Reconfigurable Timed Discrete-Event Systems

The paper’s Theorem 1 states that any timed forcible path set P solving a centralized reconfiguration problem (TCRS) also solves the corresponding decentralized problem (TDRS = timed_localize(Δ)), and it asserts the equality TRS(@B,@A,!(Δ_TCRS)) = TRS(@′,@′′, timed_localize(Δ)) = P for suitable @′,@′′. The proof appeals to timed supervisor localization, using !(G) ∩ !(TDRS) = !(TCRS), then decomposes TDRS into localized controllers and shows that after any c ∈ P, the reconfiguration event f_A is eligible in both localized components, hence in TDRS (Theorem 1 and its proof; see the statement and derivation around eq. (12) and subsequent discussion ). The TRS algorithm and its path-set definition are given earlier (Problem 1, Definitions 2–5, and Algorithm 1), establishing that TRS returns the set of timed forcible paths P for a given problem . The candidate’s solution instead argues via language-driven invariance of the TRS backtracking forcibility tree (BFT): since timed localization preserves the closed behavior !(TCRS) = !(G) ∩ !(TDRS), the eligible/forcible next-event sets coincide at every prefix, making the centralized and decentralized BFTs label-isomorphic and thus yielding the same returned set P; it also explicitly transports the attraction field Z by a prefix-state mapping. This is a correct alternative proof strategy consistent with the definitions and with the localization equality used in the paper . The paper’s proof is somewhat terse about how TRS on TDRS reproduces exactly P (and about how Z is transported), but the intended conclusion is standard given that TRS returns the forcible-path solution set by definition; the candidate fills these details with a constructive isomorphism argument.