Formal Techniques For Safety-Critical Systems (Ftscs 2018) Preface

We investigate collections of reversible gates closed under parallel and serial composition, with and without borrow and ancilla closure. In order to better understand the structure of these collections of reversible gates, we investigate the lattice of closed sets and the maximal members of this lattice, that is, collections that are not all gates, but the addition of a single new gate will allow us to construct all gates. From previous work, we know the maximal closed sets over a finite alphabet.In this paper we extend this work to ancilla and borrow closure for reversible gates. Here we find several structural results, where we know that there are restricted possibilities for all maximal borrow and ancilla closed collections of gates. We determine several classes of maximal gates and determine their dual structures as weights as defined by Jeřábek. These maximal closed collections of gates correspond to minimal weight permutation co-clones.Maximality has several applications, one of the interesting ones being that a gate in no maximal closed set must be universal. Thus an understanding of maximality is intimately tied to an understanding of universality.