at least one state is mark ed in its autom aton. Assume that ϕ is satisfiable. As sho wn abov e, this is if and only if L ( A ) , 0 ∗ . W e hav e fu r ther sho wn tha t L ( A ) is infi- nite and that the o bser ver of A con sists o f a sing le trajector y with a non- marked state, X , in its cycle p a r t, i.e. , X co ntains only no n-marked states o f the automata A 0 , . . . , A k . Since T s p e c consists of pairs of states of tw o diff e rent autom ata A i and A j , 0 ≤ i , j ≤ k , with at least one state mar ked in its automato n, we hav e that ( X × X ) ∩ T s p e c = ∅ . Ther ef ore, A is strongly per iodically D-detectable with respect to the specification T s p e c . On the other hand , assume that ϕ is n ot satisfiable. A s sho wn abov e, this is if and o nly if L (A ) = 0 ∗ . But then ev er y state of the obser ver of A must b e marked, i.e., ev er y state, X , of the obser ver contains a marked state o f some A i , 0 ≤ i ≤ k , and hen ce ( X × X ) ∩ T s p e c , ∅ . Ther ef ore, A is no t strongly per iodically D-detectable with respect to T s p e c .  a b c d e 0 0 0 0 0 Figure 9. A utomaton A 1 , 0 f g h i j k 0 0 0 0 0 0 Figure 10. Automaton A 1 , 1 l m n o p q r s t 0 0 0 0 0 0 0 0 0 Figure 11. Automaton A 1 , 2 The f ollowing example illustrates the constru ction. Example 5. Let ϕ 1 = ( x ∨ y ) ∧ (¬ x ∨ y ) and ϕ 2 = x ∧ ¬ x . Obviousl y , ϕ 1 is satisfiable a n d ϕ 2 is not. For both form ulae, we can constru ct the u nar y au tomata A 1 and A 2 , r espec- tiv ely , and show that A 1 is strongly pe r io dically D-detect- able while A 2 is not. In this pap er , we co nstruct on ly the automaton A 1 ; the constru c tion of th e automato n A 2 can be f ound in the conference v ersion [3]. The formula ϕ 1 = ( x ∨ y ) ∧ (¬ x ∨ y ) has two var iables, and therefore w e set p 1 = 2 an d p 2 = 3, the first two pr ime num- bers. Th e automato n A 1 , 0 , d epicted in Fig. 9, recogn ize s the lan g uage 0 2 ( 0 3 ) ∗ of all str ings tha t d o n o t enco de the assignment to ϕ 1 . Since ϕ 1 consists o f tw o clauses, we f ur - ther co nstru ct two automata : A 1 , 1 recogn izin g the lang u age ( 0 6 ) ∗ , and A 1 , 2 recogn izin g the lan guage 0 3 ( 0 6 ) ∗ ; the rea d er can v er ify that if 0 z ∈ L ( A 1 , 1 ) and z is a n assignment, then z assigns tr ue neither to x nor to y , and if 0 z ∈ L ( A 1 , 2 ) and z is an assignment, then z assigns tru e to x (tha t is, it assigns false to the literal ¬ x ) and false to y . The autom a ta are depicted in Figs. 1 0 and 1 1, respectivel y . The specificatio n T s p e c = { c } × { f , g , h , i , j , k , l , m , n , o , p , q , r , s , t } ∪ { a , b , c , d , e , l , m , n , o , p , q , r , s , t } × { f } ∪ { a , b , c , d , e , f , g , h , i , j , k } × { o } . Let A 1 be the NF A consisting of the au tomata A 1 , 0 , A 1 , 1 , and A 1 , 2 . T h e obser ver of A 1 is depicted in Fig. 12 . The reader can see that the obser ver co n tains a cy cle where the state { e , j , p } appear s per io dically . Sin c e this st ate d o es not contain a ny pair from the specification T s p e c , the NF A A 1 is strongly p er iodically D-detectab le as claimed. ⋄ As alr eady pointed out, the ob ser v er of a DES with a sing le obser vable ev ent consists of a single trajector y en d ing w ith 8 { a , f , l } { b , g , m } { c , h , n } { d , i , o } { e , j , p } { c , k , q } { d , f , r } { e , g , s } { c , h , t } 0 0 0 0 0 0 0 0 0 Figure 12. T he observ er of the NF A A 1 ; states marked by a double circle contain indistinguishable states of A 1 a cy cle . W e now for m ulate the c onsequences o f this obser - vation. Theorem 6. F or a DES with a sing le obser vable event, de- ciding weak detectability coincide s with decidin g str on