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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 calculations on 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.
Many concepts of combinatorial analysis have their natural Boolean counterpart. In particular, since every 0–1 point with n coordinates can be viewed as the characteristic vector of a subset of N = {1, 2, . . . ,n}, the set of points at which a Boolean function takes value 1 corresponds to a collection of subsets of N, or a hypergraph on N. (When all subsets have cardinality 2, then the function corresponds exactly to a graph.) Structural properties relating to the transversals, stable sets, or colorings of the hypergraph, for instance, often translate into interesting properties of the Boolean function.
Boolean functions are ubiquitous in theoretical computer science, where they provide fundamental models for the most basic operations performed by computers on binary digits(or bits). Turing machines and Boolean circuits are prime examples illustrating this claim. Similarly, electrical engineers rely on the Boolean formalism for the description, synthesis, or verification of digital circuits.
In operations research or management science, binary variables and Boolean functions are frequently used to formulate problems where a number of go – no go decisions are to be made these could be, for instance, investment decisions arising in a financial management framework, or location decisions in logistics, or assignment decisions for production planning. In most cases, the variables have to be fixed at values that satisfy constraints expressible as Boolean conditions and that optimize an appropriate real-valued objective function. This leads to – frequently difficult – Boolean equations (satisfiability problems) or integer programming problems.
Voting games and related systems of collective choice are frequently represented by Boolean functions, where the variables are associated with (binary) alternatives available to the decision makers, and the value of the function indicates the outcome of the process.
Various branches of artificial intelligence rely on Boolean functions to express deductive reasoning processes (in the above-mentioned propositional framework), or to model primitive cognitive and memorizing activities of the brain by neural networks, or to investigate efficient learning strategies, or to devise storing and retrieving mechanisms in databases, and so on.
We could easily extend this list to speak of Boolean models arising in reliability theory, in cryptography, in coding theory, in multicriteria analysis, in mathematical biology, in image processing, in theoretical physics, in statistics, and so on. The main objective of the present monograph is to introduce the reader to the fundamental elements of the theory of Boolean functions. It focuses on algebraic representations of Boolean functions, especially disjunctive or conjunctive normal form expressions, and it provides a very comprehensive presentation of the structural, algorithmic, and applied aspects of the theory in this framework.
Foundations
Fundamental concepts and applications 3
Boolean equations
Prime implicants and minimal DNFs
Duality theory
Special Classes
Quadratic functions
Horn functions
Orthogonal forms and shellability
Regular functions
Threshold functions
Read-once functions
Characterizations of special classes by functional equations
Generalizations
Partially defined Boolean functions
Pseudo-Boolean functions
A Graphs and hypergraphs
B Algorithmic complexity
C JBool: A software tool