On a quasi-ordering on Boolean functions

It was proved few years ago that classes of Boolean functions definable by means of functional equations [9], or equivalently, by means of relational constraints [16], coincide with initial segments of the quasi-ordered set (, ) made of the set of Boolean functions, suitably quasi-ordered. Furthermore, the classes defined by finitely many equations [9] coincide with the initial segments of (, ) which are definable by finitely many obstructions. The resulting ordered set (/ , ) embeds into ([]<, ), the set -ordered by inclusion- of finite subsets of the set of integers. We prove that (/ , ) also embeds ([]<, ). From this result, we deduce that the dual space of the distributive lattice made of finitely definable classes is uncountable. Looking at examples of finitely definable classes, we show that classes of Boolean functions with a bounded number of essential variables are finitely definable. We provide a concrete equational characterization of the subclasses made of linear fu...

• Type: Journal
• Year: 2008
• Where: TCS
• Authors: Miguel Couceiro, Maurice Pouzet
• Keywords: Boolean Functions, Functional Equations, Initial Segments, TCS 2008, Theoretical Computer Science