To appear in IEEE Transactions on Automatic Control, 2016. Online at: http://dx.doi.org/10.1109/TAC.2015.2447193

On Distributed and Parameterized Supervisor Synthesis Problems Liyong Lin, Alin Stefanescu, Rong Su

Abstract—It is shown that the problem whether an arbitrary regular language has a non-empty decomposable sublanguage with respect to a fixed distribution is decidable if and only if the independence relation induced by the distribution is transitive. A sufficient condition on the distributed control architecture is then derived, under which there exist some fixed non-blocking local generators such that the distributed supervisor synthesis problem is undecidable. We also show that a natural formulation of the parameterized supervisor synthesis problem is undecidable for a fixed non-blocking generator template, so long as the template alphabet has at least two private events and one global event that are controllable. In particular, all the undecidability results are still valid even if star free specification languages are considered. Index Terms – discrete-event systems, distributed synthesis, parameterized synthesis, supervisory control, trace theory

I. I NTRODUCTION A language L ⊆ Σ∗ is said to be decomposable with respect to a distribution, i.e., a collection of non-empty subalphabets, of Σ if L is equal to the synchronous product of its projections onto the respective subalphabets. The notion of decomposability has been well studied and applied in the literature, see for example [1], [2], [3] and [4]. We study in this note the problem of determining whether L has a non-empty decomposable sublanguage with respect to a given distribution, i.e., the decomposable sublanguage problem, and its variant. We also study their applications to the distributed and parameterized supervisor synthesis problems. The main results of our paper are as follows. We show that the decomposable sublanguage problem is decidable if and only if the independence relation, a symmetric and irreflexive relation induced by the given distribution, is transitive. This characterization relies on the result obtained for the universality problem studied in regular trace language theory [5], [6]. Then, we obtain a sufficient condition for the undecidability of the distributed supervisor synthesis problem, using the above characterization result. A decidability result for a class of distributed control architectures is then presented. We establish the undecidability of a parameterized version of the decomposable sublanguage problem, referred to as the specification template synthesis problem. This is a key step towards showing the undecidability of a parameterized supervisor synthesis problem. Indeed, we show that the specification template synthesis problem is undecidable if the template alphabet has at least two private events and the parameterized supervisor synthesis problem is undecidable for a fixed non-blocking L. Lin, R. Su are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore ([email protected], [email protected]). Alin Stefanescu is with the Department of Computer Science, University of Bucharest, Romania ([email protected]). The first and the third authors are supported by Singapore Ministry of Education Tier 1 Academic Research Fund RG84/13. The second author was partially supported by research grant MUVET no. PN-II-ID-PCE-2011-3-0688.

generator template if the template alphabet has at least two private events and one global event that are all controllable, even when the schematic specification language is required to be symmetric. In particular, it will be shown that all the undecidability results are still valid even if (schematic) star free specification languages or, equivalently, (schematic) linear temporal logic formulas are considered. The previous conference version has appeared in [7]. This work differs from [7] in the following aspects: 1) we reformulate the distributed and parameterized supervisor synthesis problems; 2) most results are strengthened or generalized, and complete proofs are provided; 3) Lemma 4 and Proposition 1 are new. The paper is organized as follows. Section II is devoted to preliminaries. In Section III, we provide a sufficient condition for the undecidability of the distributed supervisor synthesis problem. We establish a sufficient condition for the undecidability of a parameterized supervisor synthesis problem in Section IV and draw the conclusions in the last section. II. P RELIMINARIES The notations used here are standard in the theory of supervisory control and mostly follow [8], [9]. In the following, additional notations and terminologies are introduced. Let [1, n] denote the set {1, 2, . . . , n}. For a given alphabet Σ, a distribution of Σ of size n is an n-tuple ∆ = (Σ1 , Σ2 , .S . . , Σn ) of non-empty subalphabets of Σ n such that Σ = i=1 Σi and the subalphabets are pairwise incomparable with respect to set inclusion. Given a distribution ∆ = (Σ1 , Σ2 , . . . , Σn ) of Σ, we have n projections Pi from ∗ Σ∗ to Σ∗i and n inverse projections Pi−1 from Σ∗i to 2Σ . Also, both projections and inverse projections are naturally extended to mappings between languages. We say a language L is over Σ if L ⊆ Σ∗ . The synchronous product kni=1 Li Tn −1 of languages Li over Σi is defined as i=1 Pi (Li ). Given a language L over Σ, kni=1 Pi (L) is said to be the decomposition closure of L with respect to ∆. We denote the complement of a language L by Lc . L denotes the prefix closure of L. A − B denotes the set theoretic difference of A and B. A generator G over Σ is a five tuple (Q, Σ, δ, q0 , Qm ), where Q is the set of states, δ : Q × Σ 7→ Q is the partial transition function, q0 ∈ Q is the start state and Qm ⊆ Q is the subset of marker states [8]. When δ is complete, G is often called an automaton [9]. A supervisor S for G is another generator over Σ with the constraint that each state of S is marked; at each state of S, there is an outgoing transition for each event that S can not control and if there is an outgoing transition labeled by an event that S can not observe, then this transition is a selfloop. The controlled closed (respectively, marked) behavior of G under S is L(SkG) (respectively, Lm (SkG)), where SkG is the standard automata synchronous product operation [8], [9]. SkG is non-blocking iff Lm (SkG) = L(SkG).

