Lambda the Ultimate

The network approach to computation is more direct and "physical" than the one based on some specific computing devices (like Turing machines). However, the size of a usual -- e.g., Boolean -- network does not reflect the complexity of computing the corresponding function, since a small network may be very hard to find even if it exists. A history of the work of a particular computing device can be described as a network satisfying some restrictions. The size of this network reflects the complexity of the problem, but the restrictions are usually somewhat arbitrary and even awkward. Causal nets are restricted only by determinism (causality) and locality of interaction. Their geometrical characteristics do reflect computational complexities. And various imaginary computer devices are easy to express in their terms. The elementarity of this concept may help bringing geometrical and algebraic (and maybe even physical) methods into the theory of computations. This hope is supported by the grouptheoretical criterion given in this paper for computability from symmetrical initial configurations.

No doubt, modeling the history of computation rather than the current state is nothing new, but this paper tries to find new applications for that.