# On the Complexity of Universality for Partially Ordered NFAs

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Markus Krötzsch, Tomáš Masopust, Michaël Thomazo

On the Complexity of Universality for Partially Ordered NFAs

**Abstract.** Partially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace-complete.

*Published at MFCS2016 (Conference paper) *

*Download PDF* *(last update: July 2 2016)*

## Citation details

- Markus Krötzsch, Tomáš Masopust, Michaël Thomazo. On the Complexity of Universality for Partially Ordered NFAs. In Proceedings of the 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik 2016.

author = {Markus Kr\"{o}tzsch and

Tom{\'{a}}{\v{s}} Masopust and

Micha{\"{e}}l Thomazo},

title = {On the Complexity of Universality for Partially

Ordered {NFAs}},

booktitle = {Proceedings of the 41st International Symposium

on Mathematical Foundations of Computer Science

(MFCS'16)},

publisher = {Schloss Dagstuhl - Leibniz-Zentrum für

Informatik},

year = {2016},

note = {to appear}

}