The purpose of this introductory chapter is three fold. First, it contains the main definitions, terminology, and notations that are used throughout the book. After the introduction of our main feature characters–namely, Boolean functions–several sections are devoted to a discussion of alternative representations, or expressions, of Boolean functions. Disjunctive and conjunctive normal forms, in particular, are discussed at length in Sections 1.4–1.11. These special algebraic expressions play a very central role in our investigations, as we frequently focus on the relation between Boolean functions and their normal forms. Section 1.12, however, also provides a short description of different types of function representations, namely, representations over GF(2), pseudo-Boolean polynomial expressions, and binary decision diagrams.

A second objective of this chapter is to introduce several of the topics to be investigated in more depth in subsequent chapters, namely: fundamental algorithmic problems (Boolean equations, generation of prime implicants, dualization, orthogonalization, etc.) and special classes of Boolean functions (bounded-degree normal forms, monotone functions, Horn functions, threshold functions, etc.). Finally, the chapter briefly presents a variety of applications of Boolean functions in such diverse fields as logic, electrical engineering, reliability theory, game theory, combinatorics, and so on. These applications have often provided the primary motivation for the study of the problems to be encountered in the next chapters.

The solution of Boolean equations is arguably the most fundamental problem arising in the theory of Boolean functions. Actually, the quote at the beginning of Chapter 1 shows that an important aspect of Boole's original research program was essentially to reduce logic to the solution of Boolean equations. Although his hopes eventually proved overly optimistic, it will become clear in subsequent chapters of this book that Boolean equations often arise as subproblems to be solved in the course of tackling more complex problems. Therefore, their solution is a cornerstone of many Boolean algorithms.

In this chapter, we present some representative models involving Boolean equations and describe various algorithmic procedures for their solution: branching, variable elimination, the consensus method, and mathematical programming approaches. In view of the importance of this topic, we spend quite a lot of time discussing the details of classical procedures, their interrelations, respective merits, and complexity. In the last section, we generalize the basic consistency-testing problem in several ways: We examine the problems of counting and of generating all solutions of a Boolean equation and briefly discuss the maximum satisfiability (Max Sat) problem.

Definitions and applications

Definition 2.1.A Boolean equation is an equation of the form ϕ(X) = ψ(X), where X = (x1,x2,…, xn) is a vector of Boolean variables, and ϕ,ψ are Boolean expressions in these variables. A solution of the equation is a point X* ∈ Bn such that ϕ(X*) = ψ(X*).

By and large, we assume that the readers of this book have at least some intuitive knowledge about algorithms and complexity. For the sake of completeness, however, we provide in this appendix an informal introduction to fundamental concepts of computational complexity: problems, algorithms, running time, easy and hard problems, etc. For a more thorough and rigorous introduction to this topic, we refer the reader to the classical monograph by Garey and Johnson [371], or to other specialized books like Aho, Hopcroft and Ullman [11], Papadimitriou [725], or Papadimitriou and Steiglitz [726]. Note that Cook et al. [211] and Schrijver [814] also provide gentle introductions to the topic, much in the spirit of this appendix.

In Section B.8, we propose a short primer on the complexity of list-generating algorithms; such algorithms are usually not discussed in basic textbooks on complexity theory, but they arise naturally in several chapters of our book.

Decision problems

Intuitively speaking, an algorithmic or computational problem is a generic question whose formulation contains a number of undetermined parameters. For instance, we can think of “addition” as the generic problem of adding two numbers (the numbers themselves must be specified before any specific computation can be performeed). Similarly, “solving quadratic equations” of the form ax2+bx+c = 0 is a problem which can be handled by an appropriate algorithm as soon as the numerical values of the parameters a,b, and c are known.

This appendix proposes a short primer on graph and hypergraph theory. It sums up the basic concepts and terminology used in the remainder of the monograph. For (much) more information, we refer the reader to numerous excellent books dealing in-depth with this topic, such as Bang-Jensen and Gutin [51]; Berge [71, 72]; Brandstädt, Le, and Spinrad [152]; Golumbic [398]; Mahadev and Peled [645]; or Schrijver [814].

Undirected graphs

An undirected graph, or graph for short, is a pair of finite sets G = (V, E) in which V is the set of vertices of the graph, and E is a set of unordered pairs of vertices called edges of the graph. Abiding by widespread conventions, we often use the notation (u,v), or even simply uv, for an edge {u,v}. Occasionally, we consider undirected graphs with loops, where a loop is an edge of the form (v,v) for v ∈ V (we may view a loop as an edge of cardinality 1).