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An independence relation I ⊆ Σ × Σ is a symmetric and irreflexive relation. Two words w, w0 over Σ, i.e., w, w0 ∈ Σ∗ , are said to be trace equivalent with respect to I, denoted by w ∼I w0 , if there exist words v0 , . . . , vn such that w = v0 , w0 = vn and for each i ∈ [1, n], there exist ui , u0i and ai , bi such that (ai , bi ) ∈ I, vi−1 = ui ai bi u0i and vi = ui bi ai u0i . The set of trace equivalent words of s for an independence relation I is called the trace closure of s, denoted by [s]I or [s] if I is clear from the context. The S trace closure [L] of a language L is defined to be the set s∈L [s]. A language L is said to be trace closed if L = [L]. A distribution ∆ = (Σ1 , Σ2 , . . . , Σn ) of Σ naturally induces an independence relation I(∆) ⊆ Σ × Σ in the following way. The reflexive, symmetric relation D(∆) = {(a, b) ∈ Σ × Σ | ∃i ∈ [1, n], a, b ∈ Σi } is called the dependence relation induced by ∆. Then I(∆) = Σ×Σ−D(∆) is the independence relation induced by ∆. In the rest of the paper, whenever we are given a distribution ∆, we also construct its independence relation I(∆). An independence relation I ⊆ Σ × Σ is said to be transitive if for all a, b, c ∈ Σ, a 6= c, (a, b) ∈ I, (b, c) ∈ I implies (a, c) ∈ I. The following lemmas [10], [11] are useful. Lemma 1. The trace closure and the decomposition closure of a word are equal, i.e., [s]I(∆) = kni=1 Pi (s), for any word s over Σ and any distribution ∆ = (Σ1 , Σ2 , . . . , Σn ) of Σ.  Lemma 2. Let ∆ = (Σ1 , Σ2 , . . . , Σn ) be any distribution of Σ. Let Li be any language over Σi , for each i ∈ [1, n]. Then kni=1 Li is trace closed with respect to I(∆).  The class of star free languages over Σ is defined as the smallest class of languages containing ∅, {σ} for each σ ∈ Σ, and being closed under union, intersection, complement and concatenation [12]. Clearly each star free language is regular. An example of star free language is (ab)∗ over {a, b}. The star freeness of (ab)∗ follows from (ab)∗ = (b∅c ∪ ∅c aa∅c ∪ ∅c bb∅c ∪ ∅c a)c . It is known that, when interpreted over finite words, the class of star free languages corresponds exactly to the class of linear temporal logic (LTL) or, equivalently, the class of first order definable languages [13]. III. D ECOMPOSABLE S UBLANGUAGE P ROBLEM AND ITS A PPLICATION Formally, a language L over Σ is said to be decomposable with respect to a distribution ∆ = (Σ1 , Σ2 , ..., Σn ) of Σ if L = ||ni=1 Pi (L). We study the decomposable sublanguage problem (DSP) stated below.

I(∆). It follows from the undecidability of the existence of a non-empty trace closed sublanguage that DSP is undecidable for the distribution ∆ = ({a, b}, {c}) of {a, b, c} [11], [14]. In this paper, we present a characterization of the decidability of DSP. The key observation is that there exists a closed form expression for the unique maximal trace closed sublanguage of any language [5]. Lemma 3. For any independence relation I ⊆ Σ × Σ and any language L over Σ, the unique maximal trace closed sublanguage of L with respect to I is [Lc ]c .  It is straightforward to see that ∃s ∈ L, [s] ⊆ L iff [Lc ]c 6= ∅ iff [Lc ] 6= Σ∗ . Thus DSP is equivalent to the non-universality problem (NUP) in trace theory [6]. Given a fixed independence relation I ⊆ Σ × Σ, the universality problem (UP) asks, for an arbitrary regular language L over Σ, whether the trace closure [L] of L is equal to Σ∗ . It is known that UP is decidable iff I is transitive [6]. Furthermore, in view of the undecidability of UP for star free languages [15], we have the following result. Theorem 1. The decomposable sublanguage problem is decidable iff I(∆) is transitive. If I(∆) is non-transitive, the undecidability result is still valid even if L is required to be star free.  We shall now apply Theorem 1 to the problem of distributed supervisor synthesis presented below. Before that, we need to define our notion of a distributed control architecture. Definition 1. A distributed control architecture over Σ is a tuple A = (Σ1 , Σ1,c , Σ2 , Σ2,c , . . . , Σn , Σn,c ) of subalphabets of Σ, such that ∆(A) := (Σ1 , Σ2 , . . . , Σn ) is a distribution of Σ and Σk,c ⊆ Σk for each k ∈ [1, n].  Problem 2 (Distributed Supervisor Synthesis Problem). Let A = (Σ1 , Σ1,c , Σ2 , Σ2,c , . . . , Σn , Σn,c ) be a fixed distributed control architecture over Σ. Let Gk be a fixed finite state, not necessarily non-blocking, generator over Σk , for each k ∈ [1, n]. For an arbitrary regular language L over Σ, determine whether there exist n supervisors Sk , k ∈ [1, n], with each supervisor Sk controlling all of Σk,c and observing all of Σk , such that: 1) knk=1 (Sk kGk ) is non-blocking, and 2) ∅= 6 knk=1 Lm (Sk kGk ) ⊆ L. 

Problem 1 (Decomposable Sublanguage Problem). Let ∆ be a fixed distribution of Σ. For an arbitrary regular language L over Σ, determine whether L has a non-empty sublanguage that is decomposable with respect to ∆. 

