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

2011 ◽  
Vol 110 (1-4) ◽  
pp. 231-239 ◽  
Author(s):  
Michal Kunc ◽  
Alexander Okhotin
Keyword(s):  
Finite Automata ◽  
State Complexity ◽  
Nondeterministic Finite Automata

NONDETERMINISTIC FINITE AUTOMATA — RECENT RESULTS ON THE DESCRIPTIONAL AND COMPUTATIONAL COMPLEXITY

2009 ◽  
Vol 20 (04) ◽  
pp. 563-580 ◽  
Author(s):  
MARKUS HOLZER ◽  
MARTIN KUTRIB
Keyword(s):  
Computational Complexity ◽  
Finite Automata ◽  
Size Estimation ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Nondeterministic Finite Automata ◽  
Vast Literature ◽  
Recent Developments ◽  
Big Picture ◽  
Main Ideas

Nondeterministic finite automata (NFAs) were introduced in [68], where their equivalence to deterministic finite automata was shown. Over the last 50 years, a vast literature documenting the importance of finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss recent developments relevant to NFAs related problems like, for example, (i) simulation of and by several types of finite automata, (ii) minimization and approximation, (iii) size estimation of minimal NFAs, and (iv) state complexity of language operations. We thus come across descriptional and computational complexity issues of nondeterministic finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.

MAGIC NUMBERS FOR SYMMETRIC DIFFERENCE NFAS

2005 ◽  
Vol 16 (05) ◽  
pp. 1027-1038 ◽  
Author(s):  
LYNETTE VAN ZIJL
Keyword(s):  
Finite Automaton ◽  
Finite Automata ◽  
Symmetric Difference ◽  
Magic Numbers ◽  
Deterministic Finite Automaton ◽  
Nondeterministic Finite Automata ◽  
Binary Case ◽  
Nondeterministic Finite Automaton

Iwama et al. showed that there exists an n-state binary nondeterministic finite automaton such that its equivalent minimal deterministic finite automaton has exactly 2n - α states, for all n ≥ 7 and 5 ≤ α ≤ 2n-2, subject to certain coprimality conditions. We investigate the same question for both unary and binary symmetric difference nondeterministic finite automata. In the binary case, we show that for any n ≥ 4, there is an n-state symmetric difference nondeterministic finite automaton for which the equivalent minimal deterministic finite automaton has 2n - 1 + 2k - 1 - 1 states, for 2 < k ≤ n - 1. In the unary case, we consider a large practical subclass of unary symmetric difference nondeterministic finite automata: for all n ≥ 2, we argue that there are many values of α such that there is no n-state unary symmetric difference nondeterministic finite automaton with an equivalent minimal deterministic finite automaton with 2n - α states, where 0 < α < 2n - 1. For each n ≥ 2, we quantify such values of α precisely.

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