The structure and complexity of minimal NFA's over a unary alphabet

  • Tao Jiang
  • Edward McDowell
  • B. Ravikumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 560)


Many difficult open problems in theoretical computer science center around nondeterminism. We study the fundamental problem of converting a given deterministic finite automaton (DFA) to a minimal nondeterministic finite automaton (NFA). Despite extensive work on finite automata, this fundamental problem has remained open. Recently, in [Ji91] we studied this problem and showed that this (and related) problems are computationally hard. Here we study the restriction of this problem to the case when the input DFA is over a one-letter alphabet. Even in this restricted case the problem is computationally hard even though our evidence of hardness is different from (and is weaker than) the standard ones such as NP-hardness. Let A → B denote the problem of converting a finite automaton of type A to a minimal finite automaton of type B. Our main result is that DFA → NFA, when the input is a unary cyclic DFA (a DFA whose graph is a simple cycle), is in NP but not in P unless NP ⊑ DTIME(nO(log n)). Our work was also motivated by the problem of finding structurally simple ‘normal forms’ of NFA's over a unary alphabet. We present some normal forms for minimal NFA's over a unary alphabet and present an application to lower bounds on the size complexity of an NFA. In fact, the normal form result is used in a nontrivial manner to show the NP membership result stated above.


Normal Form Start State Vertex Cover Regular Language Finite Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Tao Jiang
    • 1
  • Edward McDowell
    • 2
  • B. Ravikumar
    • 3
  1. 1.Department of Computer Science and SystemsMcMaster UniversityHamiltonCanada
  2. 2.Department of Mathematics and Computer ScienceRhode Island CollegeProvidenceUSA
  3. 3.Department of Computer Science and StatisticsUniversity of Rhode IslandKingstonUSA

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