In above formulation of the distributed supervisor synthesis problem (DSSP), the distributed control architecture A and the local (finite state) generators Gk , k ∈ [1, n], are fixed a priori, while L is the only input of the decision problem. We say A satisfies condition (C) if there are i, j ∈ [1, n] and four letters a, b, c, d ∈ Σ such that {a, b, d} ⊆ Σi,c , {c, d} ⊆ Σj,c , (a, c), (b, c) ∈ I(∆(A)) and ∀σ ∈ Σ − {a, b, c, d}, ∃k ∈ [1, n], σ ∈ Σk,c . We have the following result.

It is not difficult to see that L has a non-empty decomposable sublanguage with respect to ∆ if and only if there is a word s in L such that the decomposition closure of s is a subset of L. Since the decomposition closure and the trace closure of a word coincide, by Lemma 1, L has a non-empty decomposable sublanguage with respect to ∆ if and only if L has a non-empty trace closed sublanguage with respect to

Theorem 2. Let A = (Σ1 , Σ1,c , Σ2 , Σ2,c , . . . , Σn , Σn,c ) be a fixed distributed control architecture over Σ that satisfies condition (C) with the four letters a, b, c, d as above. For each k ∈ [1, n], if d ∈ Σk , then let Lm (Gk ) = (Σk − {d})∗ d / Σk , then let Lm (Gk ) = and L(Gk ) = Lm (Gk ); if d ∈ L(Gk ) = Σ∗k . The distributed supervisor synthesis problem is undecidable for the given distributed control architecture A

and local generators Gk , k ∈ [1, n]. The undecidability result is still valid even if L is required to be star free.  Proof: ∆(A) is not transitive and each Gk is non-blocking. It will be shown that for any language L0 ⊆ {a, b, c}∗ , L0 has a non-empty decomposable sublanguage with respect to ∆(A) iff there exists a solution for the instance of DSSP with L = L0 d, for the given A and local generators Gk , k ∈ [1, n]. It then follows that for any distributed control architecture A that satisfies condition (C), there exist n fixed non-blocking generators Gk , k ∈ [1, n], such that DSSP is undecidable. From Theorem 1 and the above reduction, the undecidability result is still valid even if L is required to be star free. Indeed, if for the given A and Gk , k ∈ [1, n], the instance of DSSP with L = L0 d has a solution, then there exist n supervisors Sk , k ∈ [1, n], with each supervisor Sk controlling all of Σk,c and observing all of Σk , such that ∅ 6= knk=1 Lm (Sk kGk ) ⊆ L. And then knk=1 Lm (Sk kGk ) is a nonempty decomposable sublanguage of L = L0 d with respect to ∆(A). It follows that there exists a word s ∈ {a, b, c, d}∗ such that [s] ⊆ L0 d. Since (a, d), (b, d), (c, d) ∈ D(∆(A)), it follows that s = s0 d for some word s0 ∈ {a, b, c}∗ and [s] = [s0 d] = [s0 ]d ⊆ L0 d. It follows that [s0 ] ⊆ L0 , i.e., L0 has a non-empty decomposable sublanguage with respect to ∆(A). On the other hand, if L0 has a non-empty decomposable sublanguage with respect to ∆(A), then there exists a word s ∈ {a, b, c}∗ such that knk=1 Pk (s) ⊆ L0 . Let sk = Pk (s)d if d ∈ Σk and sk = Pk (s) if d ∈ / Σk . It is easy to see that ∅= 6 knk=1 sk ⊆ L0 d = L. We construct the supervisors Sk , k ∈ [1, n], as follows: for each event σ ∈ Σ − {a, b, c, d}, let one of the supervisors Sk with σ ∈ Σk,c always disable it, which is possible due to condition (C). It follows that the closed loop system can only execute events in {a, b, c, d}. For the particular i, j ∈ [1, n] such that {a, b, d} ⊆ Σi,c , {c, d} ⊆ Σj,c , let supervisor Si enforces the execution of closed behavior Pi (s)d by Gi and let supervisor Sj enforces the execution of closed behavior Pj (s)d by Gj . All the other supervisors Sk are not allowed to disable events in {a, b, c, d} ∩ Σk,c . By construction, ∅ 6= knk=1 Lm (Sk kGk ) = knk=1 sk ⊆ L. Also  knk=1 (Sk kGk ) is non-blocking. The negative result established above is also applicable to the more general problem formulation, where both the local generators and the specification are the inputs of the decision problem. The distributed supervisor synthesis problem, where the global generator G is given locally as interacting generators over local subalphabets, may be considered as a special case of the decentralized supervisor synthesis problem [1], [16], [17], [18]. It follows that the above undecidability result immediately implies the undecidability of the decentralized supervisor synthesis problem1 . In fact, condition (C) easily translates to a sufficient condition for the undecidability of the decentralized supervisor synthesis problem, for the fixed centralized generator kni=1 Gi given in Theorem 2. The undecidability results of [19], [20] also consider LTL specifications, 1 However, this does not imply that our result is stronger than the results of [16], [17], [18]. In [16], the supervisor is not required to be finite state. In [17], uncontrollable events are used. In [18], both the supervisors and the closed loop system are required to satisfy modal (loop) formulas.

