## Abstract

Generalized strings describe various biological motifs that arise in molecular and computational biology. In this manuscript, we introduce an alternative but efficient algorithm to construct the minimal deterministic finite automaton (DFA) associated with any generalized string. We exploit this construction to characterize the typical growth of the minimal DFA (i.e., with the least number of states) associated with a random generalized string of increasing length. Even though the worst-case growth may be exponential, we characterize a point in the construction of the minimal DFA when it starts to grow linearly and conclude it has at most a polynomial number of states with asymptotically certain probability. We conjecture that this number is linear.

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## Acknowledgements

We are thankful to two anonymous referees for their careful reading of this paper and valuable suggestions. We are also very thankful to Dr. Dougherty for partially funding this research through her NSF EXTREEMS training grant.

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This work has been partially funded by the NSF EXTREEMS Grant No. DMS 1407340, and the NSF Graduate Research Fellowship Program under Grant No. DGE 1252522. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

## Appendix

### Appendix

Here we show that our random generalized string model, *G* = *G*[1],…,*G*[*n*] with *G*[1],*G*[2],*G*[3]… i.i.d. uniform non-empty subsets of {0, 1}, does not fit in the *low correlation* framework in AitMous et al. (2012).

Following the notation of AitMous et al. (2012), let \(\mathbb {P}_{N}\) with *N* = *n*2^{n} (the largest possible “size” of *G*) denote the probability mass function of *G*. Condition (1) in Definition 1 is then satisfied with *C* = 1.

Next, sort words in *G* lexicographically so that *u*_{1} is its smallest word, *u*_{2} is the second smallest (when *G* contains at least two words), and so on. Condition (2) in the Definition requires that \(\mathbb {P}_{N}(u_{1}[1,\ell ]=u_{2}[1,\ell ])=O(\beta ^{-\ell })\) for some *β* > 1. Since the event *G*[1] = ⋯ = *G*[*ℓ*] = {0} and *G*[*ℓ* + 1] = {0, 1} is contained in the event *u*_{1}[1,*ℓ*] = *u*_{2}[1,*ℓ*], and the probability of the former is 3^{−(ℓ+ 1)}, we must have *β* ≤ 3. On the other hand, condition (3) in the Definition requires that \(n\ge \frac {8\log (n2^{n})}{\log \beta }\) asymptotically—which is possible only if *β* ≥ 256. Conditions (2) and (3) in Definition 1 of AitMous et al. (2012) are therefore incompatible under our i.i.d. model.

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Char, I.G., Lladser, M.E. Stochastic Analysis of Minimal Automata Growth for Generalized Strings.
*Methodol Comput Appl Probab* **22, **329–347 (2020). https://doi.org/10.1007/s11009-019-09706-8

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### Keywords

- Aho-Corasick algorithm
- Deterministic finite automaton
- Generalized string
- Minimization
- Motif
- Polynomial growth

### Mathematics Subject Classification (2010)

- 68Q25
- 68Q45
- 68Q87
- 68W40