g de- tectability , and deciding weak periodic detectability coin- cides with deciding str ong periodic detectability. The same holds true for D-de tectab ility. Proof. Let G be a DES with n s tates and a single obser vable ev ent Σ o = { a } . Then the obser ver of G consists of a se- quence of k states f ollow ed by a cycle consisting of ℓ states, that is, the language of G is a k ( a ℓ ) ∗ with k + ℓ ≤ 2 n . The e xistence of the cycle f ollow s f r om th e tw o assump tions o n P age 3 . Sin ce the obser ver of G consis ts of a single infinite trajectory , deciding strong (per iod ic) detectability coin c ides with deciding weak (per io dic) d etectability; the same hold s when we replace detectability with D-dete c ta b ility .  5 A T ract able Case In the previous sections, we hav e shown that d eciding strong per iodic D-detectability is a diffic u lt problem for DES mod- eled b y NF As. In this section, w e discu ss a special case of sys tems for which the verification of strong period ic D-de- tectability is polynomial. Lookin g f or a class o f DESs, f or which th e prob lem is tractable, w e first inspec t the proof of Th eorem 3 . This r e- v eals that the proo f is based on the intersection e mptiness problem f or DF As that was sho wn by Kozen [1 1] to be PSp a ce -com plete. His proo f heavil y re lies on DF As with cy- cles. Allowing o nly self-lo ops instead of cycles in the DF As makes the pro blem ea sier [ 16]. Therefore, we co nsider DESs modeled by NF As where all cycles in the tran sition g raph a re only self-lo o ps. A self - loop may b e a dded to any state, and hence the NF A m ay fulfill the deadlock -fr e e require m ent. Such NF As reco gnize a str ict subclass of regular languages that are str ictly inclu ded in star -free lan guages [4,12]. Star - free langu ages are languages d e finable by linea r tempor al logic , which is a logic widely u sed as a specificatio n lan- guage in automated verification. A prac tical motivation f or such systems co mes fr om the f ol- lo wing ob ser vation. Every infinite trajectory in a system de- scr ibes a task that is p ossibly repeated ad infinitu m. In deed, ev er y task is a fin ite sequen ce of ev ents, th ough for the mod- eling pur po ses som e repetition s o f subtasks may b e mod eled a a Figure 13. T he forbidd en patter n of r poNF As as cycles. In some cases a nd on sam e lev el of ab straction, these inter nal cycles could be seen as self-loo p s. This results in a sys tem with only self-lo ops. No w we need to mo del the situation that th e task has be e n finished and the who le process can b e res tar ted. This can b e done b y the re peated generation o f a special ev ent telling the system to restart the specific task. Let A = ( Q , Σ , δ, I , F ) be an NF A. The re achability relatio n ≤ on the state set Q is defined by p ≤ q if there is w ∈ Σ ∗ such th at q ∈ δ ( p , w ) . Th e NF A A is restrict ed par tia lly order e d (rp oNF A) if the r eachability relation ≤ is a partial order and A is self- loop d eter ministic in the sense that the patter n of Fig. 13 does not appear. For mally , f or ev er y state q and ev er y ev en t a , if q ∈ δ ( q , a ) th e n δ ( q , a ) = { q } . W e now f or mally d e fin e so-called r poDES. The name comes from restricted par tially order e d DES. Definition 7. L et G = ( Q , Σ , δ, I ) be a DES with Σ o being the set o f obser vable ev ents. Let P : Σ ∗ → Σ ∗ o be the cor - respond in g projection. W e say that G is a n r p oDES if the NF A P ( G ) = ( Q , Σ o , δ ′ , I ) o btained from G by replacin g ev- er y tra n sition ( p , a , q ) by ( p , P ( a ) , q ) , and by eliminating the ε -transition s [9] is an r po NF A. Notice tha t P ( G ) is an NF A that can