Characterize the class of PGA-definable distributions

Characterize precisely the set of discrete (sub-)distributions over non-negative integer vectors ^V whose probability generating functions can be realized as the behavior IM^*F of a probability generating automaton, i.e., an $$-automaton over the formal power series semiring $#1{}{^V}$ whose coefficients sum to at most 1. Provide a comprehensive characterization beyond the currently identified inclusion of discrete-phase distributions.

Background

The paper introduces probability generating automata (PGA) as %%%%0%%%%#1{}{^V} whose behavior is a probability generating function (PGF) describing a (sub-)distribution over V. PGAs serve as the core representation for exact inference in the proposed framework.

The authors note that PGAs capture at least the discrete-phase distributions, but they do not provide a full description of the expressive power of PGAs. They explicitly defer a comprehensive characterization of the class of distributions definable by PGAs.

References

It follows from the definitions that the class of PGA-definable distributions includes the so-called discrete-phase distributions~(see, e.g.,); a more thorough characterization of PGA-definable distributions is left for future work.

— Weighted Automata for Exact Inference in Discrete Probabilistic Programs (2509.15074 - Geißler et al., 18 Sep 2025) in Section 2: Background on Weighted Automata, paragraph “Probability Generating Functions and Automata”