Timed Discrete-Event Systems are Synchronous Product Structures

Timed Discrete-Ev en t Systems are Sync hr onous Pro duct Structures Liy ong Lin ∗ ∗ Contemp or ary Amp er ex T e chnolo gy Limite d (e-mail: l lin5@e.ntu .e du.sg). Abstract: Timed discrete-even t systems (TDES), which is a modelling for malism prop osed by Brandin and W onham, can b e used for modelling scheduling and productio n planning problems. This pap er aims to show that TDES are essentially synchronous pro duct structure s . The pro of is constructive in the sense that a generalize d synchronous pro duct rule is provided to generate a TDES fro m the a ctivity automaton and the timer automata (that is, the syntactic description of the TDES) after some model tra nsformation. W e then also explain how t he generalized synchronous pro duct opera tio n can be reduced int o the standard sync hr onous pro duct op er ation and ho w to reduce the num b er of (refined) even ts in tro duced in the mo del transformation. Th us, any softw a re that ca n compute synchronous pro ducts can b e used to compute a TDES fro m its activity automaton and its timer a utomata, after the mo del trans formation. Keywor ds: timed discr e te- even t sys tems, timed state-tree structures, s ynchronous pr o duct, sup e rvisor y control 1. INTRODU CTION The theory of control for real- time discre te- even t sys tems is a very rich sub ject that has b een in vestigated in s e v- eral different for malisms, e.g., the c lo ck automa ta Brav e [1988], the timed transition mo dels Ostroff [199 0], the timed discrete-even t systems Bra ndin [1 994], with dis- crete time semantics, and the timed automata T o i [19 91], with dense time semantics. One of the mo st widely used mo dels of rea l-time discr ete-even t systems in s upe rvisor y control th eory is the timed discrete-even t systems (TDES). TDES is attractive since many existing techniques for the co nt rol of untimed discr ete-even t sys tems (DES) can often be natura lly extended to this timed counterpart. In- deed, a few num ber of structured sup er visor synthesis a p- proaches ha v e b e en extended to the TDES formalism (e.g. Schafasc hek [201 7 ], Saada tpo or [2 009]). Some recen t ex- tensions of Bra ndin [1994 ] include the superv isor loc a liza- tion Zhang [2 013], Ostroff [201 9 ], relative observ ability and relative co o bs erv ability Cai [2016], netw or ked super visor synthesis Zhang [201 6], Ra shidinejad [2018 ], Pr uekpraser t [2020], sta te-based c ontrol Rahna mo on [201 8] and s o on. Quite a few num b er of pr actical applications have be en developed based on the framew ork of TDES (see, for ex- ample, W a re [201 7], Zhao [201 7], Monteiro [2017] for some recent w orks) and interested reader may also se e Seow [2020] fo r a n envisioned applica tion of control of TDES. The mo deling framework of Br andin [1994 ] s ta rts with a (finite state) in terv al mo del, where ea ch event is asso ciated with a fixed discrete time interv al. Then, the int erv al mo del has to b e co nv erted into a (finite state) tic k mo del, which e xplicitly enumerates the tick even ts to mo del time int erv als in the orig inal mo del. Howev er, a n inherent dif- ⋆ This w ork is financially suppor ted by National Key R&D Program of China (Grant 2022YFB47024 00). ficult y of TDES, co mpared with the untimed count erpart, is precisely due to the explicit enumeration of ticks that only makes the state explosion problem much worse, e s - pec ially