  In this paper, for symplectic and split odd special orthogonal groups, we
develop an account of theory on the intersection problem of local Arthur
packets. Specifically, following Atobe's reformulation on M{\oe}glin's
construction of local Arthur packets, we give a complete set of operators on
the construction data, based on which, we provide algorithms and Sage codes to
determine whether a given representation is of Arthur type. Furthermore, for
any representation $\pi$ of Arthur type, we give a precise formula for the set
$$ \Psi(\pi)=\{ \text{local Arthur parameter }\psi \ | \ \text{the local Arthur
packet } \Pi_{\psi} \text{ contains } \pi\}.$$
  Our results have many applications, including the precise counting of
tempered representations in any local Arthur packet, specifying and
characterizing "the" local Arthur parameter in $\Psi(\pi)$ for $\pi$,
especially when $\pi$ belongs to several local Arthur packets but does not
belong to any local $L$-packet of Arthur type.
