Comparing the size of NFAs with and without <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>ε</mml:mi></mml:math>-transitions

International audienceWe relate two complexity notions of bipartite graphs: the minimal weight biclique covering number Cov(G) and the minimal rec-tifier network size Rect(G) of a bipartite graph G. We show that there exist graphs with Cov(G) ≥ Rect(G) 3/2−ǫ . As a corollary, we estab-lish that there exist nondeterministic finite automata (NFAs) with ε-transitions, having n transitions total such that the smallest equivalent ε-free NFA has Ω(n 3/2−ǫ) transitions. We also formulate a version of previous bounds for the weighted set cover problem and discuss its con-nections to giving upper bounds for the possible blow-up

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Biclique Coverings, Rectifier Networks and the Cost of ε-Removal

Iván

Lelkes

Nagy-György

et al. 2014

Descriptional Complexity of Formal Systems

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Probabilism versus Alternation for Automata

Schnitger

2018

Adventures Between Lower Bounds and Higher Altitudes

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We compare the number of states required for one-way probabilistic finite automata with a positive gap (1P2FAs) and the number of states required for one-way alternating automata (1AFAs). We show that a 1P2FA P can be simulated by 1AFA with an at most polynomial increase in the number s(P) of states of P , provided only inputs of length at most poly(s(P)) are considered. On the other hand we gather evidence that the number of states grows super-polynomially if the number of alternations is bounded by a fixed constant. Thus the behavior of one-way automata seems to be in marked contrast with the behavior of polynomial-time computations. Many thanks to Juraj for many ideas, many questions, many answers and lots of hiking.

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The Tractability Frontier for NFA Minimization

Björklund

Martens

Automata, Languages and Programming

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Abstract. We essentially show that minimizing finite automata is NPhard as soon as one deviates from the class of deterministic finite automata. More specifically, we show that minimization is NP-hard for all finite automata classes that subsume the class that is unambiguous, allows at most one state q with a non-deterministic transition for at most one alphabet symbol a, and is allowed to visit state q at most once in a run. Furthermore, this result holds even for automata that only accept finite languages.

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Comparing the size of NFAs with and without ε-transitions

Cited by 11 publications

References 7 publications

Biclique Coverings, Rectifier Networks and the Cost of ε-Removal

Biclique Coverings, Rectifier Networks and the Cost of ε-Removal

Probabilism versus Alternation for Automata

The Tractability Frontier for NFA Minimization

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