On the Structure and the Number of Prime Implicants of 2-CNFs

Let $m(n, k)$ be the maximum number of prime implicants that any $k$-CNF on n variables can have. We show that $3^{n/3} \le m(n,2) \le (1+o(1))3^{n/3}$

Disjunctive analogues of submodular and supermodular pseudo-Boolean functions

AbstractWe consider classes of real-valued functions of Boolean variables defined by disjunctive analogues of the submodular and supermodular functional inequalities, obtained by replacing in these inequalities addition by disjunction (max operator). The disjunctive analogues of submodular and supermodular functions are completely characterized by the syntax of their disjunctive normal forms. Classes of functions possessing combinations of these properties are also examined. A disjunctive representation theory based on one of these combination classes exhibits syntactic and algorithmic analogies with classical DNF theory

Translating between Horn Representations and their Characteristic Models

Characteristic models are an alternative, model based, representation for Horn expressions. It has been shown that these two representations are incomparable and each has its advantages over the other. It is therefore natural to ask what is the cost of translating, back and forth, between these representations. Interestingly, the same translation questions arise in database theory, where it has applications to the design of relational databases. This paper studies the computational complexity of these problems. Our main result is that the two translation problems are equivalent under polynomial reductions, and that they are equivalent to the corresponding decision problem. Namely, translating is equivalent to deciding whether a given set of models is the set of characteristic models for a given Horn expression. We also relate these problems to the hypergraph transversal problem, a well known problem which is related to other applications in AI and for which no polynomial time algorithm is known. It is shown that in general our translation problems are at least as hard as the hypergraph transversal problem, and in a special case they are equivalent to it.Comment: See http://www.jair.org/ for any accompanying file

Minimizing DNF Formulas and AC 0 Circuits Given a Truth Table

For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek [31], which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than logN γ, for some constant γ 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for Set Cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o logN remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is Ω logN larger than optimal. Finally, we extend known hardness results for Min-TC0 d to obtain new hardness results for Min-AC0 d, under cryptographic assumptions

A New Class of Explanations for Classifiers with Non-Binary Features

Two types of explanations have received significant attention in the literature recently when analyzing the decisions made by classifiers. The first type explains why a decision was made and is known as a sufficient reason for the decision, also an abductive or PI-explanation. The second type explains why some other decision was not made and is known as a necessary reason for the decision, also a contrastive or counterfactual explanation. These explanations were defined for classifiers with binary, discrete and, in some cases, continuous features. We show that these explanations can be significantly improved in the presence of non-binary features, leading to a new class of explanations that relay more information about decisions and the underlying classifiers. Necessary and sufficient reasons were also shown to be the prime implicates and implicants of the complete reason for a decision, which can be obtained using a quantification operator. We show that our improved notions of necessary and sufficient reasons are also prime implicates and implicants but for an improved notion of complete reason obtained by a new quantification operator that we define and study in this paper

Testing systems of identical components

We consider the problem of testing sequentially the components of a multi-component reliability system in order to figure out the state of the system via costly tests. In particular, systems with identical components are considered. The notion of lexicographically large binary decision trees is introduced and a heuristic algorithm based on that notion is proposed. The performance of the heuristic algorithm is demonstrated by computational results, for various classes of functions. In particular, in all 200 random cases where the underlying function is a threshold function, the proposed heuristic produces optimal solutions

Fast subsumption checks using anti-links

The concept of "anti-link" is defined, and useful equivalence-preserving operations based on anti-links are introduced.These operations eliminate a potentially large number of subsumed paths in a negation normal form formula.Those anti-links that directly indicate the presence of subsumed paths are characterized. The operations have linear time complexity in the size of that part of the formula containing the anti-link. The problem of removing all subsumed paths in an NNF formula is shown to be NP-hard, even though such formulas may be small relative to the size of their path sets. The general problem of determining whether there exists a pair of subsumed paths associated with an arbitrary anti-link is shown to be NP-complete. Additional techniques based on "strictly pure full blocks" are introduced and are also shown to eliminate redundant subsumption checks. The effectiveness of these techniques is examined with respect to some benchmark examples from the literature