Networks of relations

• Authors: Jehoshua Bruck;Matthew Cook
• Venue: Networks of relations
• Year: 2005

This thesis explores the composition of relations into larger relational structures. We define several forms of networks of relations, including exclusion networks, zipper networks, and matrix networks. We prove quick confluent convergence for exclusion networks, and present a parsimonious duality between relations and variables for networks of relations. Consideration of zipper networks makes it clear that "fan-out," i.e., the ability to duplicate information, is most naturally itself represented as a relation along with everything else. This is a notable departure from the traditional lack of representation for this ability. This deconstruction of fan-out provides a more general model for combining relations than was provided by previous models, since we can examine both the traditional case where fan-out (the equality relation on three variables) is available and the more interesting case where its availability is subject to the same limitations as the availability of other relations. As we investigate the composition of relations in this model where fan-out is explicit, what we find is very different from what has been found in the past. We examine the lower portions of the lattice of implementability among relations, and find that in contrast with Post's lattice of implementability among functions, the lattice is quite irregular and positive implementability results are rare. While portions of these lattices are known to be equivalent via a Galois connection, this is only a small portion of the relation lattice, even though it is most of the function lattice. We go on to prove undecidability of the general implementability question for relations, despite its decidability when fan-out is present. Looking at functions without fan-out, we are led to the study of finite chemical reaction networks, for which we find the implementability question turns out to be decidable. We show that certain computations in finite chemical reaction networks are limited to primitive recursive computation, but when standard chemical kinetic reaction rates are used to stochastically determine the next reaction, the system becomes capable of universal computation. This is the first example we know of where adding probabilistic behavior to a natural model of computation increases the range of deterministic functions it can compute.