State Complexity of Union and Intersection for Two-way Nondeterministic Finite Automata
Michal Kunc, Alexander Okhotin · Fundamenta Informaticae · 2011
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.