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Efficient execution of dynamically controllable simple temporal networks with uncertainty

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Abstract

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 \(O(N^3)\)-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 \(O(N^3)\)-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.

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Notes

  1. They also highlighted interesting connections between the dispatchability of simple temporal networks (STNs) [21, 29] and the dynamic controllability of STNUs, but did not formally define dispatchability for STNUs. Similar remarks apply to a follow-up paper [27].

  2. Nilsson et al. [25] have observed that the M-14 and EIDC2 algorithms, although derived independently, employ many similar techniques. They conjecture that the algorithms are, in fact, equivalent.

  3. Many researchers prefer to automatically include trivial self-loops of length 0 at each time-point in an STN, corresponding to constraints of the form, \(X-X\le 0\), in which case, the diagonal entries of a consistent distance matrix will all necessarily be zero, instead of being merely non-negative.

  4. Agents are not part of the semantics of STNUs. They are used here for expository convenience.

  5. The notation presented in Definition 4 is equivalent to the original notation introduced by Morris et al. [18]. It is used in this paper primarily to facilitate access to the constituents of the contingent links—for example, \((A,x,y,C)\) instead of the more cumbersome \(( start (e), \ell (e), u(e), finish (e))\).

  6. Note that there is no prohibition against a contingent time-point \(C\) for one contingent link serving as the activation time-point for some other contingent link. In this way, contingent links may form chains or trees.

  7. The equivalence of the MMV-01 and RTED-based semantics is addressed in detail elsewhere [10].

  8. Context usually makes clear to which STNU a given space of situations applies.

  9. Note that \(C_i - A_i = \omega _i\) is shorthand for the pair of constraints, \(C_i - A_i \le \omega _i\) and \(A_i - C_i \le -\omega _i\).

  10. Morris et al. defined execution strategies as mappings from projections to schedules, which is equivalent.

  11. Allowing \(t\) to be equal to \(\mathtt {now}_\xi \) would enable a form of instantaneous reactivity. For example, an agent might observe the execution of a contingent time-point at time 3, and then instantaneously react by deciding to execute an executable time-point at that same time 3.

  12. In the original definition of STNU graphs [19], each contingent link also gave rise to two ordinary edges, \(A\mathop {\longrightarrow }\limits ^{\ y}C\) and \(A\mathop {\leftarrow }\limits ^{-x}C\), representing the known fact that the duration of the contingent link must fall within the interval \([x,y]\). However, it has been shown that including these extra edges is not necessary—because the execution semantics for STNUs ensures that they will be satisfied [13].

  13. In Table 1, applicability conditions of the form, \(X \not \equiv Y\), should be construed as requiring that \(X\) and \(Y\) be distinct time-points, not as a constraint on their values.

    Table 1 The edge-generation rules for STNUs from Morris and Muscettola [19]
    Full size table

  14. Note that a semi-reducible path might contain multiple occurrences of the same lower-case edge [13].

  15. Definitions 30 and 31, and the proof technique for Theorem 5, below, were originally presented by Morris [16], but only in the context of an execution semantics that allowed a form of instantaneous reactivity. This author subsequently modified them to conform to the standard STNU execution semantics [11, 13].

  16. To avoid confusion between the letters \(w\) and \(\omega \), the proof uses the letter \(u\) instead of \(w\).

  17. This is an application of the Upper Case rule from Table 1 where \(E \mapsto C, B \mapsto C,\) and \(F \mapsto A\).

  18. The letter \(u\) is used instead of the \(w\) from the Upper Case rule in Table 1 to avoid confusion with \(\omega \).

  19. Morris showed that extension sub-paths within the same path \(\mathcal {P}\) must either be disjoint or nested, one inside the other [13, 16]. An outermost extension sub-path is one that is not nested inside any other.

  20. In view of Theorem 7, the set \(\mathcal {F}_k\) from Theorem 6 can be taken to be \(\mathcal {F}^*\) with no loss of generality.

  21. Recall that, for any contingent link, \((A,x,y,C)\), the execution semantics for STNUs ensures that the duration of each contingent link must stay within its upper bound, \(y\). This, together with the ordinary constraint, \(C-A \ge y\), represented by the edge, \(C\mathop {\longrightarrow }\limits ^{-y}A\) in the AllMax graph, effectively forces \(C-A=y\). For an alternative view, note that if the additional ordinary edges discussed in Footnote 12 were included in the STNU graph, then for each contingent link, \((A,x,y,C)\), the AllMax graph would have two ordinary edges together representing the constraint, \(C-A = y\).

  22. The distance matrix shown in Fig. 11 presumes that there are self-loops of length 0 at each time-point, which is common practice in the literature. For dynamically controllable networks, this leads to the distance matrix having zeroes down its main diagonal.

    Fig. 11
    figure 11

    The APSSRP matrix for the sample STNU from Figs. 9 and 10 (blank entries are \(\infty \))

    Full size image

  23. As a consequence, during execution, the AllMax graph only forces all as-yet-unexecuted contingent links to take on their maximum durations.

  24. The Proof of Theorem 8 will show that \(t_U > \mathtt {now}_\xi \) invariably holds when following the strategy \({\hat{R}}\).

  25. For the purposes of the proof, it is convenient to ignore issues of computational efficiency. Thus, the graph \(\mathcal {G}_x^*\), and the distance matrix \(\mathcal {D}_x^*\) are thought of as being effectively re-computed from scratch after each update to the OU-graph \(\mathcal {G}_\mathrm {ou}^*\). The FAST-EX algorithm presented in Sect. 5 uses incremental techniques to efficiently update \(\mathcal {G}_x^*\) and \(\mathcal {D}_x^*\) after each execution event.

  26. For complete generality, each outcome involves a set of time-points that executes. However, in practice, these sets are often singleton sets.

  27. Being rigidly connected is an equivalence relation; so, the notion of a rigid component is well defined.

  28. If desired, the hash tables used by FAST-EX can be replaced by vectors and arrays.

  29. \(C\) is used as an index into the \(\mathtt {UC}\) matrix instead of \(A\), since multiple contingent links could have the same activation time-point, \(A\).

  30. Only the shortest edge between each pair of time-points is stored in the \( Ins \) and \( Outs \) hash tables.

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    Luke Hunsberger

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Hunsberger, L. Efficient execution of dynamically controllable simple temporal networks with uncertainty. Acta Informatica 53, 89–147 (2016). https://doi.org/10.1007/s00236-015-0227-0

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