Boolean formulae, hypergraphs and combinatorial topology

TL;DR

This paper introduces the theta complex, a simplicial complex associated with hypergraphs, to study the homotopy types of spaces of Boolean formulae, providing new insights into their topological structure.

Contribution

It defines the theta complex as the Alexander dual of the independence complex and relates it to the homotopy type of Boolean formula spaces, offering partial calculations for cubical hypergraphs.

Findings

The theta complex of certain hypergraphs has the homotopy type of spaces related to Boolean formulae.

Partial calculations of the homotopy type for cubical hypergraphs are provided.

Examples of theta complexes for various hypergraphs are presented.

Abstract

With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph, which is the Alexander dual of the more well-known independence complex. In particular, the set of satisfiable formulae in k-conjunctive normal form with less than or equal to n variables has the homotopy type of Theta(Cube(n,n-k)), where Cube(n,n-k) is a hypergraph associated to the (n-k)-skeleton of an n-cube. We make partial progress in calculating the homotopy type of theta for these cubical hypergraphs, and we also give calculations and examples for other hypergraphs as well. Indeed studying the theta complex of hypergraphs is an interesting problem in its own right.