when the interv a ls have lar ge upp er b ounds 1 . A symbolic approach for the super visory control of TDES has bee n s uccessfully developed in Mirema di [2015], by us ing the timed ex tended finite automata (TEF A) a nd bina r y decision diagra ms (BDD). An appr oach that attempts to av o id an explicit en umeratio n of tic ks can b e found in Lin [2019 ], Brandin [2 020], which op er ates o n interv al automata direc tly (by using interv a l ar ithmetic) for b oth synchronous pro duct constructio n and sup erv isor synthe- sis. A tim ed state-tree structures (TSTS) based superviso r synthesis framework has be e n develop ed in Saadatpo o r [2009] to cop e with the state explosion issue e nc o untered in TDES, encoura ged b y the success of the (untimed) state-tree structur es based sup ervis o r s ynthesis appro a ch in Ma [2004]. How ever, the TSTS framework developed in Saadatp o or [2009] seems far less success ful, and only deals with systems o f state sizes o f the order ≤ 10 12 , even if (the same) BDD based enco ding of state trees is used. W e conjecture one o f the main reasons b ehind this is b ecause the mo delling p ow er of the state-tree structures (STS) forma lism is not well utilized. Indee d, Saadatp o o r [2009] uses each (mono lithic) TDES as a holo n, instead of using the (constituent) a c tivit y automato n and the timer automata as the holo ns. Th us , state explosion has implicitly o ccur r ed in the computation or mo del building of the TDES holons 2 . 1 An int erv al model could hav e an un bounded “compression r ate” compared wi th the (flat) tick mo del Lin [2019]. F or example, to simulate a transition in the interv al model lab eled by ( σ , [1 , k ]), k tic ks-lab eled transitions need to be created in the tick model. 2 W e shall remark that this is not the only reason, as an explicit en umeration of tic ks still cannot be av oided in the timer automaton holons. F or example, if the interv als f or ev ents ha ve large upp er In this pap er, we shall show that this source of inefficiency of TSTS co uld b e avoidable. Indeed, the aim of this pap er is to s how that each TDES of Brandin [1994 ] is a synchronous pro duct structure, which can be built from the activity automa ton a nd the timer a utomata after some mo del tra nsformation; th us, it is p oss ible for one to use the activit y automaton and the timer auto ma ta, instead of their product, that is, TDES, a s the holons, which will hop efully help improv e the efficiency of TSTS based sup erviso r syn thesis pro cedure by increasing the horizontal mo dularity . This result may b e of independent int erest as w ell. W e remark that, b efor e this w ork, it is an open problem, to the b est of our knowledge, whether a TDES can b e built from its activity automato n and timer automata Brandin [1994]. Indeed, it is stated in the s econd para graph of Sectio n V in Bra ndin [199 4] that “Unfortunately ther e is no simple way to obtain G by straightforward combination of G act with the S P E C σ ” . The pap er is organized as follows. In Sectio n 2, we review the ba sics o f TDES. In Se c tion 3, we show the main result that TDES ar e synchronous product structures. Finally , we provide conclusion a nd future works in Section 4. 