Relative definability of boolean functions via hypergraphs

Abstract

The aim of this work is to show how hypergraphs can be used as a systematic tool in the classification of continuous boolean functions according to their degree of parallelism. Intuitively f is “less parallel” than g if it can be defined by a sequential program using g as its only free variable. It turns out that the poset induced by this preorder is (as for the degrees of recursion) a sup-semilattice. Although hypergraphs have already been used in Bucciarelli (Theoret. Comput. Sci., to appear) as a tool for studying degrees of parallelism, no general result relating the former to the latter has been proved in that work. We show that the sup-semilattice of degrees has a categorical counterpart: we define a category of hypergraphs such that every object “represents” a monotone boolean function; finite coproducts in this category correspond to lubs of degrees. Unlike degrees of recursion, where every set has a recursive upper bound, monotone boolean functions may have no sequential upper bound. However the ones which do have a sequential upper bound can be nicely characterised in terms of hypergraphs. These subsequential functions play a major role in the proof of our main result, namely that f is less parallel than g if there exists a morphism between their associated hypergraphs.