A Topological Approach to Non-Uniform Complexity

Abstract We investigate the feasibility of a topological method for proving separations of non-uniform circuit classes. Thereto, we chose a rather simple class of circuits: non-uniform constant size circuit classes with gate types underlying certain restrictions. In particular, we consider gate types admitting for a description through regular and commutative varieties of languages. Given a variety of regular languages V describing the gate types, we proceed by giving an alternative characterisation of the class of languages recognised by constant size circuits with the respective gates. This alternative description mainly relies on the block product principle (or substitution principle), which we extend to work with non-regular languages. The extended version of the block product principle is then used as the main tool to derive ultrafilter equations for the languages recognised by non-uniform constant size circuits with gates described by V , depending on the profinite equations that define the variety of regular languages V .