I F I G Institut fur¨ Informatik R esearch R eport On the Computational Complexity of Partial Word Automata Problems Markus Holzer Sebastian Jakobi Matthias Wendlandt IFIG Research Report 1404 May 2014 Institut f¨ur Informatik JLU Gießen Arndtstraße 2 35392 Giessen, Germany Tel: +49-641-99-32141 Fax: +49-641-99-32149
[email protected] www.informatik.uni-giessen.de IFIG Research Report IFIG Research Report 1404, May 2014 On the Computational Complexity of Partial Word Automata Problems Markus Holzer,1 Sebastian Jakobi,2 and Matthias Wendlandt3 Institut f¨ur Informatik, Universit¨at Giessen Arndtstraße 2, 35392 Giessen, Germany Abstract. We consider computational complexity of problems related to partial word au- tomata. Roughly speaking, a partial word is a word in which some positions are unspecified, and a partial word automaton is a finite automaton that accepts a partial word language— here the unspecified positions in the word are represented by a “hole” symbol ⋄. A partial word language L′ can be transformed into an ordinary language L by using a ⋄-substitution. In particular, we investigate the complexity of the compression or minimization problem for partial word automata, which is known to be NP-hard. We improve on the previously known complexity on this problem, by showing PSPACE-completeness. In fact, it turns out that almost all problems related to partial word automata, such as, e.g., equivalence and universality, are already PSPACE-complete. Moreover, we also study these problems under the further restriction that the involved automata accept only finite languages. In this case, the complexity of the studied problems drop from PSPACE-completeness down to P coNP-hardness and containment in Σ2 depending on the problem investigated.