b e con structed from G in polynomial time [9]. Therefore, the question whether a DES is an r p oDES is decidable in po lynomial time . What do we k n o w abo ut r poDES? Deciding weak (per i- odic) detectability for r p oDESs is PSp ace -complete [14], and h ence so is decid ing weak (p er iodic) D-d etectability . W e now show th at the comp le xity of decid ing strong per iod ic D-detectability f or r poDESs coincides with the complexity of deciding strong D-dete c ta b ility . Theorem 8. Deciding str on g p e riodic D-detectab ility for rpoDES s is NL -complete. Proof. T o pro ve the th eorem, we sho w that the obser ver of an r poDES is a pa r tiall y or dered DF A. Th en, since there are no nontr iv ial cy cles in the o bser v er, strong per io dic D-de- tectability coincid es with strong D-detectability . Theor em 1 then fin ishes the proof . Let A = ( Q , Σ o , δ, I , F ) b e an r poNF A. W e sho w that the DF A D com puted from A by the standard subset c onstru c- tion (i.e. , the obser ver) is par tially ordered. T o this aim, let X = { p 1 , . . . , p n } with p i < p j f or i < j be a state of D , and let w ∈ Σ ∗ be a nonem pty string such that δ D ( X , w ) = X . Firs t, we sh o w th at δ ( p i , w ) = { p i } f or all i . For the sake 9 of contrad iction, let 1 ≤ k ≤ n be the minimal in te ger such that δ ( p k , w ) , { p k } . Since X = δ ( X , w ) = ∪ n i = 1 δ ( p i , w ) , δ ( p i , w ) = { p i } f or all i < k , and p k < p i ≤ δ ( p i , w ) f or all i > k , we hav e that p k < ∪ n i = 1 δ ( p i , w ) = X , which is a contrad iction. 2 T herefore, p k ∈ δ ( p k , w ) , and the defini- tion of r p oNF A s implies that e v er y e v ent of w is in a self- loop in state p k . Because r p oNF As hav e no choice b e tw een sta ying in the state and leaving it u nder the same ev ent, δ ( p k , w ) = { p k } . Thus, f or i = 1 , . . . , n , δ ( p i , a ) = { p i } f or ev er y ev ent a occur r ing in w . Consequently , for any state Y of D and any str ings w 1 and w 2 , if δ ( X , w 1 ) = Y an d δ ( Y , w 2 ) = X , the pre viou s argu ment gives th at X = Y , and hence D is par tiall y ordered .  Finall y , we hav e the follo win g corollar y of Theo rem 1. Corollary 9. Decidin g whether an rp oDES with a sing le obser vable event is strong ly ( D-)detectable is N L -complete. Proof. This result is an im mediate co n sequence of th e proo f of Th eorem 1, since in the hardness par t we actually con- struct a unar y r poNF A, cf. Fig. 1.  6 Conclusions In this paper, we answered the open question con cer ning the complexity o f dec iding wheth er a DES satisfies strong per i- odic D- detectability , a nd p ro vide d a f ull complexity pictur e of th is pro blem. Since the resu lts for DES are main ly ne ga- tiv e, we also discussed a class of DESs, so-c alled r poDESs, f or which the comp le xity of d e ciding strong (p e r iodic) D-de- tectability is tractable. A ckno wledgements Suppor ted b y the Min istry of Education, Y outh and Spor ts under the INT E R -EX CELLENCE pro je c t L T A US A190 98, by th e Czech Science Foundation p roject GC19 -061 7 5J, by IGA PrF 202 0 01 9, an d by R V O 6 7985 840. Ref erences [1] M. V . S . Alv es and J. C. Basilio. State esti mation and detectabil ity of netw orked discrete e vent sys tems with multi-cha nnel communicat ion netw orks. In American Contr ol Conf er ence (ACC) , pages 5602–5607. IEEE, 2019. [2] S. Arora and B. Barak. Computa tional Complexity – A Modern Approa ch . Cambridge Univ ersity Press, 2009. [3] J. Balun and T . Masopust . On v erification of strong periodic d- detec tability f or discrete e ve nt sys tems. In W orkshop on Discr et e Eve nt Sys tems (WODES) , 2020. T o appear . Online a vai lable at http:/ /arxiv .org/abs/1 912.07312. [4] J. A. Brzoz ow ski and F . E. Fich. L anguag es of R -trivial monoids. Journal of Computer and Syste ms Sc iences , 20(1):32–49, 198 0. 