Efficient execution of dynamically controllable simple temporal networks with uncertainty

A simple temporal network with uncertainty (STNU) is a data structure for representing and reasoning about temporal constraints where the durations of certain temporal intervals—the contingent links—are only discovered during execution. The most important property of an STNU is whether it is dynamically controllable (DC)—that is, whether there exists a strategy for executing its time-points that will guarantee that all of its constraints will be satisfied no matter how the durations of the contingent links turn out. The literature on STNUs includes a variety of DC-checking algorithms and execution algorithms. The fastest DC-checking algorithm reported so far is the O(N3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^3)$$\end{document}-time algorithm due to Morris (Integration of AI and OR techniques in constraint programming—11th international conference, CPAIOR 2014, volume 8451 of Lecture Notes in Computer Science. Springer, Berlin, pp 464–479, 2014). The fastest execution algorithm for dynamically controllable STNUs is the O(N3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^3)$$\end{document}-time algorithm due to Hunsberger (Proceedings of the 20th international symposium on temporal representation and reasoning (TIME-2013). IEEE Computer Society, Washington, 2013). This paper begins by providing the first comprehensive, rigorous, and yet streamlined treatment of the theoretical foundations of STNUs, including execution semantics, dynamic controllability, and a set of results that have been collected into what has recently been called the fundamental theorem of STNUs. The paper carefully argues from basic definitions to proofs of the major theorems on which all of the important algorithmic work on STNUs depends. Although many parts of this presentation have appeared in various forms, in various papers, the scattered nature of the STNU literature has allowed too many holes in the theory to persist, and has relied all too often on proof sketches that leave important details unexamined. The presentation combines results from many sources, while also introducing novel approaches and proofs. The paper concludes by presenting a modified version of a recent algorithm for managing the execution of dynamically controllable STNUs, the fastest reported so far in the literature. The modified version organizes its computations more efficiently and corrects an oversight in the original algorithm.