2. BASICS OF TIME D DISCRETE-E VENT SYSTEMS In this section, we shall pr esent some basics of the TDE S formalism Br andin [199 4], W onham [2 021] to make this pap er more self-co ntained. T o that end, we need to fir st recall some gener al notation and terminology . Let N denote the set { 0 , 1 , 2 , 3 , . . . } o f natural n umbers . W e wr ite [ k 1 , k 2 ], where k 1 , k 2 ∈ N and k 1 ≤ k 2 , to denote the s e t { k ∈ N | k 1 ≤ k ≤ k 2 } , and wr ite [ k 1 , ∞ ) to deno te the set { k ∈ N | k ≥ k 1 } . Let I denote the collection of int erv als of the ab ove tw o t yp es , and le t I f ⊂ I denote the collection of interv als of the first t ype . F o r any interv al I , w e define I .l = k 1 , I .r := k 2 if I = [ k 1 , k 2 ]; w e define I .l := k 1 , I .r := ∞ if I = [ k 1 , ∞ ). F or brevity , w e use “ t ” , instead of “tick”, to denote the tick even t. 2.1 Synt ax The syntax of a TDES is given by a tuple ( G act , T ), where G act = ( A, Σ act , δ act , a 0 , A m ) is a finite state automaton (ov er Σ act ) and T : Σ act 7→ I maps each σ ∈ Σ act to an in ter v al T ( σ ) ∈ I . G act is the activity automaton , w he r e A is the finite set of activities; Σ act is the finite set of (activity) even ts; δ act : A × Σ act 7→ A is the (partia l) a ctivity transition function 3 ; a 0 ∈ A is the initial activity and A m ⊆ A the subs e t of mar ker activities. T is the timer map which na turally induces a partition of the even t set Σ act = Σ spe ˙ ∪ Σ r em , where 4 σ ∈ Σ spe if and only if T ( σ ) ∈ I f . bounds, then eac h timer automaton holon can ha v e a large state size. 3 W e also wri te δ act ⊆ A × Σ act × A . As usual, we can extend δ act to the partial function δ act : A × Σ ∗ act 7→ A . 4 The subscript “sp e” denotes “prospective”, while “rem” denot es “remote”. 2.2 Semantics Recall that the semantics of the tuple ( G act , T ) is a finite state automa ton G = ( Q, Σ , δ, q 0 , Q m ) ov er Σ := Σ act ˙ ∪{ t } , often referred to as a timed discrete-event sy stem (TDES). In the r e st o f this subsection, w e shall ex pla in how the four comp onents Q, δ, q 0 and Q m are gener ated from the tuple ( G act , T ). F or an inf ormal description of the semantics, the reader is referred to Bra ndin [19 94]. A timer int erv al T σ is defined for each even t σ as follows: T σ :=  [0 , T ( σ ) .r ] , if σ ∈ Σ spe [0 , T ( σ ) .l ] , if σ ∈ Σ r em The state se t is defined to b e Q := A × Q σ ∈ Σ act T σ , whe r e without loss of genera lity we ar bitrarily fix an order ing in the enumeration of Σ act in the Ca rtesian pro duct. A s tate is a tuple of the form q = ( a, ( t σ ) σ ∈ Σ act ), where a ∈ A a nd, for each σ ∈ Σ act , t σ ∈ T σ . The t σ comp onent of q is the timer v alue of σ in q . The default timer v alue t σ 0 for each σ ∈ Σ act is defined as follows: t σ 0 :=  T ( σ ) .r, if σ ∈ Σ spe T ( σ ) .l , if σ ∈ Σ r em The initial state is q 0 = ( a 0 , ( t σ 0 ) σ ∈ Σ act ). And, the set of marker states is defined to b e 5 Q m = A m × Q σ ∈ Σ act { t σ 0 } . The definition of the partial transition function δ is mor e tedious. Let q = ( a, ( t σ ) σ ∈ Σ act ) and q ′ = ( a ′ , ( t ′ σ ) σ ∈ Σ act ) be any tw o states. W e have the following three cases: A) for an y σ ∈ Σ spe , δ ( q , σ ) = q ′ if and only if (a) δ act ( a, σ )! and 0 ≤ t σ ≤ T ( σ ) .r − T ( σ ) .l (b) a ′ = δ act ( a, σ ) and for each τ ∈ Σ act , (i) if τ 6 = σ , then t ′ τ :=  t τ 0 , if ¬ δ act ( a ′ , τ )! t τ , if δ act ( a ′ , τ )! (ii) if τ = σ , then t ′ τ = t σ 0 B) for any σ ∈ Σ r em , δ ( q , σ ) = q ′ if and only if (a) δ act ( a, σ )! and t σ = 0 (b) a ′ = δ act ( a, σ ) and for each τ ∈ Σ act , (i) if τ 6 = σ , then t ′ τ :=  t τ 0 , if ¬ δ act ( a ′ , τ )! t τ , if δ act ( a ′ , τ )! (ii) if τ = σ , then t ′ τ = t σ 0 C) δ ( q , t ) = q ′ if a nd o nly if (a) ∀ τ ∈ Σ spe , ( δ act ( a, τ )! = ⇒ t τ > 0) (b) a ′ = a a nd for each τ ∈ Σ act , (i) if τ ∈ Σ spe , then t ′ τ :=  t τ 0 , if ¬ δ act ( a, τ )! t τ − 1 , if δ act ( a, τ )! ∧ t τ > 0 (ii) if τ ∈ Σ r em , then t ′ τ :=    t τ 0 , if ¬ δ act ( a, τ )! t τ − 1 , if δ act ( a, τ )! ∧ t τ > 0 0 , if δ act ( a, τ )! ∧ t τ = 0 5 In Brandin [1994], it is only required there that Q m ⊆ A m × Q σ ∈ Σ act T σ . The softw are TTCT pro vides t wo options: Q m = A m × Q σ ∈ Σ act T σ or Q m = A m × Q σ ∈ Σ act { t σ 0 } . The second option i s adopted in this wo rk, without loss of gene rality , as the marker states set can b e easily adapted accordingly . 2 W e have now completed the des cription of the semant ics of the tuple ( G act , T ) as a timed disc r ete-even t system G . W e finally re mark that it is required that all TDES must be activity-lo op-fr e e , namely , ∀ q ∈ Q, s ∈ Σ + act , δ ( q , s ) 6 = q . 3. TDES ARE SYNCHRONOUS PRODUCT STR UCTURES W e note that the description of the semantics of ( G act , T ) in Section 2.2 is a bit in volv ed and ma y ev en be difficult to visualize (for beginner s). While all the o ther compo nent s of G suggest that G could b e a synchronous pro duct structure (for example, the state space Q is a Cartesian pro duct), the definition of δ is not presented in a n explicitly structured manner no r is it clear that δ has a pr o duct structur e; and, it is difficult to o btain muc h insight fro m the def- inition of δ . A structured construction of G by combin- ing 6 G act and T is an open pro blem that still r e mains not addres sed Brandin [199 4]. In the following, we s how that a generaliz ed sync hronous pro duct op er ation for the construction of a TDES fro m its activity automaton and timer map description, after so me mo del transformatio n, is feasible. Later, w e als o show ho w the gener alized pr o duct op eration can b e mapped into the standard synchronous pro duct op er ation. F urther more, we will show that the synchronous pro duct based construction exa ctly matches the co nstruction provided in Section 2.2. The overall constr uction consists o f four steps; the de ta ils will be explained in the rest o f this section. 3.1 Step 1: Timer Map to Timer Automata In order for the synchronous pro duct to be co mputed, the first step is to trans form the timer ma p to a c ollection of timer auto ma ta. F or each pair ( σ , T ( σ )), we shall define a timer automaton G σ = ( Q σ , Σ σ , δ σ , q 0 ,σ , Q m,σ ). Before we provide the formal definition of G σ , we shall use the next example a s a n illustratio n. The formal cons tr uction is as follows. 1) If σ ∈ Σ spe , then we let Q σ = [0 , T ( σ ) .r ], Σ σ = { t, σ } , q 0 ,σ = T ( σ ) .r and Q m,σ = { T ( σ ) .r } . δ σ is sp ecified by its graph 7 , which is the union of { ( i + 1 , t, i ) | i ∈ [0 , T ( σ ) .r − 1] } a nd { ( i , σ , T ( σ ) .r ) | i ∈ [0 , T ( σ ) .r − T ( σ ) .l ] } . 2) If σ ∈ Σ r em , then we let Q σ = [0 , T ( σ ) .l ], Σ σ = { t, σ } , q 0 ,σ = T ( σ ) .l a nd Q m,σ = { T ( σ ) .l } . δ σ is sp ecified by its graph, whic h is the union of { ( i + 1 , t, i ) | i ∈ [0 , T ( σ ) .l − 1 ] } and { (0 , σ, T ( σ ) .l ) , (0 , t, 0) } . Each timer automaton G σ constructed ab ove cor resp onds to the specificatio n S P E C σ over { t, σ } in Sec tion V of B randin [1 994]. Despite of the fact that the timer au- tomata have already b een co ns tructed in Bra ndin [199 4], it is not known whether a nd how they could b e used in the co mputation o f G in Bra ndin [199 4]. In the following subsections, we provide the remaining three steps to com- plete the construction o f G b y us ing a synchronous pro duct op eration, after so me mo del transfor mation. 