however, interpreted over ω-words. It turns out that only four letters a, b, c, d are needed in order to establish the undecidability of DSSP. It is still open, when Σ has three letters, whether DSSP is decidable for any given A over Σ and local generators. However, it is possible to show that with Σ = {a, b}, DSSP is always decidable. In fact, it will be shown that for any fixed A that satisfies the condition that ∃i, (∀j 6= i, |Σj | = 1), DSSP is decidable for any fixed local generators. The following lemma is needed. Lemma 4. Given any regular language L3 over Σ3 and any regular language L2 over Σ2 $ Σ3 , the supremal language L↑1 over Σ1 := Σ3 − Σ2 such that L↑1 kL2 ⊆ L3 is effectively regular. Indeed L↑1 = Σ∗1 \P1 (P2−1 (L2 )\L3 ), where P1 : Σ∗3 7→ Σ∗1 and P2 : Σ∗3 7→ Σ∗2 are the natural projections.  Proof: It is clear that the supremal element L↑1 of {L1 ⊆ | L1 kL2 ⊆ L3 } exists and is unique. By definition, L↑1 = {s ∈ Σ∗1 | skL2 ⊆ L3 } = {s ∈ Σ∗1 | P1−1 (s) ∩ P2−1 (L2 )\L3 = ∅} = Σ∗1 \{s ∈ Σ∗1 | P1−1 (s) ∩ P2−1 (L2 )\L3 6= ∅} = Σ∗1 \P1 (P2−1 (L2 )\L3 ).  Σ∗1

Proposition 1. The distributed supervisor synthesis problem is decidable, for any fixed distributed control architecture A that satisfies the condition ∃i, (∀j 6= i, |Σj | = 1) and for any fixed local generators Gk , k ∈ [1, n].  Proof: Let Σ = {ak | k ∈ [1, n + m]}, where m ≥ 0, and we assume, without loss of generality, Σk = {ak } for k ∈ [1, n − 1] and Σn = {an , . . . , an+m }. Let Gk be any fixed local generator over Σk , k ∈ [1, n]. Let L be an arbitrary regular language over Σ. If there is any k ∈ [1, n − 1] such that ak is uncontrollable and Gk is blocked, then there is no solution to the distributed supervisor synthesis problem. Otherwise the test for the existence of non-blocking supervisors could be carried out as follows: For each k ∈ [1, n−1], if ak is uncontrollable, let Sk be a non-blocking supervisor such that Lm (Sk kGk ) = Lm (Gk ); if ak is controllable, let Sk be a nonblocking supervisor such that Lm (Sk kGk ) = {sk }, where sk is the shortest word of Lm (Gk ). Let Lk := Lm (Sk kGk ) for k ∈ [1, n−1]. We compute the supremal language L↑n such that L1 kL2 k. . . kLn−1 kL↑n ⊆ L, with Lemma 4. It is not difficult to see that the instance of DSSP has a solution iff the supremal controllable and Lm (Gn )-closed sublanguage of L↑n ∩Lm (Gn ) is non-empty, the latter of which is decidable [21].  Remark: An implication of Proposition 1 is that DSSP is decidable for any A, with ∆(A) = ({a, b}, {c}), and any local generators G1 , G2 . The decomposable sublanguage problem is undecidable for this distribution.  The standard setup for the distributed supervisor synthesis problem assumes no structural relations between the local generators and supervisors. In next section, an important subclass of the distributed supervisor synthesis problem, where the given local generators are assumed to be isomorphic and the supervisors are required to be isomorphic, is considered. IV. PARAMETERIZED S UPERVISOR S YNTHESIS P ROBLEM The idea of parameterized supervisor synthesis is to use instantiated supervisors from a designed supervisor template

to control a parameterized family of finite state generators in a uniform manner. A natural top-down approach for parameterized supervisor synthesis starts with a schematic regular language (user-given specification) L(n) (see [22]), parameterized by the number of local generators n, and a generator template G (a generator over the template alphabet) to be controlled; then, a non-empty specification template T (a regular language over the template alphabet) is synthesized such that the composition of n isomorphic copies of T satisfies the specification L(n), for any n ≥ 2; finally, a supervisor template S (a supervisor over the template alphabet) is synthesized according to the specification template T and generator template G. Our first result in this section is to show that the problem of determining whether there exists a non-empty specification template T as above is undecidable. To that end, we introduce the following terminologies and notations, before we present a formal statement of the problem. Let ΣT = Σg ∪ Σp be the template alphabet [23], where Σg is the global event set and Σp is the private event set. A distribution ∆T (n) = (Σ1 , Σ2 , . . . , Σn ) of size n based Sn on template alphabet ΣT is a distribution of Σ(n) := i=1 Σi , where Σi = Σg ∪ (Σp × {i}) for every i ∈ [1, n]. Let hi : ΣT 7→ Σi be a bijective map that maps events in ΣT to events in Σi for each i ∈ [1, n], such that hi (σ) = σ, if σ ∈ Σg and hi (σ) = (σ, i), if σ ∈ Σp . To simplify the notation, (σ, i) is denoted by σi . Depending on the context, hi may also be regarded as a bijective map from Σ∗T to ∗ ∗ Σ∗i or a bijective map from 2ΣT to 2Σi . A language L over Σ(n) is said to be symmetric with respect to ∆T (n) if −1 ∀i, j ∈ [1, n], h−1 i (Pi (L)) = hj (Pj (L)). Problem 3 (Specification Template Synthesis Problem). Let ΣT = Σg ∪ Σp be a fixed template alphabet. Sn For an arbitrary schematic regular language L(n) over i=1 Σi parameterized by n, where Σi ’s are isomorphic copies of the template alphabet ΣT , i.e., Σi = hi (ΣT ), determine whether there exists a non-empty specification template T ⊆ Σ∗T such that the composition kni=1 hi (T ) of n isomorphic copies of T satisfies the specification L(n), i.e., kni=1 hi (T ) ⊆ L(n), for all n ≥ 2.  The specification template synthesis problem (STSP) is essentially a parameterized version of the decomposable sublanguage problem. By modifying the reduction used in [15] to work in the parameterized setting, it will be shown that STSP is undecidable. The construction is provided below. Let A, B be two disjoint alphabets. Let ΣT = A∪B and hi : ΣT 7→ Σi be a bijective map that maps events in ΣT to events in Σi such that hi (σ) = σi for any σ ∈ ΣT , for any i ∈ [1, n]. That is, ΣT = Σp and ∀i ∈ [1, n], Σi = Ai ∪ Bi . Clearly the independence relation I(n) induced by ∆T (n) is {(x, y) ∈ Σ(n) × Σ(n) | x ∈ Σi and y ∈ Σj for different i and j}. Let f, g : A∗ 7→ B ∗ be two given homomorphisms such that |f (σ)| ≥ 1 and |g(σ)| ≥ 1 for any σ ∈ A, and fi , gi be two homomorphisms from A∗i to Bi∗ such that fi (σi ) = hi (f (σ)) and gi (σi ) = hi (g(σ)) for each σ ∈ A and i ∈ [1, n]. For any w ∈ A+ , let sfi (w) = hi (wf (w)) ∈ (Ai ∪ Bi )∗ and sgi (w) = hi (wg(w)) ∈ (Ai ∪ Bi )∗ , for each i ∈ [1, n]. Wf (n) = {sf1 (w)sf2 (w) . . . sfn (w) | w ∈ A+ } and Wg (n) =

