  Arithmetic circuits (AC) are circuits over the real numbers with 0/1-valued
input variables whose gates compute the sum or the product of their inputs.
Positive AC -- that is, AC representing non-negative functions -- subsume many
interesting probabilistic models such as probabilistic sentential decision
diagram (PSDD) or sum-product network (SPN) on indicator variables. Efficient
algorithms for many operations useful in probabilistic reasoning on these
models critically depend on imposing structural restrictions to the underlying
AC. Generally, adding structural restrictions yields new tractable operations
but increases the size of the AC. In this paper we study the relative
succinctness of classes of AC with different combinations of common
restrictions. Building on existing results for Boolean circuits, we derive an
unconditional succinctness map for classes of monotone AC -- that is, AC whose
constant labels are non-negative reals -- respecting relevant combinations of
the restrictions we consider. We extend a small part of the map to classes of
positive AC. Those are known to generally be exponentially more succinct than
their monotone counterparts, but we observe here that for so-called
deterministic circuits there is no difference between the monotone and the
positive setting which allows us to lift some of our results. We end the paper
with some insights on the relative succinctness of positive AC by showing
exponential lower bounds on the representations of certain functions in
positive AC respecting structured decomposability.
