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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 learned. We discuss various classes of
bounded action theories. Then we show that verification of a powerful first-order variant
of the μ-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
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