In this doctoral dissertation, we study three basic problems in machine learning and two new hypothesis spaces with corresponding learning algorithms. The problems we investigate are: accuracy estimation, feature subset selection, and parameter tuning. The latter two problems are related and are studied under the wrapper approach. The hypothesis spaces we investigate are: decision tables with a default majority rule (DTMs) and oblivious read-once decision graphs (OODGs).

OBDD's are the state-of-the-art data structure for Boolean function manipulation since basic tasks of Boolean manipulation such as testing equivalence, satisfiability, or tautology, and performing single Boolean synthesis steps can be done efficiently. In the following we show that the efficient manipulation of OBDD's can be extended to a more general data structure, so-called FBDD's. In detail, the advantages of using FBDD's instead of OBDD's are ffl FBDD's are generally more (sometimes even exponentially more) succinct than OBDD's, ffl FBDD's provide, similarly to OBDD's, canonical representations of Boolean functions, and ffl in terms of FBDD's basic tasks of Boolean manipulation can be performed similarly efficient as in terms of OBDD's. The power of the FBDD-concept is demonstrated by showing that the verification of the benchmark circuit design for the hidden weighted bit function HWB proposed by Bryant can be carried out efficiently in terms of FBDD's while, for princip...

. We prove that read-once branching programs computing integer multiplication require size 2 ## # n) . This is the first nontrivial lower bound for multiplication on branching programs that are not oblivious. By the appropriate problem reductions, we obtain the same lower bound for other arithmetic functions. Key words. multiplication, read-once, branching programs, BDD, verification AMS subject classifications. 68Q05, 68Q25, 68M15 PII. S0097539795290349 1. Introduction and background. It is well known that many functions, some of them very simple, cannot be computed by read-once branching programs of polynomial size [We88, Za84, Du85, We87, BHST87, Ju88, Kr88]. Interest in whether integer multiplication can be so computed has been created by recent developments in the field of digital design and hardware verification. 1.1. Hardware verification and branching programs. The central problem of verification is to check whether a combinational hardware circuit has been correctly designe...

Abstract. A central issue in the solution of many computer aided design problems is to find concise representations for circuit designs and their functional specification. Recently, a restricted type of branching programs (OBDDs) proved to be extremely useful for representing Boolean functions for various CAD applications [Bry92]. Unfortunatelly, many circuits of practical interest provably require OBDD--representations of exponential size. In the following we systematically study the question up to what extend more concise BP-representations can be successfully used in symbolic Boolean manipulation, too. We prove, in very general settings, -- The frontier of efficient (deterministic) symbolic Boolean manipulation on the basis of BP--representations are read--once--only branching programs (BP1). -- The frontier of efficient probabilistic manipulation with BP--based data structures are parity read--once--only branching programs (\Phi-- BP1). Since BP1s and \Phi--BP1s are generally mor...

We present a data structure for Boolean functions, which we call Parity--OBDDs or \Phi-- OBDDs, which combines the nice algorithmic properties of the well--known ordered binary decision diagrams (OBDDs) with a considerably larger descriptive power. Beginning from an algebraic characterization of the \Phi--OBDD complexity we prove in particular that the minimization of the number of nodes, the synthesis, and the equivalence test for \Phi--OBDDs, which are the fundamental operations for circuit verification, have efficient deterministic solutions. Several functions of pratical interest, i.e. the indirect storage access function, have exponential ODBB--size but are of polynomial size if \Phi--OBDDs are used. Keywords: data structures for Boolean functions, BDDs, circuit verification 1 Introduction Formal circuit verification is a fundamantal task. The following approach for verification is often used (for a survey see [8] and [21]). A data structure for representing Boolean functions is...

In this paper, we are concerned with randomized OBDDs and randomized read-ktimes branching programs. We present an example of a Boolean function which has polynomial size randomized OBDDs with small, one-sided error, but only nondeterministic read-once branching programs of exponential size. Furthermore, we discuss a lower bound technique for randomized OBDDs with two-sided error and prove an exponential lower bound of this type. Our main result is an exponential lower bound for randomized read-k-times branching programs with two-sided error. 1 Introduction Branching programs are a theoretically and practically interesting data structure for the representation of Boolean functions. In complexity theory, among other problems, lower bounds for the size of branching programs for explicitly defined functions and the relations of the various branching program models are investigated. A branching program (BP) on the variable set fx 1 ; : : : ; x n g is a directed acyclic graph with one sour...

Bryant [5] has shown that any OBDD for the function MULn−1,n, i.e. the middle bit of the n-bit multiplication, requires at least 2 n/8 nodes. In this paper a stronger lower bound of essentially 2 n/2 /61 is proven by a new technique, using a universal family of hash functions. As a consequence, one cannot hope anymore to verify e.g. 128-bit multiplication circuits using OBDD-techniques because the representation of the middle bit of such a multiplier requires more than 3 · 10 17 OBDD-nodes. Further, a first non-trivial upper bound of 7/3 · 2 4n/3 for the OBDD-size of MULn−1,n is provided.

In this paper, a simple technique which unifies the known approaches for proving lower bound results on the size of deterministic, nondeterministic, and randomized OBDDs and kOBDDs is described.

This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps read-once branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...

. We investigate the question whether and to what extend the solution of central tasks of digital logic circuit design of a given Boolean function f benefits from a representation of f in terms of certain restricted decision graphs or branching programs. Introduction One of the fundamental problems in computer-aided circuit design is the task of representing logic functions. Although, in principle, any valid representation is allowed, some representations may be preferred because they are -- more efficient in memory, -- more efficient to manipulate, or -- more indicative of the complexity of the final implementation. The search of an optimal trade--off between these competing objectives -- succinct representation of Boolean functions and feasible manipulation algorithms -- is a central theme of logic synthesis. The most fundamental concept in the description of logic functions is that of the truth table. Truth tables define a function by listing the output value for each possible inp...