Minimizing the Average Query Complexity of Learning Monotone Boolean Functions

This paper addresses the problem of completely reconstructing deterministic monotone Boolean functions via membership queries. The minimum average query complexity is guaranteed via recursion, where partially ordered sets (posets) make up the overlapping subproblems. For problems with up to 4 variables, the posets’ optimality conditions are summarized in the form of an evaluative criterion. The evaluative criterion extends the computational feasibility to problems involving up to about 20 variables. A frameworkfor unbiased average case comparison of monotone Boolean function inference algorithms is developed using unequal probability sampling. The unbiased empirical results show that an implementation of the subroutine considered as a standard in the literature performs almost twice as many queries as the evaluative criterion on the average. It should also be noted that the first algorithm ever designed for this problem performed consistently within two percentage points of the evaluative criterion. As such, it prevails, by far, as the most efficient of the many preexisting algorithms

Minimizing the Average Query Complexity of Learning Monotone Boolean Functions

This paper addresses the problem of completely reconstructing deterministic monotone Boolean functions via membership queries. The minimum average query complexity is guaranteed via recursion, where partially ordered sets (posets) make up the overlapping subproblems. For problems with up to 4 variables, the posets ’ optimality conditions are summarized in the form of an evaluative criterion. The evaluative criterion extends the computational feasibility to problems involving up to about 20 variables. A framework for unbiased average case comparison of monotone Boolean function inference algorithms is developed using unequal probability sampling. The unbiased empirical results show that an implementation of the subroutine considered as a standard in the literature performs almost twice as many queries as the evaluative criterion on the average. It should also be noted that the first algorithm ever designed for this problem performed consistently within two percentage points of the evaluative criterion. As such, it prevails, by far, as the most efficient of the many preexisting algorithms

List, Sample, and Count

Counting plays a fundamental role in many scientific fields including chemistry, physics, mathematics, and computer science. There are two approaches for counting, the first relies on analytical tools to drive closed form expression, while the second takes advantage of the combinatorial nature of the problem to construct an algorithm whose output is the number of structures. There are many algorithmic techniques for counting, they cover the explicit approach of counting by listing to the approximate approach of counting by sampling. This thesis looks at counting three sets of objects. First, we consider a subclass of boolean functions that are monotone. They appear naturally in great variety of contexts including combinatorics, cryptography, voting theory, and game theory. Next, we consider permutations of n pairs of numbers, called Skolem sequences. These sequences are employed in several areas including construction of Steiner triple systems, binary sequences with controllable complexity, interference resistant codes, and graph labeling. Finally, we consider a variation of the n-queens problem, called the queens of the night. This constraint satisfaction problem is not just a recreational puzzle, but rather it is useful in designing conflict free access in parallel systems. In each case we verify previously known values and provide the next unknown exact value(s) in the counting sequence. Furthermore, we approximate the count for the next unknown values in the sequence by employing a sampling procedure

Data mining and knowledge discovery: a guided approach base on monotone boolean functions

This dissertation deals with an important problem in Data Mining and Knowledge Discovery (DM & KD), and Information Technology (IT) in general. It addresses the problem of efficiently learning monotone Boolean functions via membership queries to oracles. The monotone Boolean function can be thought of as a phenomenon, such as breast cancer or a computer crash, together with a set of predictor variables. The oracle can be thought of as an entity that knows the underlying monotone Boolean function, and provides a Boolean response to each query. In practice, it may take the shape of a human expert, or it may be the outcome of performing tasks such as running experiments or searching large databases. Monotone Boolean functions have a general knowledge representation power and are inherently frequent in applications. A key goal of this dissertation is to demonstrate the wide spectrum of important real-life applications that can be analyzed by using the new proposed computational approaches. The applications of breast cancer diagnosis, computer crashing, college acceptance policies, and record linkage in databases are here used to demonstrate this point and illustrate the algorithmic details. Monotone Boolean functions have the added benefit of being intuitive. This property is perhaps the most important in learning environments, especially when human interaction is involved, since people tend to make better use of knowledge they can easily interpret, understand, validate, and remember. The main goal of this dissertation is to design new algorithms that can minimize the average number of queries used to completely reconstruct monotone Boolean functions defined on a finite set of vectors V = {0,1}^n. The optimal query selections are found via a recursive algorithm in exponential time (in the size of V). The optimality conditions are then summarized in the simple form of evaluative criteria, which are near optimal and only take polynomial time to compute. Extensive unbiased empirical results show that the evaluative criterion approach is far superior to any of the existing methods. In fact, the reduction in average number of queries increases exponentially with the number of variables n, and faster than exponentially with the oracle\u27s error rate