Abstract
To get a more comprehensive understanding of the branching complexity of nondeterministic finite automata (NFA), we introduce and study the string path width and depth path width measures. The string path width on a string w counts the number of all complete computations on w, and the depth path width on an integer \(\ell \) counts the number of complete computations on all strings of length \(\ell \). We give an algorithm to decide the finiteness of the depth path width of an NFA. Deciding finiteness of string path width can be reduced to the corresponding question on ambiguity.
An NFA is nearly acyclic if any computation can pass through at most one cycle. The class of nearly acyclic NFAs consists of exactly all NFAs with finite depth path width. Using this characterization we show that the finite depth path width of an m-state NFA over a k-letter alphabet is at most \((k+1)^{m-1}\) and that this bound is tight. The nearly acyclic NFAs recognize exactly the class of constant density regular languages.
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- 1.
Note that this is not the same as the graph theory notion of tree width.
- 2.
Here and in the title of the paper by “branching” we mean an informal notion of path expansion in computations. A specific technical notion called branching is considered by Goldstine et al. [7].
- 3.
This is a conservative upper bound chosen to keep the argument simple. If A were to have m cycles, the length of the cycles naturally could not be m.
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Acknowledgments
Research supported by NSERC grant OGP0147224. Full version of the work can be found in [12].
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Keeler, C., Salomaa, K. (2017). Branching Measures and Nearly Acyclic NFAs. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal Systems. DCFS 2017. Lecture Notes in Computer Science(), vol 10316. Springer, Cham. https://doi.org/10.1007/978-3-319-60252-3_16
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DOI: https://doi.org/10.1007/978-3-319-60252-3_16
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