State Complexity of Union and Intersection for Two-way Nondeterministic Finite Automata

  • Authors:

  • Michal Kunc;Alexander Okhotin

  • Affiliations:

  • (Correspd.) (Supported by grant 201/09/1313 of the Grant Agency of the Czech Republic) Department of Mathematics, Masaryk University, Brno, Czech Republic. kunc@math.muni.cz;(Additional affiliation: Academy of Finland. Supported by the Academy of Finland under grant 134860) Department of Mathematics, University of Turku, Turku, FI-20014, Finland. alexander.okhotin@utu ...

  • Venue:
  • Fundamenta Informaticae - Theory that Counts: To Oscar Ibarra on His 70th Birthday
  • Year:
  • 2011

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Abstract

The number of states in a two-way nondeterministic finite automaton (2NFA) needed to represent intersection of languages given by an m-state 2NFA and an n-state 2NFA is shown to be at least m+n and at most m+n+1. For the union operation, the number of states is exactly m+n. The lower bound is established for languages over a one-letter alphabet. The key point of the argument is the following number-theoretic lemma: for all m, n ≥ 2 with m, n ≠ 6 (and with finitely many other exceptions), there exist partitions m = p 1 +. . .+ p k and n = q 1 +. . .+q l, where all numbers p 1, . . . , p k, q 1, . . . , q l ≥ 2 are powers of pairwise distinct primes. For completeness, an analogous statement about partitions of any two numbers m, n ∉ {4, 6} (with a few more exceptions) into sums of pairwise distinct primes is established as well.