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

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 +. . .+ p_k and n = q₁ +. . .+q_l, where all numbers p₁, . . . , p_k, q₁, . . . , 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.