6 T is only a timer map here; th us, it is exp ected that w e need to transform T into some automata. The de tails will b e presented so on. 7 The graph of a partial function f with domain D om is the relation { ( x, f ( x )) | x ∈ Dom } . 3.2 Step 2: Automata T r ansformation The automata ( G act , ( G σ ) σ ∈ Σ act ) cannot be use d fo r syn- chronous product. The main idea to reso lve the difficulties is to 1) add lab els to tick tra nsitions and even t transi- tions in the activit y automaton to r eflect, resp ectively , the status of enablemen t of even ts and the effect of ev ent tra n- sitions on the enablement of even ts, and 2) split and add lab els to tic k transitions and even t transitions in the timer automata to reflect the effects of different tr a nsitions 8 . Let Σ act = { σ 1 , σ 2 , . . . , σ n } . The transformation can then be summariz ed as follows: [Computing T ransformed Activity Automaton] : Given G act (befor e a dding the self-lo ops), we p erform the following oper ations. (1) for each state a ∈ A in G act , a dd a self- lo op lab eled by ( t, ph 1 , ph 2 , . . . , p h n ), wher e ph i is a place holder that is to b e replace d by σ i ! if σ i is defined at state a , and r e placed by ¬ σ i ! if σ i is not defined at state a . (2) for each tr ansition ( a, σ, a ′ ), where σ ∈ Σ act , replace it with the transition ( a, ( σ, ph ′ 1 , ph ′ 2 , . . . , ph ′ n ) , a ′ ), i.e., add the la b e l ( ph ′ 1 , ph ′ 2 , . . . , p h ′ n ) to σ in the transition ( a, σ, a ′ ). Here, ph ′ i is a place holder that is to b e replaced by E σ i if σ i is defined at state a ′ , and replaced b y D σ i if σ i is not defined at state a ′ . Int uitively , the meaning of ( t, α ! , ¬ β !) a t state 0 ∈ A is that α is enabled and β is disabled at state 0 ∈ A . The meaning of the trans ition (0 , ( α, D α, E β ) , 1 ) is tha t after firing even t α at state 0, α is disabled and β is enabled (in the next state). The other cases can b e explained in a similar w ay . W e shall use the nex t exa mple to illustrate the tra nsfor- mation pr o cedure fo r the timer automata . The trans formation can b e summarized as follows: [Computing T ransformed Tim er Automaton] : Given G σ (befor e adding the self-lo ops), w e p er form the following oper ations. Recall that t σ 0 :=  T ( σ ) .r if σ ∈ Σ spe T ( σ ) .l if σ ∈ Σ r em is the initial state of G σ . (1) for each state i ∈ Q σ and each even t σ ′ ∈ Σ act other than σ , a dd the transition ( i, ( σ ′ , E σ ) , i ), i.e., add a self-lo op lab eled b y ( σ ′ , E σ ), and add the tr ansition ( i, ( σ ′ , D σ ) , t σ 0 ). (2) for each transition ( i, σ, t σ 0 ), r eplace it with the tw o transitions ( i, ( σ , E σ ) , t σ 0 ) a nd ( i, ( σ, D σ ) , t σ 0 ). (3) for each transition labeled by t , r eplace the lab el with ( t, σ !). (4) for the state t σ 0 ∈ Q σ , add the tra nsition ( t σ 0 , ( t, ¬ σ !) , t σ 0 ), i.e., add a self-lo o p lab eled by ( t, ¬ σ !). Int uitively , lab els are added to reflect the effects o f differ- ent tr ansitions to the timer v alue for ea ch timer auto ma- ton. In the ab ove pro cedure, we split tick transitions and even t transitions for exactly that purp ose. 8 Here, ev ent transitions refer to those transitions lab eled by even ts in Σ act . 