A1 Refereed original research article in a scientific journal
State Complexity of Union and Intersection for Two-Way Nondeterministic Finite Automata
Authors: Kunc M, Okhotin A
Publisher: IOS PRESS
Publication year: 2011
Journal: Fundamenta Informaticae
Journal name in source: FUNDAMENTA INFORMATICAE
Journal acronym: FUND INFORM
Volume: 110
Issue: 1-4
First page : 231
Last page: 239
Number of pages: 9
ISSN: 0169-2968
DOI: https://doi.org/10.3233/FI-2011-540
Abstract
The number of states in a two-way nondeterministic finite automaton (2NFA) needed to represent the 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 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 is not an element of {4, 6} (with a fewmore exceptions) into sums of pairwise distinct primes is established as well.
The number of states in a two-way nondeterministic finite automaton (2NFA) needed to represent the 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 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 is not an element of {4, 6} (with a fewmore exceptions) into sums of pairwise distinct primes is established as well.
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