Covering space theory for directed topology
Abstract
The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a "local preorder" encoding control flow. In the case where time does not loop, the "locally preordered" state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a "locally monotone" covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes.
Cite
@article{arxiv.0812.1157,
title = {Covering space theory for directed topology},
author = {Eric Goubault and Emmanuel Haucourt and Sanjeevi Krishnan},
journal= {arXiv preprint arXiv:0812.1157},
year = {2011}
}
Comments
14 pages, 2 figures, results partially presented at ATMCS III 2008; deleted false Lem 2.2; corrected proof of Lem 3.7; deleted wrong formula for \lambda(\theta,d_{-1}\theta) on p10; deleted Prop 2.20 and all mention of op-fibration (confusing); generalized Lem 3.6; weakened claim of Eg. 2.2 b/c homotopies need not lift; added defs and fixed typos throughout; main results unchanged