Logic of Simultaneity

A logical model of spatiotemporal structures is pictured as a succession of processes in time. One usual way to formalize time structure is to assume the global existence of time points and then collect some of them to form time intervals of processes. Under this set-theoretic approach, the logic that governs the processes acquires a Boolean structure. However, in a real distributed system or a relativistic universe where the messagepassing time between different locations is not negligible, the logic has no choice but to accept time interval instead of time point as a primitive concept. From this modeling process of spatiotemporal structures, orthologic, the most simplified version of quantum logic, emerges naturally. 1 General Assumptions We consider a simple model of concurrent processes in computer systems, or more generally, a logical representation of the physical universe that consists of causal sequences of events occurring at different spatial locations. These locations are here referred to as sites . Formally, a site is a sequence of processes ordered by the happened-before relation. The term process denotes one of the continuing states in a site, including in particular nothing happening states. It is also assumed for simplicity that (i) the processes in each site occur consecutively with no time gaps, (ii) each process lasts for some non-zero time duration. Any change of processes is referred to as an event (Fig. 1). ∗Department of Computer and Control Engineering, Oita National College of Technology, 1666 Maki, Oita 870-0152, Japan. Email: [email protected]