  Recent years have seen a great deal of progress in our understanding of
bootstrap percolation models, a particular class of monotone cellular automata.
In the two dimensional lattice there is now a quite satisfactory understanding
of their evolution starting from a random initial condition, with a strikingly
beautiful universality picture for their critical behaviour. Much less is known
for their non-monotone stochastic counterpart, namely kinetically constrained
models (KCM). In KCM each vertex is resampled (independently) at rate one by
tossing a p-coin iff it can be infected in the next step by the bootstrap
model. In particular infection can also heal, hence the non-monotonicity.
Besides the connection with bootstrap percolation, KCM have an interest in
their own as they feature some of the most striking features of the
liquid/glass transition, a major and still largely open problem in condensed
matter physics. In this paper we pave the way towards proving universality
results for KCM similar to those for bootstrap percolation. Our novel and
general approach establishes a close connection between the critical scaling of
characteristic time scales for KCM and the scaling of the critical length in
critical bootstrap models. Although the full proof of universality for KCM is
deferred to a forthcoming paper, here we apply our general method to the
Friedrickson-Andersen k-facilitated models, amongst the most studied KCM, and
to the Gravner-Griffeath model. In both cases our results are close to optimal.
