Classical Dynamic Controllability Revisited - A Tighter Bound on the Classical Algorithm

Abstract

Abstract:Simple Temporal Networks with Uncertainty (STNUs) allow the representation of temporal problems where some durations are uncontrollable (determined by nature), as is often the case for actions in planning. It is essential to verify that such networks are dynamically controllable (DC) -executable regardless of the outcomes of uncontrollable durations -and to convert them to an executable form. We use insights from incremental DC verification algorithms to re-analyze the original verification algorithm. This algorithm, thought to be pseudo-polynomial and subsumed by an O(n 5 ) algorithm and later an O(n 4 ) algorithm, is in fact O(n 4 ) given a small modification. This makes the algorithm attractive once again, given its basis in a less complex and more intuitive theory. Finally, we discuss a change reducing the amount of work performed by the algorithm. BACKGROUNDTime and concurrency are increasingly considered essential in planning and multi-agent environments, but temporal representations vary widely in expressivity. For example, Simple Temporal Problems (STPs, (Dechter et al., 1991)) allow us to efficiently determine whether a set of timepoints (events) can be assigned real-valued times in a way consistent with a set of constraints bounding temporal distances between timepoints. The start and end of an action can be represented as timepoints, but its possible durations can only be represented as an STP constraint if the execution mechanism can choose durations arbitrarily within the given bounds. Usually, exact durations are instead chosen by nature and agents must generate plans that work regardless of the eventual outcomes. STPs with Uncertainty, STPUs (Vidal and Ghallab, 1996), capture this aspect by introducing contingent timepoints that correspond to the end of an action, associated with contingent temporal constraints corresponding to possible durations to be decided by nature. One must then find a way to assign times to ordinary controlled timepoints (determine when to start actions) so that for every possible outcome for the contingent constraints (action durations), there exists some solution for the ordinary requirement constraints (corresponding to STP constraints).If an STPU allows us to schedule controlled timepoints (actions to be started) incrementally given that we receive information when a contingent timepoint occurs (an action ends), it is dynamically controllable (DC) and can be efficiently executed by a dispatcher (Muscettola et al., 1998). Conversely, guaranteeing that constraints are satisfied when executing a non-DC plan is impossible, as it would require information about future duration outcomes.Three algorithms for verifying the dynamic controllability of a complete STPU have been published: 1. MMV (Morris et al., 2001), here also called the classical algorithm. It is a simple algorithm that derives and tightens constraints using specific rules. It is easily implemented, captures the intuition behind STNUs and has a direct correctness proof. Its run-time is pseudo-polynomial...

Efficient Temporal Reasoning with Uncertainty

“…This can be read "i can never be later than −a ij after j", which can also be expressed as 'i is at least a ij before j'. 25] and similarly for [13, ∞]. In a consistent STN, a ij ≤ b ij due to the definition of the constraints.…”

“…Theorem 4 (Nilsson et al [25]). The classical MMV algorithm for deciding dynamic controllability of an STNU can, with the small modifications shown in Algorithm 11, decide dynamic controllability in time O(n 4 ).…”

“…Lemma 4 (Nilsson et al [25]). Suppose all rules in Figure 4.6 are applied to the EDG of a dynamically controllable STNU until no more rules are applicable.…”

“…Lemma 6 (Nilsson et al [25]). The length of the longest critical chain in GlobalDC with APSP is ≤ 7n.…”

“…A contingent constraint is used to describe the uncertainty of a process that is started at one event, which may or may not be controllable, and ends in a contingent event. Dinner Ready [25,30] …”