Computation of moments for probabilistic finite-state automata

Abstract

[EN] The computation of moments of probabilistic finite-state automata (PFA) is researched in this article. First, the computation of moments of the length of the paths is introduced for general PFA, and then, the computation of moments of the number of times that a symbol appears in the strings generated by the PFA is described. These computations require a matrix inversion. Acyclic PFA, such as word graphs, are quite common in many practical applications. Algorithms for the efficient computation of the moments for acyclic PFA are also presented in this paper.This work has been partially supported by the Ministerio de Ciencia y Tecnologia under the grant TIN2017-91452-EXP (IBEM), by the Generalitat Valenciana under the grant PROMETE0/2019/121 (DeepPattern), and by the grant "Ayudas Fundacion BBVA a equipos de investigacion cientifica 2018" (PR[8]_HUM_C2_0087).Sánchez Peiró, JA.; Romero, V. (2020). Computation of moments for probabilistic finite-state automata. Information Sciences. 516:388-400. https://doi.org/10.1016/j.ins.2019.12.052S388400516Sakakibara, Y., Brown, M., Hughey, R., Mian, I. S., Sjölander, K., Underwood, R. C., & Haussler, D. (1994). Stochastic context-free grammers for tRNA modeling. Nucleic Acids Research, 22(23), 5112-5120. doi:10.1093/nar/22.23.5112Álvaro, F., Sánchez, J.-A., & Benedí, J.-M. (2016). An integrated grammar-based approach for mathematical expression recognition. Pattern Recognition, 51, 135-147. doi:10.1016/j.patcog.2015.09.013Mohri, M., Pereira, F., & Riley, M. (2002). Weighted finite-state transducers in speech recognition. Computer Speech & Language, 16(1), 69-88. doi:10.1006/csla.2001.0184Casacuberta, F., & Vidal, E. (2004). Machine Translation with Inferred Stochastic Finite-State Transducers. Computational Linguistics, 30(2), 205-225. doi:10.1162/089120104323093294Ortmanns, S., Ney, H., & Aubert, X. (1997). A word graph algorithm for large vocabulary continuous speech recognition. Computer Speech & Language, 11(1), 43-72. doi:10.1006/csla.1996.0022Soule, S. (1974). Entropies of probabilistic grammars. Information and Control, 25(1), 57-74. doi:10.1016/s0019-9958(74)90799-2Justesen, J., & Larsen, K. J. (1975). On probabilistic context-free grammars that achieve capacity. Information and Control, 29(3), 268-285. doi:10.1016/s0019-9958(75)90437-4Hernando, D., Crespi, V., & Cybenko, G. (2005). Efficient Computation of the Hidden Markov Model Entropy for a Given Observation Sequence. IEEE Transactions on Information Theory, 51(7), 2681-2685. doi:10.1109/tit.2005.850223Nederhof, M.-J., & Satta, G. (2008). Computation of distances for regular and context-free probabilistic languages. Theoretical Computer Science, 395(2-3), 235-254. doi:10.1016/j.tcs.2008.01.010CORTES, C., MOHRI, M., RASTOGI, A., & RILEY, M. (2008). ON THE COMPUTATION OF THE RELATIVE ENTROPY OF PROBABILISTIC AUTOMATA. International Journal of Foundations of Computer Science, 19(01), 219-242. doi:10.1142/s0129054108005644Ilic, V. M., Stankovi, M. S., & Todorovic, B. T. (2011). Entropy Message Passing. IEEE Transactions on Information Theory, 57(1), 375-380. doi:10.1109/tit.2010.2090235Booth, T. L., & Thompson, R. A. (1973). Applying Probability Measures to Abstract Languages. IEEE Transactions on Computers, C-22(5), 442-450. doi:10.1109/t-c.1973.223746Thompson, R. A. (1974). Determination of Probabilistic Grammars for Functionally Specified Probability-Measure Languages. IEEE Transactions on Computers, C-23(6), 603-614. doi:10.1109/t-c.1974.224001Wetherell, C. S. (1980). Probabilistic Languages: A Review and Some Open Questions. ACM Computing Surveys, 12(4), 361-379. doi:10.1145/356827.356829Sanchez, J.-A., & Benedi, J.-M. (1997). Consistency of stochastic context-free grammars from probabilistic estimation based on growth transformations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(9), 1052-1055. doi:10.1109/34.615455Hutchins, S. E. (1972). Moments of string and derivation lengths of stochastic context-free grammars. Information Sciences, 4(2), 179-191. doi:10.1016/0020-0255(72)90011-4Heim, A., Sidorenko, V., & Sorger, U. (2008). Computation of distributions and their moments in the trellis. Advances in Mathematics of Communications, 2(4), 373-391. doi:10.3934/amc.2008.2.373Vidal, E., Thollard, F., de la Higuera, C., Casacuberta, F., & Carrasco, R. C. (2005). Probabilistic finite-state machines - part I. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(7), 1013-1025. doi:10.1109/tpami.2005.147Sánchez, J. A., Rocha, M. A., Romero, V., & Villegas, M. (2018). On the Derivational Entropy of Left-to-Right Probabilistic Finite-State Automata and Hidden Markov Models. Computational Linguistics, 44(1), 17-37. doi:10.1162/coli_a_0030