arXiv:2610.23694v1 Announce Type: new Abstract: We study alternation hierarchies for distributed decision in anonymous networks with constant-size certificates. A prover and a disprover alternately assign certificates to nodes over a constant number of rounds, after which each node decides whether to accept based on its own input and the certificates observed in its neighborhood. An instance is accepted if all nodes accept. Unlike previous work on distributed alternation hierarchies, which considered restricted local computation (Reiter, PODC 2024) or logarithmic-size certificates (Feuilloley et al., ICALP 2016), we focus on constant-size certificates and the role of neighborhood information in determining expressive power. We consider three models: SET, where nodes observe only the set of certificate sequences of their neighbors; MULTISET, where multiplicities are also visible; and PORTS, where neighbors can additionally be distinguished by local port numbers. We establish that SET is strictly weaker than MULTISET, which is in turn strictly weaker than PORTS, and that all three alternation hierarchies are strict at every level. Despite their severe restrictions, these models are remarkably expressive: LEADER belongs to the third level of SET. Our main characterization shows that, over finite input alphabets, SET coincides exactly with monadic second-order logic (MSO$_1$). Moreover, every language definable in counting monadic second-order logic (CMSO$_1$) belongs to PORTS, while MULTISET already contains languages that are not CMSO$_1$-definable. Nevertheless, this expressive power has fundamental limitations. PORTS does not capture MSO$_2$: neither Hamiltonicity nor the existence of a perfect matching belongs to PORTS. Furthermore, over unbounded input alphabets, PORTS cannot verify that every node has been given the correct network size.