P Figure 1: Lookup Table Figure 2: Decomposition

a t t e r n T h e o r e t i c L e a r n i n g Abstract The goal of learning from sample data is to extract a concept that captures the underlying pattern while still representing it in a way useful to the investigator. A new approach based on function decomposition in the Pattern Theory framework is presented here. The objective of this extended abstract is threefold. The rst is to provide an overview of our new approach to learning. Speciically, we wish to show the applicability to discovery. Second, we will demonstrate the correlation of decomposed function cardinality (DFC) and \patterned." Finally, we demonstrate the robust-ness of this approach by exhibiting experimental results on binary functions with C4.5. This new approach to discovery and learning is a powerful method for nding patterns in a robust manner. 1 The Pattern Theory Approach Pattern Theory is a discipline that arose out of machine learning 2] 6] and switching theory 5]. The original goal was to develop formal methods of algorithm design from speciications. The approach is based on a technique called function decomposition and a measure called decomposed function cardinality (DFC). Since Pattern Theory is able to extrapolate on available information based on the inherent structure in the data, it is directly related to scientiic discovery. Decomposing a function involves breaking it up into smaller subfunctions. These smaller functions are further broken down until all subfunctions will no longer decompose. For a given function, the number of ways to choose two sets of variables (the partition space) is exponential. The decomposition space is even larger, since there are several ways the subfunctions can be combined and there are several levels of subfunctions possible. The complexity measure that we use to determine the relative predictive power of diierent function decompositions is called DFC. DFC is calculated by adding the cardinalities of each of the subfunctions in the decomposition. The cardinality of an n-variable binary function is 2 n : We illustrate the measure in the above gures. In Figure 1, we have a function on four variables with cardinality 2 4 = 16: In Figure 2, we show the same function after it has been decomposed. The DFC of this representation for the original function is 2 2 + 2 2 + 2 2 = 12: The DFC measures the relative complexity of a …