An overview of reconstructability analysis <s> Set-theoretic reconstruction <s> System reconstruction refers to the following problem: given a behaviour system, viewed as an overall system, determine which sets of its subsystems are adequate for reconstructing the given system with an acceptable degree of approximation. The reconstruction problem can be solved by finding multiple variable relationships in complex data. To deal with this problem, we apply the idea of qualitative reasoning as discussed in artificial intelligence. The basic idea is to convert continuous data into discrete data (e.g. through clustering) so that qualitative rules can be constructed. In this paper a method of compressing data (called the Lebesgue discretization) is proposed to establish relationships between the dependent variable and other variables. Qualitative rules can be constructed from this relationship and then stored in a knowledge base. This technique also facilitates automated knowledge acquisition. <s> BIB001 </s> An overview of reconstructability analysis <s> Set-theoretic reconstruction <s> In the past few years, several authors have presented methods of using functional decomposition as applied to machine learning. These authors explore the ideas of functional decomposition, but left the concepts of machine learning to the papers that they reference. In general, they never fully explain why a logic synthesis method should be applied to machine learning. This paper explores and presents the basic concepts of machine learning, and how some concepts match nicely with multi-valued logic synthesis, while others pose great difficulties. The main reason for using multi-valued synthesis is that many problems are naturally multi-valued (i.e., values taken from a discrete set). Thus, mapping the problem directly to a multi-valued set of inputs and outputs is much more natural than encoding the problem into a binary form. The paper also shows that any multi-valued logic synthesis method could be applied to the machine learning problem. But, this paper focuses on multivalued functional decomposition because of its generality of minimizing a given data set. <s> BIB002
In SRA, data is a set-theoretic relation --a subset of a Cartesian product --the set of states actually observed, without regard to frequency. Set-theoretic reconstruction is similar to information-theoretic reconstruction, as can be seen by comparing Table 2 to Table 7 . In Table 7 , the two values of A, B, and C are indicated as 0 and 1, but these are only arbitrary labels. As in IRA, ABC AB:BC is the maximum uncertainty relation, given the model constraints, i.e., the model-specified projections. U(ABC AB:BC ) is now the Hartley (rather than the Shannon) entropy, U(R) = log 2 |R|, where | | is cardinality. This maximization is achieved in composition by (1) expanding the projections by Cartesian products with variables omitted in them (maximizing uncertainty), and (2) = {00*,01*,11*}∩{*00,*10,*11} = {000,010,011,110,111} = ABC (• is a place-holder for absent variable(s); * means "don't care"). The second line of this equation shows that SRA resembles a product of sums of products. Generalizing the above equation, for model P 1 :P 2 :...P n , where P j is a projection of relation R (the data), and where M j is the Cartesian product of variables absent (projected on) in P j , the calculated reconstructed R is given by Relations for cyclic models are calculated in the same way; no iterative procedure is needed. In the present case, ABC AB:BC has no error. Imperfect decomposition can, however, be allowed, and again transmission (error) is defined as T(AB:BC) = U(ABC AB:BC ) -U(ABC). No model simpler than AB:BC agrees fully with the data, as shown below in Table 8 . The precise relationship between SRA and LDL methods has not yet been fully elucidated. Of particular interest would be a comparison of SRA to LDL multi-valued functional decomposition  BIB002 ). An initial examination of an enhanced version of SRA shows that it is superior for 3-variable binary functions to Ashenhurst-Curtis decomposition, a well-known and widely used LDL technique which utilizes latent variables (Al-Rabadi, AlRabadi & Zwick, 2003) . IRA problems can be approximated by SRA. In Table 2 , if frequency is discretized into ranges BIB001   , it becomes a nominal variable and the mapping A ⊗ B ⊗ C → F can be decomposed by SRA. In doing so, the different frequency bins become nominal states and their order cannot be exploited. This also precludes the assessment of statistical significance. Discretization of frequencies is qualitatively different from discretization of variables, and the error consequences of this approach, in comparison to IRA/LL statistical analyses, is under investigation. Conversely, SRA problems can be treated by IRA by giving equal probability to observed tuples; IRA then generates results equivalent to SRA but in a different form. Just as a Fourier transform of a function may illuminate particular properties of the function previously obscure, even though the transform is fundamentally equivalent to the function, so IRA computation may be informative in ways not directly achievable by set-theoretic analysis  .