Decomposition of multiple-valued relations

Abstract

| This paper presents a new decomposition problem: decomposition of multi-valued (MV) relations, and a method of its solution. Decomposition is non-disjoint and multi-level. A fundamental di erence in decomposition of MV functions and MV relations is discussed: the column (cofactor) pair compatibility translates to the group compatibility for functions, but not for relations. This makes the decomposition of relations more di cult. The method is especially e cient for strongly unspeci ed data typical for Machine Learning (ML). It is implemented in program GUD-MV. 1 I. Introduction. Functional Decomposition of switching functions has applications in binary and multiple-valued circuit design, Machine Learning (ML), and Knowledge Discovery from Data Bases (KDD). Despite the fundamental nature of the MV decomposition problem and many possible applications of its solutions, e cient MV decomposers do not exist yet, with the exception of 1]. (The Curtis decomposition of binary functions is presented in detail in 4], Curtis-like decomposition of multi-valued functions based on graph coloring was presented in 6]). In this paper we will focus on a new problem of Curtis-like Decomposition of MV Relations. We present also an e cient computer program for this task. The solution of the MV Relation Decomposition Problem nds numerous applications in Machine Learning, binary circuits and Finite State Machine design. An example of a relation with binary inputs and a single MV output is shown in Table 1. Observe, that only the care minterms (care cubes) are present in the relation table as its rows. Standard don't cares ("unknown data samples" in ML) are represented by the remaining, implicit, minterms. The values in the column for output variable f include also the so-called "generalized d o n 't

A decomposition of fuzzy relations

“…Thus, our encoding method introduces the don't cares and in general the relations to the MV data for decomposition. It proves thus that the concept of decomposing relations, introduced by us for Machine Learning and circuit design applications in program GUD-MV [12,13], is also useful for fuzzy logic. Currently we keep looking for more fuzzy logic benchmarks, especially large ones, but unfortunately all examples from books and conference proceedings that we were able to find are too small for the power of our decomposers.…”

A software architecture for reconstructability analysis

“…1980), where the two overlap. Set-theoretic RA is related to techniques used in logic design and machine learning (Perkowski et al 1997;Files and Perkowski 1998). …”

An error reducing approach to machine learning using multi-valued functional decomposition

“…For more on complexity measures see [7,8,16,17]. This complexity measure, called Decomposed Functional Cardinality (DFC) is the sum of the cardinalities of the component functions in a combinational representation, when the sum has been minimized.…”

“…Several papers written on decomposition of multi-valued networks have shown an interest in using their decomposers in the field of machine learning [7,16,14,13]. Each of these papers had a slight twist to the decomposition process and the type of data that they were able to handle.…”