State complexity of insertion

Yo Sub Han, Sang Ki Ko, Timothy Ng, Kai Salomaa

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn-m-1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

Original languageEnglish
Pages (from-to)863-878
Number of pages16
JournalInternational Journal of Foundations of Computer Science
Issue number7
StatePublished - 1 Nov 2016


  • Insertion operation
  • regular languages
  • state complexity


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