  CSTNs is a constraint-based graph-formalism for conditional temporal
planning. In order to address the DC-Checking problem, in [Comin and Rizzi,
TIME 2015] we introduced epsilon-DC (a refined, more realistic, notion of DC),
and provided an algorithmic solution to it. The epsilon-DC notion is
interesting per se, and the epsilon-DC-Checking algorithm in [Comin and Rizzi,
TIME 2015] rests on the assumption that the reaction-time satisfies epsilon >
0; leaving unsolved the question of what happens when epsilon = 0. In this
work, we introduce and study pi-DC, a sound notion of DC with an instantaneous
reaction-time (i.e. one in which the planner can react to any observation at
the same instant of time in which the observation is made). Firstly, we
demonstrate by a counter-example that pi-DC is not equivalent to 0-DC, and that
0-DC is actually inadequate for modeling DC with an instantaneous
reaction-time. This shows that the main results obtained in our previous work
do not apply directly, as they were formulated, to the case of epsilon=0.
Motivated by this observation, as a second contribution, our previous tools are
extended in order to handle pi-DC, and the notion of ps-tree is introduced,
also pointing out a relationship between pi-DC and HyTN-Consistency. Thirdly, a
simple reduction from pi-DC-Checking to DC-Checking is identified. This allows
us to design and to analyze the first sound-and-complete pi-DC-Checking
procedure. Remarkably, the time complexity of the proposed algorithm remains
(pseudo) singly-exponential in the number of propositional letters. Finally, it
is observed that the technique can be leveraged to actually reduce from pi-DC
to 1-DC, this allows us to further improve the exponents in the time complexity
of pi-DC-Checking.