{sg1 (w)sg2 (w) . . . sgn (w) | w ∈ A+ } are constructed for each n ≥ 2. Then the following key lemma holds. Lemma 5. There exist two star free languages Lf (n) and Lg (n) over Σ(n) such that [Lf (n)]c = [Wf (n)] and [Lg (n)]c = [Wg (n)] with respect to I(n); and moreover, L(n) = Σ(n)∗ − (Lf (n) ∪ Lg (n)) is symmetric with respect to ∆T (n) and star free, for each n ≥ 2.  Proof: The construction spirit described in [15] is adopted here and the following sets are constructed, whose union defines Lf (n). The basic idea is that there are a finite number of rules which determine whether a word s ∈ Σ(n)∗ is in [Wf (n)]. They are listed below. A word s ∈ [Wf (n)] if and only if the following conditions hold, 1) for each i, letters from each Ai must appear in s; 2) for each i, no letter of Bi comes before letters of Ai ; −1 3) for each i, j, h−1 i (PAi (s)) = hj (PAj (s)), where PAi : ∗ ∗ Σ(n) → Ai is the natural projection; −1 4) for each i, j, h−1 i (PBi (s)) = hj (PBj (s)), where PBi : ∗ ∗ Σ(n) → Bi is the natural projection; 5) for each i, PAi (s) matches PBi (s) in the form of hi (wf (w)) with hi (w) ∈ A∗i and hi (f (w)) ∈ Bi∗ . A regular language Li (n) is then constructed such that each word in its trace closure [Li (n)] violates condition (i), for each i ∈ [1, 5]. It follows that the complement of the union of [Li (n)]’s (i ∈ [1, n]) will be the set of words satisfying allSconditions, i.e., the set [Wf (n)]. Then by setting 5 Lf (n) = i=1 Li (n), in view of the property that trace closure distributes it follows that [Lf (n)]c = S5 T5over union, c c [ i=1 Li (n)] = i=1 [Li (n)] = [Wf (n)]. The constructions of Li (n)’s are given below. Sn S Sn 1) Let L1 (n) := i=1 ( j∈[1,n],j6=i Aj ∪ ( k=1 Bk ))∗ , the complement of whose trace closure ensures that each word contains letters Sn from each Ai . 2) Let L2 (n) := i=1 Σ(n)∗ Bi Σ(n)∗ Ai Σ(n)∗ , the complement of whose trace closure ensures that, for each i ∈ [1, n], no letter of Bi comes before letters of Ai . 3) Regarding the third condition, let L3 (n) := S S + ∗ ∗ ( i,j∈[1,n]:i6=j k∈[1,n]:k6=i∧k6=j Ak ) [(Ai Aj ) (Ai ∪ + Aj ) ∪ (Ai Aj )∗ {yy 0 | y ∈ Ai ∧ y 0 ∈ Aj ∧ y 0 6= Sn ∗ ∗ hj (h−1 i (y))}(Ai Aj ) ]( k=1 Bk ) . The meaning of L3 (n) can be explained as follows. Two integers i and j in [1, n] with i 6= j are arbitrarily picked, and due to the independence relation I(n), all other Ak ’s can be moved to the leftmost position, and all Bk ’s (including Bi ’s and Bj ’s) to the rightmost position. The expression in the square bracket is to enumerate −1 two cases which lead to h−1 i (PAi (s)) 6= hj (PAj (s)). These are: (1) more letters from Ai or more letters from Aj appear in s, which is captured by the expression + (Ai Aj )∗ (A+ i ∪Aj ); (2) an equal number of letters from both Ai and Aj but some do not match, i.e., there exist y ∈ Ai and y 0 ∈ Aj such that y is paired with y 0 but y 0 6= hj (h−1 i (y)). 4) Similarly a regular language L4 (n) could be constructed, the complement of whose trace closure ensures the

