Betti numbers and minimal free resolutions for multi-state system reliability bounds

The paper continues work on monomial ideals in system reliability began by Giglio and Wynn [GW04] following work in discrete tube theory by Naiman and Wynn [NW92, NW97]. The key component is that of multigraded Betti numbers, and an algorithm using MayerVietoris trees by the first author [dC06] is the main tool. First a mapping must be made between the states of a multistate system and a monomial ideal, or more specifically a collection of monomials. A multi-state system is a system of n components whose states are described by real variables Y = (Y1, . . . , Yn). The (discrete) states of each system are labeled by {1, 2, . . .} = N so that Y = Nn. Then a = (a1, . . . , an) ∈ Y is encoded by xa = x1 1 · · ·xan n . Assume that the system has a distinguished subset F , called the failure set which is such that for a ≤ b,a ∈ F ⇒ b ∈ F . Then the system is said to be coherent: if the system fails, in the sense of its state being in F , it will also fail at a more extreme state. The main conceptual link between the two fields is that this correspond exactly to the monomial ideal property. Thus, if IdF = 〈xa : a ∈ F〉 then a ≤ b, xa ∈ IdF ⇒ xb ∈ IdF . In addition we may consider the minimal cut set F∗ which corresponds to the minimal generators of the ideal IdF . If the behaviour of the system is described by allowing the state to be the realisation of a random variable Y , then the failure probability is prob{Y ∈ F}. If we can find bounds or equalities for the indicator function of F then they are inherited by this probability. Equivalently, here, we bound the generating function: F(x) = ∑ a∈F x a. Consider a multigraded R-module, M, over the ring R = k[x1, . . . , xn] considered as a k vector space over each of its multigraded “pieces”, and a monomial ideal I. If an R-resolution P of I is multigraded we obtain the muligraded Hilbert series is given by