We present relaxed notions of simulation and bisimulation on Probabilistic
Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve
the usual notions of bisimulation and simulation on PAs. We give logical
characterisations of these notions by choosing suitable logics which differ
from the elementary ones, L with negation and L without negation, by the modal
operator. Using flow networks, we show how to compute the relations in PTIME.
This allows the definition of an efficiently computable non-discounted distance
between the states of a PA. A natural modification of this distance is
introduced, to obtain a discounted distance, which weakens the influence of
long term transitions. We compare our notions of distance to others previously
defined and illustrate our approach on various examples. We also show that our
distance is not expansive with respect to process algebra operators. Although L
without negation is a suitable logic to characterise epsilon-(bi)simulation on
deterministic PAs, it is not for general PAs; interestingly, we prove that it
does characterise weaker notions, called a priori epsilon-(bi)simulation, which
we prove to be NP-difficult to decide.Comment: In Proceedings QAPL 2011, arXiv:1107.074