fourth condition: S Sn L4 (n) := i,j∈[1,n]:i6=j ( k=1 Ak )∗ [(Bi Bj )∗ (Bi+ ∪ Bj+ ) ∪ (Bi Bj )∗ {yy 0 | y ∈ Bi ∧ y 0 ∈ Bj ∧ y 0 6= S ∗ ∗ hj (h−1 i (y))}(Bi Bj ) ]( k∈[1,n]:k6=i∧k6=j Bk ) . 5) To come up with L5 (n) for the last condition the following construction is performed: S S L5 (n) = i,j∈[1,n]:i6=j ( k∈[1,n]:k6=i Ak )∗ [{xs | x ∈ Ai ∧ s ∈ Bj∗ ∧ |s| = |fi (x)|}∗ ({xs | x ∈ Ai ∧ s ∈ Bj∗ ∧ |s| < |fi (x)|}A∗i ∪ Bj+ ) ∪ {xs | x ∈ Ai ∧ s ∈ Bj∗ ∧ |s| = |fi (x)|}∗ {ys0 | y ∈ Ai ∧ s0 ∈ Bj∗ ∧ |s0 | = |fi (y)| ∧ s0 6= hj (h−1 | x ∈ Ai ∧ s ∈ i (fi (y)))}{xs S Bj∗ ∧ |s| = |fi (x)|}∗ ]( k∈[1,n]:k6=j Bk )∗ . By condition (4) it is known that the word in Bi∗ and the word in Bj∗ are isomorphic. Thus the word in Bj∗ is used to compare with the word in A∗i . The main construction is similar to the constructions for L3 (n) and L4 (n), i.e., all cases that violate condition (5) are enumerated, which are described in the square bracket. That is: either more letters from Ai (but note that the matching is from one letter in Ai to a word in Bi∗ instead of one letter in Ai to one letter in Bi due to the usage of fi : A∗i → Bi∗ ) or more letters from Bj (which corresponds to Bi due to condition (4)) will violate the matching condition, or the number of letters from Ai and the number of letters from Bj match with each other, but some y ∈ Ai does not match s0 ∈ Bj∗ in the sense that s0 6= hj (h−1 i (fi (y))). It can be shown that all above five languages are indeed star free. It follows that Lf (n) is star free language. Now by suitably enlarging Lf (n) it is possible to make it become symmetric with respect to ∆T (n) while preserving [Lf (n)]c = [Wf (n)] and the star freeness of Lf (n). Indeed, the relative positions of the subwords sfi (w) in s(n) := sf1 (w)sf2 (w) . . . sfn (w) do not make a difference in [Wf (n)] due to the independence relation I(n). Thus any procedure to construct Lf (n) could be extended to a procedure that constructs a symmetric Lf (n), by taking the union of n! copies of Lf (n) with indexes permuted. The meaning of index permutation is formalized by a permutation π : [1, n] 7→ [1, n]. Abusing the notation, we write π(σi ) = σπ(i) for any σ ∈ Σp , for any i ∈ [1, n] and extend S π to a homomorphism between n words and languages over i=1 (Σp × {i}). Similarly, a symmetric and star free Lg (n) can be constructed in this manner. By construction, both of the new Lf (n) and Lg (n) are invariant under index permutation, i.e., for any permutation π, π(Lf (n)) = Lf (n) and π(Lg (n)) = Lg (n). Thus L(n) = Σ(n)∗ −(Lf (n)∪Lg (n)) is also invariant under index permutation. Invariance under index permutation is stronger than symmetry property. It follows that L(n) is also symmetric with respect to ∆T (n). Clearly L(n) is also star free.  Now it is possible to establish the undecidability of STSP. Theorem 3. The specification template synthesis problem is undecidable if ΣT has at least two private events. The undecidability holds even when L(n) is required to be symmetric with respect to ∆T (n) and star free, for each n ≥ 2.  Proof: Let A, B be two disjoint alphabets. Any instance of Post’s Correspondence Problem (PCP) [9], [15] is encoded by