2 By p i ≤ δ ( p i , w ) we mean p i ≤ q for ev er y q ∈ δ ( p i , w ) . [5] P . E. Cain es, R. Greiner , and S. W ang. Dynamical logic observers f or finite automata . In Conf erence on Decision and Control (CDC) , pag es 226– 233, 198 8. [6] S. Cho and D. T . Huynh. Finit e-automato n aperiodi city is PSP A CE- complete . Theore tical Comput er Science , 88(1):99–116, 199 1. [7] M. R. Garey and D. S. Johnson. Computer s and Intr actabili ty ; A Guide to the Theory of NP -Comple tene ss . W . H. Freeman & Co., Ne w Y ork, NY , USA, 1990. [8] C. N. Hadjicos tis. Estimatio n and Inf erenc e in Discre te Eve nt Syst ems . Spring er , 2020. [9] J. E . Hopcroft, R. Motwani, and J. D. Ullman. Intr oduction to auto mata theory, languag es, and computation . Addison - W esle y , 3rd editi on, 2007. [10] N. Immerm an. Nondet erministic space is closed under complement ation. SIAM Jour nal on Computing , 17:935–938, 1988 . [11] D. Kozen. L ow er bounds fo r natural proo f sy stems. In Symposi um on F oundation s of Compute r Science (FOC S) , pag es 254–266, 1977. [12] M. Krötzsch, T . Masopust , and M. Thomazo. Comple xity of univ ersalit y a nd relate d problems f or partiall y ordered NF As. Inf ormation and Computation , 255(1):177–192, 2017 . [13] F . Lin. Opacit y of discrete e vent sys tems and its applicat ions. Automat ica , 47(3): 496–503, 2011. [14] T . Masopust. Complexi ty of decidi ng detec tability in discrete ev ent sys tems. Automat ica , 93: 257–261, 2018 . [15] T . Masop ust. Complexi ty of verifyin g nonbloc kingness in modular supervisory control. IEEE T ra nsactions on Automatic Contr ol , 63(2):602– 607, 2018. [16] T . Masopust and M. Krötzsch. Partiall y ordered automata and piece wise testa bility . Preprint at arXiv :1907.13115, 2019. [17] T . Masopust and X. Yin. Complexi ty of detecta bility , opacity and A-diagnosa bility f or modular discrete ev ent syst ems. Aut omatica , 101:290–2 95, 2019. [18] C. M. Ozvere n and A. S. Willsky . Observa bility of discrete e ven t dynamic sy stems. IEE E T ransactio ns on Automatic Contr ol , 35(7):797– 806, 1990. [19] P . J. Ramadge. Observabi lity of discrete ev ent syst ems. In Confer ence on Decision and Contro l (CDC) , pages 1108–1112, 1986. [20] S. Shu and F . Lin. General ized detectabil ity f or discrete ev ent sys tems. Sys tems & Contr ol Lette rs , 60( 5):310–317, 2011. [21] S. Shu and F . Lin. Dela yed det ectabil ity of discrete e ven t syst ems. IEEE T ran sactions on Aut omatic Control , 58(4):862– 875, 2013. [22] S. Shu, F . Lin, and H. Ying. Detectabili ty of discrete ev ent sy stems. IEEE T ran sactions on Aut omatic Control , 52(12):235 6–2359, 2007. [23] L. J. Stockme yer and A. R. Mey er. W ord problems requiring e xponentia l time: Prelimin ary repor t. In ACM Symposium on the Theory of Computing (STOC) , page s 1–9 . AC M, 1973. [24] R. Szelepcsé nyi. The method of f orced enumeration fo r nondete rm inis tic automata. Acta Info r matica , 26:279–284, 1988. [25] X. Yin and S. Laf or tune. V erification comple xity of a class of observa tional properties f or modular discrete e ven ts syst ems. Automat ica , 83:1 99–205, 2017. [26] K. Z hang. The problem of determining the wea k (p eriodic) detec tability of d iscrete e ve nt sy stems is PSP AC E-complete. Automat ica , 81:2 17–220, 2017. [27] K. Zhang and A. Giua. R evi siting dela ye d strong detectabi lity of discrete -ev ent systems. Preprint at arXiv:1910.1376 8, 2019. [28] K. Z hang, L. Zhang, and L. Xie. Discr et e- Time and D iscret e-Space Dynamical Sys tems . Spring er , 2020. 10