Journal of Applied Probability

Pattern Markov chains: optimal Markov chain embedding through deterministic finite automata

Grégory Nuel

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In the framework of patterns in random texts, the Markov chain embedding techniques consist of turning the occurrences of a pattern over an order-m Markov sequence into those of a subset of states into an order-1 Markov chain. In this paper we use the theory of language and automata to provide space-optimal Markov chain embedding using the new notion of pattern Markov chains (PMCs), and we give explicit constructive algorithms to build the PMC associated to any given pattern problem. The interest of PMCs is then illustrated through the exact computation of P-values whose complexity is discussed and compared to other classical asymptotic approximations. Finally, we consider two illustrative examples of highly degenerated pattern problems (structured motifs and PROSITE signatures), which further illustrate the usefulness of our approach.

Article information

J. Appl. Probab. Volume 45, Number 1 (2008), 226-243.

First available in Project Euclid: 16 April 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C40: Computational Markov chains

Language regular expression exact distribution structured motif PROSITE signature


Nuel, Grégory. Pattern Markov chains: optimal Markov chain embedding through deterministic finite automata. J. Appl. Probab. 45 (2008), no. 1, 226--243. doi:10.1239/jap/1208358964.

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