Affiliations: Department of Mathematics, Masaryk University, Brno, Czech Republic. kunc@math.muni.cz | Department of Mathematics, University of Turku, Turku, FI–20014, Finland. alexander.okhotin@utu.fi
Note: [] Supported by grant 201/09/1313 of the Grant Agency of the Czech Republic. Address for correspondence: Department of Mathematics, Masaryk University, Brno, Czech Republic
Note: [] Also affiliation: Academy of Finland. Supported by the Academy of Finland under grant 134860.
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 = p1 +. . .+ pk and n = q1 +. . .+ql, where all numbers p1, . . . , pk, q1, . . . , ql ≥ 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.
Keywords: Finite automata, two-way automata, state complexity, partitions into sums of primes
