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In this paper, we study the problem of determining a minimum stateprobabilistic finite state machine capable of generating statisticallyidentical symbol sequences to samples provided. This problem is qualitativelysimilar to the classical Hidden Markov Model problem and has been studied froma practical point of view in several works beginning with the work presentedin: Shalizi, C.R., Shalizi, K.L., Crutchfield, J.P. (2002) \textit{An algorithmfor pattern discovery in time series.} Technical Report 02-10-060, Santa FeInstitute. arxiv.org/abs/cs.LG/0210025. We show that the underlying problem is$\mathrm{NP}$-hard and thus all existing polynomial time algorithms must beapproximations on finite data sets. Using our $\mathrm{NP}$-hardness proof, weshow how to construct a provably correct algorithm for constructing a minimumstate probabilistic finite state machine given data and empirically study itsrunning time.

Minimum Probabilistic Finite State Learning Problem on Finite Data Sets: Complexity, Solution and Approximations