Systems & Control Letters 17 (1991) 105-113 105 North-Holland On nonconflicting languages that arise in supervisory control of discrete event systems * Enke Chen and St6phane Lafortune so share a trace containing this prefix, i.e., Department of Electrical Engineering and Computer Science, L 1 n L 2 = L 1 r3 L2, where the overbar notation de- University of Michigan, Ann Arbor, MI 48109-2122, USA notes the prefix-closure of a set. Closed (in the sense of prefix-closed) languages are always non- Received 27 December 1990 conflicting with one another. Revised 22 April 1991 The concept of nonconflicting languages finds applications in modular supervisory control [10] Abstract: We study four classes of nonconflicting sublanguages and in nonblocking supervisor design [2] of dis- of a given language that arise in supervisory control of discrete event systems. We first present closed-form expressions for the crete event systems. For instance, it is shown in supremal nonconflicting sublanguage and for the supremal [10] that nonconflicting is a sufficient condition closed nonconflicting sublanguage of a given language. The for the intersection of two controllable languages nonconflicting condition is with respect to a second given to be a controllable language. It is also shown in language. We then present algorithms to compute the supremal [10] that the conjunction of two nonblocking su- nonconflicting controllable sublanguage and the supremal closed nonconflicting controllable sublanguage of a given lan- pervisors is nonblocking if and only if the two guage. The regularity properties of these languages are also concerned languages are nonconflicting. In a dif- investigated. ferent context, it is shown in [2] that the inner- blocking measure of a supervisor is empty if and Keywords: Discrete event systems; supervisory control; formal only if two particular languages are nonconflicting languages; nonconflicting languages; controllable languages. (see [2], Section 3.2, for details). When solving supervisor synthesis problems for discrete event systems, it is usually necessary to 1. Introduction first calculate the supremal element of a certain class of languages, e.g., supremal controllable sub- Let ~ be a non-empty finite set of events language [9], supremal normal sublanguage [6], (alphabet) and denote by X* the set of all finite etc. The same situation arises for the class of traces of elements of ~, including the empty trace nonconflicting sublanguages of a given language e. A subset L _c ,~* is a language over ,~. Lan- (with respect to another fixed language). For in- guages are used to model the logical behavior of stance, this is the case in [2], Section 3.3, where in (uncontrolled or controlled) discrete event order to synthesize the so-called 'minimally re- processes. Several properties of languages such as, strictive non-innerblocking solution' of the super- controllability, observability and normality, have visory control problem with blocking, one must been studied extensively in supervisory control of calculate the supremal closed controllable noncon- discrete event systems (see, e.g., [8]). This paper is flicting sublanguage of a particular language, an concerned with the nonconflicting property of lan- unsolved problem. The primary motivation of this guages. This property was first introduced in [10]. paper is to address this computation and find Two languages L 1 and L 2 are said to be noncon- algorithms to calculate the supremal closed con- flicting if whenever they share a prefix, they al- trollable nonconflicting sublanguage. For this pur- pose, it is necessary to first deal with the computa- * Research supported in part by the National Science Founda- tion of the supremal nonconflicting sublanguage, tion under Grant ECS-9057967. and then introduce the requirements of prefix- 0167-6911/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland) 106 E. Chen, S. Lafortune / Nonconflicting languages in supervisory control closure and controllability. From a general point cg.