Towards Synthesizing Optimal Coordination Modules for Distributed Agents

In a discrete-event framework, we define the concept of a coordinable language and show that it is the necessary and sufficient existence condition of coordination modules for distributed agents to achieve conformance to a given inter-agent constraint language. We also present a synthesis algorithm to compute near optimal coordination modules. Background & Notation We use the following notations from language and automata theory (Cassandras and Lafortune 1999): • For an event set Σ, Σ denotes the set of all finite strings over Σ, including the empty string ε; and for two event sets Σ ⊆ Σ, PΣ2,Σ1 denotes the natural projection from (Σ) to (Σ), erasing all σ ∈ Σ − Σ in s ∈ (Σ). • For a language L over an event set Σ, i.e., L ⊆ Σ, L̄ denotes the set of all prefixes of its strings. L is said to be prefix-closed if L = L̄. • For an automaton A = ( X,Σ, δ, x0 , X A m ) , L(A) andLm(A) denote its generated prefix-closed andmarked languages, respectively; and for x ∈ X, s ∈ (Σ), δ(s, x)! denotes that δ(s, x) is defined; Trim(A) denotes the procedure that computes and returns a nonblocking automatonwhich generates the samemarked language as A; and A = A1 ‖ A2 denotes that automaton A is the synchronous product of the two automata A1 and A2. Consider a system modeled by an automaton A, with the event set Σ partitioned into (i) Σ = Σc . ∪ Σuc and (ii) Σ = Σo . ∪ Σuo, where Σ A c , Σ A uc, Σ A o and Σ A uo denote the sets of controllable, uncontrollable, observable and unobservable events of A, respectively. Let K be a sublanguage of Lm(A), i.e.,K ⊆ Lm(A). The statements ‘K is controllable w.r.tA, Σc ’ and ‘K is observable w.r.tA, PΣA,ΣAo ’ refer, respectively, to the concepts of controllability (Ramadge and Wonham 1987) and observability (Lin and Wonham 1988) of a language K in supervisory control theory. Finally, for an automatonC, theSupcon(C,A,Σc ) procedure (Wonham and Ramadge 1987) computes a nonblocking automatonS such thatLm(S) is the supremal controllable subCopyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. language (Ramadge andWonham 1987) of Lm(C)∩Lm(A) w.r.t A and Σc . Discrete-Event Agents & Coordination Consider a system of two agents modeled by the respective automata Ai = (X i,Σi , δi , xi 0 , X Ai m ) (i ∈ {1, 2}), where Σ1 ∩ Σ2 = ∅. The event set Σi of agent Ai is partitioned into the controllable set Σi c and the uncontrollable set Σi uc . In enabling distributed agents to coordinate, each agentAi is equipped with a coordination module (CM) modeled by an automaton Si with the following properties: 1. Σi = Σi ∪ ComSet(Si, Aj), where ComSet(Si, Aj) ⊆ Σj , (i, j ∈ {1, 2}, i 6= j). Σi is called the coordination event set for agent Ai, and ComSet(Si, Aj) is the set of events that agent Aj needs to communicate to Ai to synchronize Si. 2. Si is Σ Ai uc -enabling, namely, (∀s ∈ (Σ i))(∀σ ∈ Σi uc ) ((s ∈ L(Si ‖ Ai) and PΣSi ,Σi (s)σ ∈ L(Ai)) ⇒ (sσ ∈ L(Si ‖ Ai))). 3. Si and Sj are cooperative, namely, (∀s ∈ (Σ))(∀σ ∈ ComSet(Si, Aj)) ((PΣA,ΣAj (s)σ ∈ L(Aj) and PΣA,ΣSj (s)σ ∈ L(Sj)) ⇒ (PΣA,ΣSi (s)σ ∈ L(Si))). Let A = A1 ‖ A2 and S12 = (S1, S2) denote the CM pair of S1 and S2. Write S12/A for the system of two agents A1 and A2 coordinating through their respective CM’s. Definition 1. Coordinated Behaviors 1. Prefix-closed coordinated behavior L(S12/A) (a) ε ∈ L(S12/A). (b) (∀s ∈ L(S12/A))(∀σ ∈ Σi) (sσ ∈ L(S12/A) ⇔ (sσ ∈ L(A) and (PΣA,ΣSi (s)σ ∈ L(Si)))). 2. Marked coordinated behavior Lm(S12/A) Lm(S12/A) = L(S12/A)∩Lm(A)∩Lm(S1)∩Lm(S2). CM pair S12 is nonblocking if Lm(S12/A) = L(S12/A). Definition 2. Coordinable Language: Let Σcom ⊆ Σ. A languageK ⊆ Lm(A) is coordinable w.r.t A and Σcom if 1. K is controllable w.r.t A and Σc = Σ A1 c ∪ Σ A2 c ; and 2. K is observable w.r.t A and PΣA,ΣAi∪Σcom (i ∈ {1, 2}). Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008)