State complexity of union and intersection for two-way nondeterministic finite automata
| Authors | |
|---|---|
| Year of publication | 2011 |
| Type | Article in Periodical |
| Magazine / Source | Fundamenta Informaticae |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.3233/FI-2011-540 |
| Field | General mathematics |
| Keywords | finite automata; two-way automata; state complexity; partitions into sums of primes |
| Description | 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 not equal to 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 not in {4,6} (with a few more exceptions) into sums of pairwise distinct primes is established as well. |
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