two homomorphisms f, g : A∗ 7→ B ∗ . The given PCP instance has a solution iff ∃w ∈ A+ , f (w) = g(w). Let ΣT = A ∪ B be a template alphabet with ΣT = Σp and hi : ΣT 7→ Σi be the bijective map. For each n ≥ 2, the star free language L(n) over Σ(n) is constructed, which is symmetric with respect to ∆T (n), from Lf (n) and Lg (n) that are constructed, respectively, from Wf (n) = {sf1 (w)sf2 (w) . . . sfn (w) | w ∈ A+ } and Wg (n) = {sg1 (w)sg2 (w) . . . sgn (w) | w ∈ A+ } as in Lemma 5. It will be shown that ∃w ∈ A+ , f (w) = g(w) iff there exists a non-empty specification template T ⊆ Σ∗T such that kni=1 hi (T ) ⊆ L(n), for any n ≥ 2. Suppose w0 ∈ A+ such that f (w0 ) = g(w0 ). Then clearly f si (w0 ) = hi (w0 f (w0 )) = hi (w0 g(w0 )) = sgi (w0 ) for each i ∈ [1, n] and it follows that sf1 (w0 )sf2 (w0 ) . . . sfn (w0 ) = sg1 (w0 )sg2 (w0 ) . . . sgn (w0 ) for any n ≥ 2. It follows that s(n) = sf1 (w0 )sf2 (w0 ) . . . sfn (w0 ) ∈ Wf (n) ∩ Wg (n) ⊆ [Wf (n)] ∩ [Wg (n)] = Σ(n)∗ − [Lf (n) ∪ Lg (n)] ⊆ L(n) and at the same time [s(n)] ⊆ [Wf (n) ∩ Wg (n)] ⊆ [Wf (n)] ∩ [Wg (n)] ⊆ L(n), for any n ≥ 2. In view of Lemma 1, this implies kni=1 Pi (s(n)) ⊆ L(n), where Pi : Σ(n)∗ → Σ∗i is the natural projection. The special form s(n) = sf1 (w0 )sf2 (w0 ) . . . sfn (w0 ) ensures that ∀i ∈ [1, n], Pi (s(n)) = hi (w0 f (w0 )). By setting T = {w0 f (w0 )} ⊆ Σ∗T , it follows that kni=1 hi (T ) ⊆ L(n), for any n ≥ 2. On the other hand, suppose T0 is a non-empty specification template over ΣT such that kni=1 hi (T0 ) ⊆ L(n), for any n ≥ 2. Since kni=1 hi (T0 ) is trace closed with respect to I(n) by Lemma 2, it follows that kni=1 hi (T0 ) is a non-empty subset of the maximal trace closed sublanguage of L(n), i.e., kni=1 hi (T0 ) ⊆ [L(n)c ]c = Σ(n)∗ − [Lf (n) ∪ Lg (n)] = [Wf (n)] ∩ [Wg (n)]. Let s be any word of [Wf (n)] ∩ [Wg (n)]. There exist w0 , w1 ∈ A+ such that Pi (s) = sfi (w0 ) = sgi (w1 ) for i ∈ [1, n]. This implies that sf1 (w0 )sf2 (w0 ) . . . sfn (w0 ) = sg1 (w1 )sg2 (w1 ) . . . sgn (w1 ) ∈ Wf (n) ∩ Wg (n) and then s ∈ kni=1 Pi (s) ⊆ [Wf (n) ∩ Wg (n)]. It follows that Wf (n) ∩ Wg (n) 6= ∅ and therefore ∃w ∈ A+ , f (w) = g(w). We finally note that the alphabets A and B can be encoded with the set {a, b} [11], [15]. Thus the above reduction works if ΣT has at least two private events.  Any reasonable modeling formalism is capable of expressing two private events. So the above undecidability result is rather strong. By reduction from the undecidability of STSP, we are now able to establish the undecidability of a natural formulation of the parameterized supervisor synthesis problem (PSSP) (without being restricted to the top-down approach), to be formally defined below. Problem 4 (Parameterized Supervisor Synthesis Problem). Let ΣT be a fixed template alphabet and let ΣT,c ⊆ ΣT be a fixed subalphabet. Let G be a fixed non-blocking generator template over ΣT . For an arbitrary schematic regular language L(n) ⊆ Σ(n)∗ , determine whether there exists a supervisor template S over ΣT , where S observes all of ΣT and controls all of ΣT,c , such that for any n ≥ 2: 1) kni=1 (Si kGi ) is nonblocking, and 2) ∅ = 6 kni=1 Lm (Si kGi ) ⊆ L(n). Here each Si (respectively, Gi ) over Σi is an isomorphic copy of S (respectively, G) obtained by relabeling each σ ∈ ΣT to hi (σ), and Σi is an isomorphic copy of ΣT . 

Theorem 4. Let ΣT and ΣT,c be given such that there are two private events a, b and one global event g with {a, b, g} ⊆ ΣT,c . Let G be a fixed generator template such that Lm (G) = (a + b)∗ g and L(G) = Lm (G). The parameterized supervisor synthesis problem is undecidable in this setting, even if L(n) is required to be symmetric with respect to ∆T (n) and star free, for each n ≥ 2.  Proof: Let Σ0T = {a, b} and h0i : {a, b} 7→ {ai , bi } be the corresponding bijective map.SLet L0 (n) be an arbitrary n schematic regular language over ı=1 {ai , bi }. Similar to the proof of Theorem 2, there exists a non-empty specification template T 0 ⊆ {a, b}∗ such that kni=1 h0i (T 0 ) ⊆ L0 (n) for each n ≥ 2 iff there exists a solution for the instance of PSSP with L(n) = L0 (n)g, for the given alphabets ΣT , ΣT,c and generator template G. From Theorem 3 and the above reduction, the undecidability is still valid even if L(n) is required to be symmetric with respect to ∆T (n) and star free, for each n ≥ 2.  Related Results: The above proof also works for the setting when n is a fixed integer, for any n ≥ 2, or there is an upper bound on the number of generators [24], [25]. It is noteworthy that only global and private events are needed in our problem formulation and each global event has to be synchronized by all the isomorphic copies Gi , Si , i ∈ [1, n]. And any supervisor template S is by definition globally consistent, i.e., for any n ≥ 1 and any i ∈ [1, n], L(Si ) = Pi (knj=1 L(Sj )) [26]. The undecidability of parameterized supervisor synthesis in general easily follows from the undecidability of checking the non-blocking property [27], even when all events are controllable. However, the work in [27] considers broadcast templates and networks of isomorphic processes that are incomparable with the templates studied in this work. In fact, the problem of verifying non-blocking properties is decidable for templates with only global and private events [23], [28]. Recently, it has been shown that parameterized synthesis for specifications in indexed LTL\X is undecidable [25]. However, their specification formula is interpreted over ω-words. As a potential tool for dealing with parameterized supervisor synthesis problems, the notion of weak invariant simulation is recently proposed in [29] for the analysis of parameterized discrete event systems. A decidable fragment of state based parameterized supervisor synthesis problem, without considering the non-blocking property, is reported in [30].  V. C ONCLUSION It is shown in this work that the distributed and parameterized supervisor synthesis problems are undecidable under some weak conditions, even if the (schematic) specification is required to be star free or, equivalently, LTL definable. In future work, we will report our current efforts in designing efficient semi-decision procedures for these supervisor synthesis problems to cope with the undecidability results. ACKNOWLEDGMENTS It is a pleasure to acknowledge the help of Prof. Wonham to the development of Lemma 3. We also thank the anonymous reviewers for observations that improved the paper.

