State Complexity of Unambiguous Operations on Deterministic Finite Automata

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10952)


The paper determines the number of states in a deterministic finite automaton (DFA) necessary to represent “unambiguous” variants of the union, concatenation, and Kleene star operations on formal languages. For the disjoint union of languages represented by an m-state and an n-state DFA, the state complexity is \(mn-1\); for the unambiguous concatenation, it is known to be \(m2^{n-1} - 2^{n-2}\) (Daley et al. “Orthogonal concatenation: Language equations and state complexity”, J. UCS, 2010), and this paper shows that this number of states is necessary already over a binary alphabet; for the unambiguous star, the state complexity function is determined to be \(\frac{3}{8}2^n+1\). In the case of a unary alphabet, disjoint union requires up to \(\frac{1}{2}mn\) states, unambiguous concatenation has state complexity \(m+n-2\), and unambiguous star requires \(n-2\) states in the worst case.


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Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovak Republic
  2. 2.St. Petersburg State UniversitySaint PetersburgRussia

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