Effect of restricting inputs to partial‑word completions

Investigate how the tractability frontier for probabilistic membership changes when inputs are restricted to distributions induced by partial words (i.e., completion counting), and determine which of the paper’s PTIME and #P‑hardness results persist in this restricted setting.

Background

Beyond general probabilistic words, the authors consider the special case of partial words (uniform‑or‑Dirac choices per position), which correspond to completion counting. While many hardness proofs already handle a domain‑constrained variant, the overall tractability landscape for this restriction is not established.

Clarifying this would connect their results to classical counting problems on partial words.