R EFERENCES [1] Y. Willner, M. Heymann. “Supervisory control of concurrent discrete event systems”, Int. Journal of Control, 54(5): 1143-1169, 1991. [2] F. Lin, W. M. Wonham. “Decentralized supervisory control of discreteevent systems”, Information Sciences, 44(3): 199-224, 1988. [3] K. Cai, W. M. Wonham. “Supervisor localization: a top-down approach to distributed control of discrete-event systems”. IEEE Trans. on Automatic Control, 55(3): 605-618, 2010. [4] J. Komenda, T. Masopust, J. H. van Schuppen. “On conditional decomposability”, Systems & Control Letters, 61(12): 1260-1268, 2012. [5] I. J. Aalbersberg, E. Welzl. “Trace languages defined by regular string languages”, R.A.I.R.O. Theoretical Informatics and Applications, 20(2): 103-119, 1986. [6] J. Sakarovitch. “The “last” decision problem for rational trace languages”. LATIN’92, LNCS, Vol. 583, pp. 460-473, 1992. [7] L. Lin, A. Stefanescu, R. Su, W. Wang, A. R. Shehabinia. “Towards decentralized synthesis: decomposable sublanguage and joint observability problems”. American Control Conference, pp. 2047-2052, 2014. [8] W. M. Wonham. Supervisory control of discrete-event systems, Syst. Control Group, Dept. of ECE, University of Toronto, Canada, 2010. [9] J. E. Hopcroft, J. D. Ullman. Introduction to automata theory, languages, and computation, Addison-Wesley, Reading, Massachusetts, 1979. [10] S. Tripakis. “Two-phase distributed observation problems”. 5th Application of Concurrency to System Design, pp. 98-105, 2005. [11] A. Stefanescu. “Automatic synthesis of distributed transition systems”, Ph.D. dissertation. University of Stuttgart, 2006. [12] J. E. Pin. Varieties of formal languages, North Oxford, London and Plenum, New-York, 1986. [13] V. Diekert, P. Gastin. “First-order definable languages”, Logic and Automata: History and Perspectives, pp. 261-306, 2008. [14] Y. Chen, X. C. Ding, A. Stefanescu, C. Belta. “Formal approach to the deployment of distributed robotic teams”, IEEE Trans. on Robotics, 28(1): 158-171, 2012. [15] A. Muscholl, H. Petersen. “A note on the commutative closure of starfree languages”, Inform Process Lett., 57(2): 71-74, 1996. [16] S. Tripakis. “Undecidable problems of decentralized observation and control on regular languages”, Inform Process Lett., 90(1): 21-28, 2004. [17] J. G. Thistle. “Undecidability in decentralized supervision”, Systems & Control Letters, 54(5): 503-509, 2005. [18] A. Arnold, A. Vincent, I. Walukiewicz. “Games for synthesis of controllers with partial observation”, Theor. Comp. Sci, 303(1): 7-34, 2003. [19] P. Madhusudan, P. S. Thiagarajan. “A decidable class of asynchronous distributed controllers”, 13th Concurrency Theory, pp. 145-160, 2002. [20] S. Schewe. “Distributed synthesis is simply undecidable”, Inform Process Lett., 114(4): 203-207, 2014. [21] R. M. Ziller, J. E. R. Cury. “On the supremal Lm -closed and the supremal Lm -closed and L-controllable sublanguages of a given language”, 11th International Conference on Analysis and Optimization of Systems: Discrete Event Systems, LNCIS, Vol. 199, pp. 80-85, 1994. [22] K. R. Apt, D. C. Kozen. “Limits for automatic verification of finite-state concurrent systems”, Inform Process Lett., 22(6): 307-309, 1986. [23] K. Rohloff and S. Lafortune. “The verification and control of interacting similar discrete event systems”, SIAM Journal on Control and Optimization, 45(2): 634-667, 2006. [24] R. Su, L. Lin. “Synthesis of control protocols for multi-agent systems with similar actions”, IEEE 52nd Conference on Decision and Control, pp. 6986-6991, 2013. [25] S. Jacobs, R. Bloem. “Parameterized synthesis”, Logical Methods in Computer Science, 10(1:12), 1-29, 2014. [26] R. Su, W. M. Wonham. “Hierarchical fault diagnosis for discrete-event systems under global consistency”, Discrete Event Dyn Syst, 16(1): 3970, 2006. [27] K. Nazari, J. G. Thistle. “Blocking in fully connected networks of arbitrary size”, IEEE Trans. on Auto. Control, 57(5): 1233-1242, 2012. [28] W. Wang, R. Su, L. Lin. “On analysis of deadlock and blocking freeness in isomorphic module systems”, American Control Conference, pp. 923928, 2013. [29] M. H. Zibaeenejad, J. G. Thistle. “Weak invariant simulation and analysis of parameterized networks”, IEEE Trans. on Automatic Control, 59(8): 2024 - 2037, 2014. [30] H. Bherer, J. Desharnais, and R. St-Denis. “Control of parameterized discrete event systems”, Discrete Event Dyn Syst, 19(2): 213-265, 2009.

On Distributed and Parameterized Supervisor Synthesis Problems

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Distributed PageRank Computation Based on Iterative ... - CiteSeerX
Oct 31, 2005 - Department of Computer. Science. University of California, Davis. CA 95616, USA .... sults show that the DPC algorithm achieves better approx-.