![](bgb.png)
In best-case scenarios, the pre-processing algo-
rithm will result in all off-diagonal entries in F be-
ing crossed out, implying that there can be no nesting
of LCR paths in any iSRN loop. In such cases, it is
only necessary to do a single, O(N
3
)-time round of
the N
4
algorithm to ascertain whether the STNU is
dynamically controllable. The benefit in such cases
can be dramatic, for if the network contains even one
semi-reducible path having K levels of nesting, then
the unaided N
4
algorithm would needlessly perform
K rounds of processing in O(N
4
) time.
6 CONCLUSIONS
This paper presented a new way of analyzing the
structure of STNU graphs with the aim of speeding
up DC checking. It proved that the number of oc-
currences of lower-case edges in any iSRN loop is
bounded above by 2
K
− 1. It presented a recursive
algorithm for constructing STNUs that contain iSRN
loops that attain this upper bound, thereby showing
that the bound is tight. In view of their highly con-
voluted structure, such loops are called magic loops.
Finally, it presented an O(N
3
)-time pre-processing al-
gorithm that exploits the 2
K
−1 bound to speed up DC
checking for some networks. Thus, the paper makes
both theoretical and practical contributions.
Other researchers have sought to speed up the
process of DC checking using incremental algo-
rithms. Stedl and Williams (2005) developed Fast-
IDC, an incremental algorithm that maintains the dis-
patchability of an STNU after the insertion of new
constraints or the tightening of existing constraints.
Shah et al. (2007) extended Fast-IDC to accommo-
date the removal or weakening of constraints. Al-
though intended to be applied incrementally, their al-
gorithm showed orders of magnitude improvement
over an earlier pseudo-polynomial DC-checking algo-
rithm when evaluated empirically, checking dynamic
controllability from scratch. It would be interesting
to see if their work could be applied to generate an
incremental version of the Morris’ N
4
algorithm.
Others have extended the concept of dynamic con-
trollability to accommodate various combinations of
probability, preference and disjunction. For exam-
ple, Tsamardinos (2002) augmented contingent dura-
tions with probability density functions and provided
a method that, under certain restrictions, finds “the
schedule that maximizes the probability of execut-
ing the plan in a way that respects the temporal con-
straints.” Tsamardinos et al. (2003) then extended that
work by developing algorithms to compute lower and
upper bounds for the probability of a legal plan exe-
cution. Morris et al. (2005) similarly used probability
density functions to represent the uncertainties associ-
ated with contingent durations, but also incorporated
preferences over event durations. Rossi et al. (2006)
presented a thorough treatment of STNUs augmented
with preferences (but not probabilities). They defined
the Simple Temporal Problem with Preferences and
Uncertainty (STPPU) and notions of weak, strong and
dynamic controllability.
Effinger et al. (2009) defined dynamic controlla-
bility for temporally-flexible reactive programs that
include the following constructs: “conditional execu-
tion, iteration, exception handling, non-deterministic
choice, parallel and sequential composition, and sim-
ple temporal constraints”. They presented a DC-
checking algorithm for temporally-flexible reactive
programs that frames the problem as an “AND/OR
search tree over candidate program executions.”
REFERENCES
Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C.
(2009). Introduction to Algorithms. MIT Press.
Dechter, R., Meiri, I., and Pearl, J. (1991). Temporal con-
straint networks. Artificial Intelligence, 49:61–95.
Effinger, R., Williams, B., Kelly, G., and Sheehy, M. (2009).
Dynamic controllability of temporally-flexible reac-
tive programs. In Gerevini, A., Howe, A., Cesta, A.,
and Refanidis, I., editors, Proceedings of the Nine-
teenth International Conference on Automated Plan-
ning and Scheduling (ICAPS 09). AAAI Press.
Hunsberger, L. (2010). A fast incremental algorithm for
managing the execution of dynamically controllable
temporal networks. In Proceedings of the 17th Inter-
national Symposium on Temporal Representation and
Reasoning (TIME-2010), pages 121–128, Los Alami-
tos, CA, USA. IEEE Computer Society.
Morris, P. (2006). A structural characterization of temporal
dynamic controllability. In Principles and Practice of
Constraint Programming (CP 2006), volume 4204 of
Lecture Notes in Computer Science, pages 375–389.
Springer.
Morris, P., Muscettola, N., and Vidal, T. (2001). Dynamic
control of plans with temporal uncertainty. In Nebel,
B., editor, 17th International Joint Conference on Ar-
tificial Intelligence (IJCAI-01), pages 494–499. Mor-
gan Kaufmann.
Morris, R., Morris, P., Khatib, L., and Yorke-Smith, N.
(2005). Temporal constraint reasoning with prefer-
ences and probabilities. In Brafman, R. and Junker,
U., editors, Proceedings of the IJCAI-05 Multidisci-
plinary Workshop on Advances in Preference Han-
dling, pages 150–155.
Rossi, F., Venable, K. B., and Yorke-Smith, N. (2006). Un-
certainty in soft temporal constraint problems: A gen-
eral framework and controllability algorithms for the
fuzzy case. Journal of Artificial Intelligence Research,
27:617–674.
MagicLoopsinSimpleTemporalNetworkswithUncertainty-ExploitingStructuretoSpeedUpDynamicControllability
Checking
167