The digraph property KERNEL is a very simple and well-known property
studied in various areas.
We previously defined a variant of this property as
a counterexample of 0-1 law for
the monadic existential second order
logic with at most two first-order
variables, over structures
with 16 binary relations.
Goranko and Kapron have defined a variant
in frames which is also expressible in the fragment
which expresses frame satisfiability of propositional modal logic,
a small fragment of the logic above over structures
with only one relation.
We prove that this variant is almost surely false and
we propose other variants of this property
and establish that one of them provides a counterexample
of the 0-1 law for frame satisfiability of propositional modal logic.
This refutes a result by Halpern and Kapron
which establishes that the 0-1 law holds for this logic.
This also strongly refines our previous counterexample.
