Complexity of universality and related problems for partially ordered NFAs

Partially ordered NFAs (poNFAs) are NFAs where cycles occur only in the form of self-loops. A poNFA is universal if it accepts all words over its alphabet. Deciding universality is PSpace -complete for poNFAs. We show that this remains true when restricting to fixed alphabets. This is nontrivial since standard encodings of symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP -complete complexity bound is obtained if all self-loops in the poNFA are deterministic. We find that such restricted poNFAs (rpoNFAs) characterize R-trivial languages, and establish the complexity of deciding if the language of an NFA is R-trivial. The limitation to fixed alphabets is essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace -complete. Consequently, we obtain the complexity results for inclusion and equivalence problems. Finally, we show that the languages of rpoNFAs are definable by deterministic (one-unambiguous) regular expressions.