A graph can be represented as a diagram consisting of points (vertices) joined by lines (edges), as in Figure A.1.

When e = (u,v) is an edge, we say that vertices u and v are adjacent, that u is a neighbor of v, that u and v are incident to e, that u and v are the endpoints of e, and so forth. The neighborhood of a vertex u ∈ V is the set N(u) = {v ∈ V : (u,v) ∈ E}.

Definitions and examples

In Chapter 1, we defined a pseudo-Boolean function to be a mapping from Bn = {0,1}n to ℝ. In other words, a pseudo-Boolean function is a real-valued function of a finite number of 0–1 variables. Identifying the Boolean symbols 0 and 1 (or T and F, Yes and No, etc.) with the corresponding integers, we see that pseudo-Boolean functions provide a proper generalization of Boolean functions. In fact, just as in the Boolean case, the deliberate ambiguity that results from this identification rarely causes any difficulties, but it is frequently the source of fruitful developments.

The systematic investigation of pseudo-Boolean functions, their theoretical properties, and their applications has been initiated by Hammer and Rudeanu in [460], building on previous ideas of Fortet [342, 343] and of Hammer, Rosenberg, and Rudeanu [458]. This field of research has given rise to countless subsequent publications over the last decades.

Since the element of {0,1}n are in one-to-one correspondence with the subsets of N = {1, 2, …, n}, every pseudo-Boolean function can also be viewed as a real-valued set function defined on P(N), the power set of N = {1, 2, …, n}. Set functions have been extensively studied because of their mathematical appeal and their presence in numerous fundamental models of mathematics and of applied sciences. By considering functions defined on {0,1}n rather than on P(N), however, the pseudo-Boolean approach provides an algebraic viewpoint, which sometimes carries clear advantages over the set-theoretic description.

Boolean functions, meaning {0,1}-valued functions of a finite number of {0,1}-valued variables, are among the most fundamental objects investigated in pure and applied mathematics. Their importance can be explained by several interacting factors.

• It is reasonable to argue that a multivariate function f:A1×A2×…×An→A is “interesting” only if each of the sets A1,A2,…,An, and A contains at least two elements, since otherwise the function either depends trivially on some of its arguments, or is constant. Thus, in a sense, Boolean functions are the “simplest interesting” multivariate functions. It may even be surprising, actually, that such primitive constructs turn out to display a rich array of properties and have been investigated by various breeds of scientists for more than 150 years.

• When the arguments of a Boolean function are viewed as atomic logical propositions, the value of the function at a 0–1 point can be interpreted as the truth value of a sentence composed from these propositions. Carrying out calculation son Boolean functions is then tantamount to performing related logical operations (such as inference or theorem-proving) on propositional sentences. Therefore, Boolean functions are at the heart of propositional logic.

• […]

The concept of orthogonal disjunctive normal form (or ODNF, sometimes called sum of disjoint products) was introduced in Chapter 1. Orthogonal forms are a classic object of investigation in the theory of Boolean functions, where they were originally introduced in connection with the solution of Boolean equations (see Kuntzmann [589], Rudeanu [795]). More recently, they have also been extensively studied in the reliability literature (see, e.g., Colbourn [205, 206]; Provan [759]; Schneeweiss [811]).

In general, however, orthogonal forms are difficult to compute, and few classes of disjunctive normal forms are known for which orthogonalization can be efficiently performed. An interesting class with this property, called the class of shellable DNFs, has been introduced and investigated by Ball and Provan [49, 760]. As these authors established, the DNFs describing several important classes of reliability problems (all-terminal reliability, all-point reachability, k-out-of-n systems, etc.) are shellable. Moreover, besides its unifying role in reliability theory, shellability also provides a powerful theoretical and algorithmic tool of combinatorial geometry, where it originally arose in the study of abstract simplicial complexes (see [96, 97, 205, 206, 254, 569, etc.]; let us simply mention here, without further details, that an abstract simplicial complex can be viewed as the set of true points of a positive Boolean function).

In this chapter, we first review some basic facts concerning orthogonal forms and describe a simple orthogonalization procedure for DNFs. Then, we introduce shellable DNFs and establish some of their most remarkable properties: In particular, we prove that shellable DNFs can be orthogonalized and dualized in polynomial time.