  We give two criteria for diagrams of Heegaard splittings of 3-manifolds.
Weaker one of them guarantees that the splitting is strongly irreducible, and
the stronger one guarantees in addition that the Goeritz group is finite. They
are generalizations of the criteria introduced by Casson-Gordon and
Lustig-Moriah. Their original ones require as input the diagram formed by a
pair of maximal disk systems. Our present ones accept as input that of
arbitrary disk systems, including minimal disk systems. In other words, our
criteria accept Heegaard diagrams.
