Directed Homotopy in Non-Positively Curved Spaces

TL;DR

This paper introduces a geometric and algebraic framework for analyzing concurrent programs with mutexes, showing they have non-positive curvature and that directed homotopy aligns with classical homotopy in these settings.

Contribution

It defines non-positive curvature for precubical sets, proves the equivalence of directed and non-directed homotopy for such programs, and relates these structures to generalized metric spaces.

Findings

Programs with mutexes have non-positive curvature.

Directed and non-directed homotopy coincide in these spaces.

Conditions on precubical sets match those for generalized metric spaces.

Abstract

A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of programs using only mutexes, which are the most widely used synchronization primitive. We show that in this case, the resulting programs have non-positive curvature, a notion that we introduce and study here for precubical sets, and can be thought of as an algebraic analogue of the well-known one for metric spaces. Using this it, as well as categorical rewriting techniques, we are then able to show that directed and non-directed homotopy coincide for directed paths in these precubical sets. Finally, we study the geometric realization of precubical sets in metric spaces, to show that our conditions on precubical sets actually coincide with those for metric spaces. Since the category of metric spaces is not cocomplete, we are lead to work with generalized metric spaces and study some of their properties.