Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T08:21:50.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilistic models for pattern statistics

Published online by Cambridge University Press:  20 July 2006

Massimiliano Goldwurm  and
Roberto Radicioni
Massimiliano Goldwurm
Affiliation:
Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, via Comelico 39/41, 20135 Milano, Italy; [email protected] & [email protected].
Roberto Radicioni
Affiliation:
Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, via Comelico 39/41, 20135 Milano, Italy; [email protected] & [email protected].
Get access

Abstract

In this work we study some probabilistic models for the random generation of words over a given alphabet used in the literature in connection with pattern statistics. Our goal is to compare models based on Markovian processes (where the occurrence of a symbol in a given position only depends on a finite number of previous occurrences) and the stochastic models that can generate a word of given length from a regular language under uniform distribution. We present some results that show the differences between these two stochastic models and their relationship with the rational probabilistic measures.

Keywords

Pattern statisticsMarkov chainsprobabilistic automatarational formal series.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 2: Alberto Bertoni: Climbing summits , April 2006 , pp. 207 - 225
Copyright
© EDP Sciences, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bernstein, I.N., Modules over a ring of differential operators, study of the fundamental solutions of equations with constant coefficients. Functional Anal. Appl. 5 (1971) 1–16 (Russian); pages 89–101 (English). CrossRef
J. Berstel and C. Reutenauer, Rational Series and their Languages. Springer-Verlag, New York, Heidelberg, Berlin (1988).
Bertoni, A., Choffrut, C., Goldwurm, M. and Lonati, V., On the number of occurrences of a symbol in words of regular languages. Theoret. Comput. Sci. 302 (2003) 431–456. CrossRef
Bertoni, A., Choffrut, C., Goldwurm, M. and Lonati, V., Local limit properties for pattern statistics and rational models. Theory. Comput. Syst. 39 (2006) 209–235. CrossRef
J. Bourdon and B. Vallée, Generalized pattern matching statistics. Mathematics and computer science II: algorithms, trees, combinatorics and probabilities, in Proc. of Versailles Colloquium. Birkhauser (2002) 249–265
Denise, A., Génération aléatoire et uniforme de mots de langages rationnels. Theoret. Comput. Sci. 159 (1996) 43–63. CrossRef
Flajolet, P., Zimmerman, P. and Van Cutsem, B., A calculus for the random generation of labelled combinatorial structures. Theoret. Comput. Sci. 132 (1994) 1–35. CrossRef
Goldwurm, M. and Lonati, V., Pattern occurrences in multicomponent models, in Proc. 22nd STACS. Lect. Notes Comput. Sci. 3404 (2005) 680–692. CrossRef
Hansel, G. and Perrin, D., Rational probability measures. Theoret. Comput. Sci. 65 (1989) 171–188 (french version in Mots, edited by M. Lothaire, Hermes (1990) 335–357). CrossRef
M. Iosifescu, Finite Markov Processes and Their Applications. J. Wiley and Son (1980).
P. Jacket and W. Szpankowski, Analytic approach to pattern matching, Chap. 7 in Applied Combinatorics on Words. Cambridge University Press (2005).
J.G. Kemeny and J.L. Snell, Finite Markov Chains. Van Nostrand (1960).
Lipshitz, L., D-finite power series. J. Algebra 122 (1989) 353–373. CrossRef
Nicodème, P., Salvy, B. and Flajolet, P., Motif statistics. Theoret. Comput. Sci. 287 (2002) 593–617. CrossRef
A. Paz, Introduction to Probabilistic Automata. Academic Press (1971).
Régnier, M. and Szpankowski, W., On pattern frequency occurrences in a Markovian sequence. Algorithmica 22 (1998) 621–649. CrossRef
A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag (1978).
E. Seneta, Non-negative Matrices and Markov Chains. Springer-Verlag (1981).
Stanley, R.P., Differentiably finite power series. Eur. J. Combin. 1 (1980) 175–188. CrossRef