Dualization in Lattices Given by Ordered Sets of Irreducibles

TL;DR

This paper explores the complexity of dualizing monotone Boolean functions within finite lattices represented by irreducibles, revealing computational limitations and special cases where efficient algorithms are possible.

Contribution

It establishes the computational hardness of dualization in general lattices given by irreducibles and identifies tractable cases for distributive lattices.

Findings

Dualization is NP-hard in general lattices given by irreducibles.

In distributive lattices, dualization can be performed in subexponential time.

Dualization is equivalent to enumerating minimal hypotheses in this setting.

Abstract

Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 to the set of its maximal 0 values. In this paper we consider finite lattices given by ordered sets of their meet and join irreducibles (i.e., as a concept lattice of a formal context). We show that in this case dualization is equivalent to the enumeration of so-called minimal hypotheses. In contrast to usual dualization setting, where a lattice is given by the ordered set of its elements, dualization in this case is shown to be impossible in output polynomial time unless P = NP. However, if the lattice is distributive, dualization is shown to be possible in subexponential time.