Between quantum logic and concurrency

TL;DR

This paper explores the relationship between quantum logic and concurrency in causal nets, showing how logical structures can be derived from concurrent process histories and identifying classical substructures within a quantum logical framework.

Contribution

It introduces a logical language based on closed sets of causal nets, connecting lines to truth assignments, and reveals classical substructures within a non-classical quantum logic.

Findings

Closed sets in the lattice correspond to truth assignments for lines.

Maximal antichains relate to Boolean substructures.

The logic captures non-classical features of concurrent processes.

Abstract

We start from two closure operators defined on the elements of a special kind of partially ordered sets, called causal nets. Causal nets are used to model histories of concurrent processes, recording occurrences of local states and of events. If every maximal chain (line) of such a partially ordered set meets every maximal antichain (cut), then the two closure operators coincide, and generate a complete orthomodular lattice. In this paper we recall that, for any closed set in this lattice, every line meets either it or its orthocomplement in the lattice, and show that to any line, a two-valued state on the lattice can be associated. Starting from this result, we delineate a logical language whose formulas are interpreted over closed sets of a causal net, where every line induces an assignment of truth values to formulas. The resulting logic is non-classical; we show that maximal antichains in a causal net are associated to Boolean (hence "classical") substructures of the overall quantum logic.