3 3.3 Step 3: Gener alize d Synchr onous Pr o duct Up to now, we hav e completed the transfor mation of the activity automaton and the timer automata. It is str aight- forward to define a genera lized synch ronous pr o duct op er - ation that combines the tra nsformed activity automa ton and the tr ansformed timer a uto mata. The sy nchronization constructs are shown in the following. (1) for ev en t ( t, ph 1 , ph 2 , . . . , p h n ) in the transfor med activity automaton, it can be sync hronized with even t ( t, ph i ) in the transformed timer automaton for σ i , for i ∈ [1 , n ], where ph i is a place holde r for σ i ! or ¬ σ i !; after the sy nchronization, the lab el is event t (2) for e ven t ( σ , ph ′ 1 , ph ′ 2 , . . . , ph ′ n ) in the transfor med activity automaton, where σ ∈ Σ act , it can b e syn- chronized with ev ent ( σ, ph ′ i ) in the tr ansformed timer automaton for σ i , for i ∈ [1 , n ], where ph ′ i is a plac e holder for E σ i or D σ i ; after the synchronization, the lab el is event σ Let G T act denote the transfor med activity automa ton and G T σ the tra nsformed timer automaton for σ ∈ Σ act . If we use ⊲ ⊳ to denote the generalized synchronous pro duct op eration, then G T act ⊲ ⊳ ( ⊲ ⊳ σ ∈ Σ act G T σ ) can b e constructed using the ab ov e synchronization constructs. 3.4 Step 4: Gener alize d Synchr onous Pr o duct to Standar d Synchr onous Pr o duct T o map the g eneralized synchronous pro duct o p er ation ⊲ ⊳ int o the standa r d sync hronous pro duct op era tion k , the last step in volv es straig htf orward r elab elling for eac h timer automaton. Thus, for ea ch timer automaton, we p er form the fo llowing. (1) replace each trans ition that is labeled with ( σ, ph i ) with the se t of transitions la be led with { ( σ , ph 1 , . . . , ph i − 1 , ph i , ph n ) | ph j = E σ j ∨ ph j = D σ j , j ∈ [1 , n ] − { i }} (2) replace eac h tra nsition that is lab ele d with ( t, ph i ) with the se t of transitions la be led with { ( t, ph 1 , . . . , p h i − 1 , ph i , ph n ) | ph j = σ j ! ∨ ph j = ¬ σ j ! , j ∈ [1 , n ] − { i }} Let R ( G T σ ) denote the r elab elled trans formed timer au- tomaton, for each σ ∈ Σ. Then, w e ca n obtain the synchronous pro duct G T act k ( k σ ∈ Σ act R ( G T σ )). E a ch ev ent of G T act k ( k σ ∈ Σ act R ( G T σ )) is o f the fo r m ( σ, ph 1 , ph 2 , . . . , ph n ) or ( t, ph ′ 1 , ph ′ 2 , . . . , ph ′ n ), which contains refined even t in- formation. It is straightforw ard to rec ov er the origi- nal even t information by hiding the placeholder infor - mation, by r eplacing ( σ, ph 1 , ph 2 , . . . , p h n ) with σ and ( t, ph ′ 1 , ph ′ 2 , . . . , ph ′ n ) with t . W e denote the resulting finite state automaton h ( G T act k ( k σ ∈ Σ act R ( G T σ ))). It app ears that an exp onential num b er of even ts need to be crea ted for the model transformatio n, to b e used in the s y nchronous product constructions. This can be easily av oide d b y relab eling the events in G act first. Then, one only need to transfor m each G σ with the even ts used in G T act . Thus, the num b er of event s used is uppe r b ounded by the num b er of transitions of G act . 4. CONCLUSIONS AND FUTURE W ORK In this work, w e hav e sho wn that TDES ar e synchronous pro duct structures, thus resolv ing a pr o blem from Brandin [1994] that has been unaddressed. 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