~ has a supremal element (w.r.t. set inclusion) of view, the results that we establish on these denoted L ~ := sup c~£~a i.e., L ~ ~ WAx' and K special classes of nonconflicting sublanguages will cg&a ~ K _c L T. L T is called the supremal con- be of interest in other contexts as well. trollable sublanguage of L. Its computation is More specifically, we introduce and study four discussed in several references, among these [9,1,4]. nonconflicting sublanguages of a given language We recall a property which is stated in [1]. L: the supremal nonconflicting sublanguage (de- noted L NC ), the supremal closed nonconflicting l~mma 2.1 [1]. If B c Z* is closed, then for VL c_ sublanguage (denote L~c), the supremal noncon- S*, the language B - L~,* is also closed. [] flicting controllable sublanguage (denoted LCNc), and the supremal closed nonconflicting controlla- The following result (whose proof is straight- ble sublanguage (denoted L~NC). Here, the non- forward) will also be needed. conflicting condition is with respect to a second given language, and the controllability condition is Lemma 2.2. Let L, R c ,~* and L n R = ~J. Then with respect to a third given language and a fixed Z A RZ* = ~J and L N RZ* = fJ. [] set of uncontrollable events. We present closed- form expressions for the first two nonconflicting sublanguages and present algorithms for the com- 3. Supremal nonconflicting sublanguages putation of the last two nonconflicting sublan- guages. We establish the finite convergence of 3.1. General case these algorithms in the regular case based on a finite-state machine implementation of these al- gorithms. Consider the following class of languages: Our presentation is organized as follows. Nec- ,,2°NC ;= {K: (KCL) A(KNP=KNP)} (3.1) essary background and preliminary results are presented in Section 2. L NC and L~Nc are defined where L, P c,~* are two fixed languages. In and studied in Section 3, while Section 4 is de- words, Z,aNC is the class of sublanguages of L voted to LCN C and L~N C. Section 5 concludes the that are nonconflicting with P. We characterize paper. the supremal element (w.r.t. set inclusion) of ZaNc by the following result. 2. Preliminaries Theorem 3.1. (i) LNC := sup,~VNC is well defined. (ii) LNC = L - (L n P - L n P)2~*. We need to introduce some necessary back- (iii) LNcGP=LNcAP=LAP. ground for the work that follows. If s, s', t ~ X* with s't = s, then s' is a prefix of s; thus both e Proof. (i) We assume that K~ ~Z~aNC for a in and s are prefixes of s. The closure L of L is the some index set, i.e., language consisting of all the prefixes of traces in L; if L=~J then L=JJ, and if L4:~ then e~L. K c_L, K~AP=K~AP. Clearly L _c L L is closed if L = i.. A language is Then, (U~K~) _ L. Also, regular if and only if it is accepted by a finite automaton [3]. (U~,K,,) n P = U~,(K,~ n P) Let M be a fixed language over X, and let Xu be a fixed subset of ~ denoting the set of 'uncon- = uo(K-T ) trollable' events (in the sense that their occurrence cannot be disabled). A language K _c ,~* is said to be controllable with respect to (w.r.t.) M and Xu = (uo ) if ~"~u n M _c K [8]. The class of controllable =U~K~ n P. sublanguages of a given language L is defined as This shows that .L~aNC is closed under arbitrary 9'.,.~:= { K: (Kc L) A (,K~', n M c: ,K)}. unions. Thus, LNC := sup&aNc is well defined. E. Chen, S. Lafortune / Nonconflicting languages in supervisory control 107 (ii) Let L. Let us proceed as in (3.2). Thus (3.3), (3.4) and (3.6) are still valid. Also, let RHS := L- (L n P- Ln P)X*. LN c = RHS © R m. (3.7) Obviously, when ~L n P = L n P (e.g., if L = ~f or P =~), then LNc = RHS = L. The equation LNc As we know, = RHS is valid in this case. The proof that LNc =RHSwhenLnPcLnP(thusL4=~and P4= LNC C L, (3.8) ~f) is organized into three steps. LNC n P c LNC n P. (3.9) Step 1. We need to show that RHS c L, which is obviously true. Since Step 2. We need to show that RHS is noncon- flicting with P, i.e., RHS = L- RX* = L- (L n RX*), it follows that RHS n P= RHS N P. Let L=[L-(LnRZ*)] 0(CnRZ*) = RHS RZ*) LnP=LnPOR (3.2) ©(Ln and where © denotes disjoint union, R ¢ ~ and L n P n R = ~. Lemma 2.2 implies that LNC = RHS tJ R m L n P n RX* = ~, (3.3) _L (by (3.8)) ( L n P ) n RX* = ~f. (3.4) Thus Then, RHS = L - RX* c L - RX*. Hence: RrnC (L n Rz~*) (3.10) RHS c L - RX* and so = L- RX* (by Lemma 2.1). RmAPC (LAP) ARz~* RHS n P_C (L- R2*) nP =~ (by (3.4)). = (LAP) -R2* Hence = + R) - RZ* R m n P =~. (3.11) =LAP-RX* (sincet~X*) Substituting (3.7) in (3.9), we have =LnP (by (3.3)). RHSORmAP (RHS U Rm) AP RHS AP= (L-RX*)AP so that =(LAP)-RX* =LnP (by (3.4)). (RHS U Rm) nPc (RHS n P) 0(Rm riP) Therefore and thus RHS n P c RHS n P. (3.5) (RHS n P) U (Rmmn P) _ RHS n P (by (3.11)). Since the reverse inclusion of (3.5) is always true, But by (3.6), we know that RHS n P = RHS n P = L n P (3.6) RHSA P= RHSAP= LAP, which completes Step 2.