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On the Complexity of Minimizing Median Normal Forms of Monotone Boolean Functions and Lattice Polynomials
Miguel Couceiro, Pierre Mercuriali, Romain Péchoux and Abdallah Saffidine

In this document, we consider a median-based calculus to represent monotone Boolean functions efficiently. We study an equational specification of median forms and extend it from the domain of monotone Boolean functions to the domain of polynomial functions over distributive lattices. This specification is sound and complete. We illustrate its usefulness when simplifying median formulas algebraically. Furthermore, we propose a definition of median normal forms (MNF), that are thought of as minimal median formulas with respect to a structural ordering of expressions. We investigate related complexity issues and show that the problem of deciding whether a formula is in MNF, that is the problem of minimizing the median form of a monotone Boolean function, is in ∑P2. Moreover, we show that it still holds for arbitrary Boolean functions, not necessarily monotone.

Keywords: Monotone boolean function, lattice polynomial, normal form, median algebra, efficient representation

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