Special classes of boolean functions

This dissertation is devoted to the study of several important classes of Boolean functions. We mainly concentrate our efforts on the recognition and on the structural description of such special classes. In the first part of the thesis, we give "functional" representations of several of the most frequently appearing classes of Boolean functions including in particular unate, Horn, quadratic and renamable Horn functions. We then consider Boolean formulae which after the proper fixation of some of the variables, or after the "removal" of some of the variables or terms result in new formulae belonging to one of the highly structured classes thoroughly examined in the literature. The complexity of finding such minimum cardinality sets of variables or terms is shown to be in general NP-hard. The second part of the thesis studies in detail the class of submodular Boolean functions (which are shown to be in particular both Horn and quadratic) and establishes a one-to-one correspondence between these functions and partial preorders. For the class of quadratic functions, a dualization algorithm which runs in output polynomial time is provided. We then focus our attention on Boolean functions which acquire some special connectivity properties on the Boolean hypercube. The last part of the thesis deals with a classification problem which has received much attention in recent years. The data of the problem consists of a set of "positive" points and a set of "negative" points representing observations from the real world. The problem entails determining the proper classification (positive or negative) of a new observation on the basis of the known data. It is shown that several distance-based algorithms appearing in the literature, as well as several algorithms proposed in this dissertation, predict with high accuracy the proper classifications.