An overview of reconstructability analysis <s> SRA Example <s> A new bi-colored miniature rose plant which has abundant flowers and good keepability. The variety successfully propagates from softwood cuttings and is suitable for year round production in commercial glasshouses. This new and distinct variety has shown to be uniform and stable in the resulting generations from such asexual propagation. <s> BIB001 </s> An overview of reconstructability analysis <s> SRA Example <s> Little is known about the behavior of dynamic systems with many intricately interacting parts, and about the factors which tend to affect their behavior in general, rather than detailed, ways. This paper describes a study of such systems built up out of unit elements which compute recursive logical functions. <s> BIB002
SRA reconstruction is exemplified by a study which attempted to predict whether discrete dynamic systems --specifically, elementary cellular automata (ECA) --are chaotic or not. Table 12 . Structures for ECA rules. The σ identifier for the 6 structural levels is given, and the 9 different specific structures are shown in bold. An ECA is a one-dimensional array of cells, s(1)...s(n), governed by a mapping, s t (i-1) ⊗ s t (i) ⊗ s t (i+1) → s t+1 (i), which specifies how the state of each cell at time t+1 depends upon its state and the state of its two adjacent neighbors at time t. The three cells at time t and the center cell at time t+1 will be labelled A, B, C, and D. This is a deterministic directed system which illustrates the use of RA for time series analysis (see also  for an IRA time series example). An example of an ECA mapping or "rule" is given in Table 10 . There are 256 mappings which preserve the identity of the 3 inputs, but in the ECA context these group into 88 equivalence classes, analyzed by SRA. Every mapping can also be converted in probability distribution by setting p(A,B,C,D) to 1/8 if an (A, B, C, D) tuple appears in the mapping and 0 if it does not. The resulting probability distributions were also analyzed by IRA. Table 11 illustrates how a rule governs ECA dynamics. Eight cells are arranged in a toroid, so s(9) = s(1). For every cell, the rule produces its next state, s t+1 (i), given its present state, s t (i), and the present state of its left and right neighbors, s t (i-1) and s t (i+1). Eventually the system reaches either a fixed point or limit cycle attractor. Such discrete dynamics can be considered "chaotic" if the time to reach the attractor goes up rapidly with the number of cells. Assignments of chaoticity or non-chaoticity were taken from BIB001 . Two "standard" parameters used to predict chaoticity were employed for comparison purposes: λ (Langton, 1992) and Z (Wuensche, 1992) . The parameter λ was actually first proposed by BIB002 , who called it "homogeneity." SRA or IRA decomposition properties of the rules were also used  Zwick & Shu, 2001) to predict chaoticity or non-chaoticity. It can be shown that the specific structures for this 3-input, 1-output problem can be reduced to the 12 specific structures shown in Table 12 . These group into 6 levels of complexity, indexed by parameter, σ. Structures whose variables are permuted have the same complexity, but for ECAs not all permutations are equivalent, because the neighbors A and C are different from B, so there are 9 general structural types (shown in bold). Table 12 indicates that while all single-predicting component models are themselves mappings, the σ = 4 and 5 models decompose the rule mapping into (stochastic) relations whose intersection yields the correct mapping. Such an approach to decomposition is quite different than what is encountered in LDL decomposition. By doing SRA on each rule, a σ is assigned to the rule, which indicates how decomposable without loss the rule is. Also, a vector parameter, τ, characterize each rule by the full set of decomposition losses (transmissions) for all 12 possible specific structures. IRA decomposition was also done on the rules, and two measures, f′′ and f′, were calculated, which are closely related to the fluency measure of BIB002 . These measures involve the transmissions for the 2 of the 3 models (shown italicized in Table 12 ) at level σ = 3. The first measure, f′, is a vector measures which preserves information about the separate losses in the models; the second, f′′, is a scalar measure which sums the decomposition losses of these two models. f′ = {T(ABC:BCD), T(ABC:ABD)} f′′ = T(ABC:BCD) + T(ABC:ABD) Table 13 indicates the predictability of chaoticity or nonchaoticity of ECA dynamics using RA measures (σ, f′, f′′, and τ) as compared to using standard ECA parameters (λ and Z). The table shows that RA measures predict better than standard parameters. Predictability is assessed information-theoretically as uncertainty reduction in attractor variable, a, which has two states, {chaotic, nonchaotic}. For rule parameter, r, which is either an RA measure or a standard parameter, the table lists ∆U a and ∆U r , which are large for good predictors, where Not only do RA measures predict chaoticity or nonchaoticity better than the standard parameters of λ and Z, the RA framework actually subsumes these standard measures in τ, the complete vector of RA losses. Specifically, U(λ | τ) = U(Z | τ) = 0, i.e., τ specifies also λ and Z. In fact, λ turns out to be isomorphic with U(D). No SRA measure comparable to fluency was apparent, and this illustrates the point made earlier that IRA analysis can be useful even for set-theoretic functions and relations, because it presents the analytical results in a form different from SRA. Analysis using the complete loss vector τ is, however, equivalent in SRA and IRA.