Boolean Functions: Theory, Algorithms, and Applications

This monograph provides the first comprehensive presentation of the theoretical, algorithmic and applied aspects of Boolean functions, i.e., {0,1}-valued functions of a finite number of {0,1}-valued variables. The book focuses on algebraic representations of Boolean functions, especially normal form representations. It presents the fundamental elements of the theory (Boolean equations and satisfiability problems, prime implicants and associated representations, dualization, etc.), an in-depth study of special classes of Boolean functions (quadratic, Horn, shellable, regular, threshold, read-once, etc.), and two fruitful generalizations of the concept of Boolean functions (partially defined and pseudo-Boolean functions). It features a rich bibliography of about one thousand items. Prominent among the disciplines in which Boolean methods play a significant role are propositional logic, combinatorics, graph and hypergraph theory, complexity theory, integer programming, combinatorial optimization, game theory, reliability theory, electrical and computer engineering, artificial intelligence, etc. The book contains applications of Boolean functions in all these areas

Incremental polynomial time dualization of quadratic functions and a subclass of degree-k functions

Cataloged from PDF version of article.We consider the problem of dualizing a Boolean function f represented by a DNF. In its most general form, this problem is commonly believed not to be solvable by a quasi-polynomial total time algorithm.We show that if the input DNF is quadratic or is a special degree-k DNF, then dualization turns out to be equivalent to hypergraph dualization in hypergraphs of bounded degree and hence it can be achieved in incremental polynomial time

On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions

AbstractLet f:{0,1}n→{0,1} be a monotone Boolean function whose value at any point x∈{0,1}n can be determined in time t. Denote by c=⋀I∈C⋁i∈Ixi the irredundant CNF of f, where C is the set of the prime implicates of f. Similarly, let d=⋁J∈D⋀j∈Jxj be the irredundant DNF of the same function, where D is the set of the prime implicants of f. We show that given subsets C′⊆C and D′⊆D such that (C′,D′)≠(C,D), a new term in (C⧹C′)∪(D⧹D′) can be found in time O(n(t+n))+mo(logm), where m=|C′|+|D′|. In particular, if f(x) can be evaluated for every x∈{0,1}n in polynomial time, then the forms c and d can be jointly generated in incremental quasi-polynomial time. On the other hand, even for the class of ∧,∨-formulae f of depth 2, i.e., for CNFs or DNFs, it is unlikely that uniform sampling from within the set of the prime implicates and implicants of f can be carried out in time bounded by a quasi-polynomial 2polylog(·) in the input size of f. We also show that for some classes of polynomial-time computable monotone Boolean functions it is NP-hard to test either of the conditions D′=D or C′=C. This provides evidence that for each of these classes neither conjunctive nor disjunctive irredundant normal forms can be generated in total (or incremental) quasi-polynomial time. Such classes of monotone Boolean functions naturally arise in game theory, networks and relay contact circuits, convex programming, and include a subset of ∧,∨-formulae of depth 3

Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

Good-for-Game QPTL: An Alternating Hodges Semantics

An extension of QPTL is considered where functional dependencies among the quantified variables can be restricted in such a way that their current values are independent of the future values of the other variables. This restriction is tightly connected to the notion of behavioral strategies in game-theory and allows the resulting logic to naturally express game-theoretic concepts. The fragment where only restricted quantifications are considered, called behavioral quantifications, can be decided, for both model checking and satisfiability, in 2ExpTime and is expressively equivalent to QPTL, though significantly less succinct