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

The number of states in a two-way nondeterministic finite automaton (2NFA) needed to represent union and 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. 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= 6 (and with finitely many other exceptions), there exist partitions m = p1 + . . .+ pk and n = q1 + . . .+ q`, where all numbers p1, . . . , pk, q1, . . . , q` > 2 are powers of pairwise distinct primes.