  In this paper, we investigate bounded action theories in the situation
calculus. A bounded action theory is one which entails that, in every
situation, the number of object tuples in the extension of fluents is bounded
by a given constant, although such extensions are in general different across
the infinitely many situations. We argue that such theories are common in
applications, either because facts do not persist indefinitely or because the
agent eventually forgets some facts, as new ones are learnt. We discuss various
classes of bounded action theories. Then we show that verification of a
powerful first-order variant of the mu-calculus is decidable for such theories.
Notably, this variant supports a controlled form of quantification across
situations. We also show that through verification, we can actually check
whether an arbitrary action theory maintains boundedness.
