  We present a general framework for the information backflow (IB) approach of
Markovianity that not only includes a large number, if not all, of IB
prescriptions proposed so far but also is equivalent to completely positive
divisibility for invertible evolutions. Following the common approach of IB,
where monotonic decay of some physical property or some information quantifier
is seen as the definition of Markovianity, we propose in our framework a
general description of what should be called a proper `physicality quantifier'
to define Markovianity. We elucidate different properties of our framework and
use them to argue that an infinite family of non-Markovianity measures can be
constructed, which would capture varied strengths of non-Markovianity in the
dynamics. Moreover, we show that generalized trace-distance measure in two
dimensions serve as a sufficient criteria for IB Markovianity for a number of
prescriptions suggested earlier in the literature.
