Minimisation of Multiplicity Tree Automata

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.

[1]  Andreas Maletti.  Minimizing deterministic weighted tree automata , 2009, Inf. Comput..

[2]  Björn Borchardt,et al.  A Pumping Lemma and Decidability Problems for Recognizable Tree Series , 2004, Acta Cybern..

[3]  James Worrell,et al.  Complexity of equivalence and learning for multiplicity tree automata , 2014, J. Mach. Learn. Res..

[4]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[5]  Symeon Bozapalidis.  Effective construction of the syntactic algebra of a recognizable series on trees , 2005, Acta Informatica.

[6]  Symeon Bozapalidis,et al.  The Rank of a Formal Tree Power Series , 1983, Theor. Comput. Sci..

[7]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[8]  L. Csanky,et al.  Fast Parallel Matrix Inversion Algorithms , 1976, SIAM J. Comput..

[9]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[10]  Oscar H. Ibarra,et al.  A Note on the Parallel Complexity of Computing the Rank of Order n Matrices , 1980, Inf. Process. Lett..

[11]  José Oncina,et al.  Learning Multiplicity Tree Automata , 2006, ICGI.

[12]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[13]  Stefan Kiefer,et al.  Stability and Complexity of Minimising Probabilistic Automata , 2014, ICALP.

[14]  Jean Berstel,et al.  Recognizable Formal Power Series on Trees , 1982, Theor. Comput. Sci..

[15]  Azaria Paz,et al.  Introduction to Probabilistic Automata , 1971 .

[16]  Wen-Guey Tzeng,et al.  A Polynomial-Time Algorithm for the Equivalence of Probabilistic Automata , 1992, SIAM J. Comput..

[17]  Jarkko Kari,et al.  Digital Image Compression , 2009 .

[18]  Walter S. Brainerd,et al.  The Minimalization of Tree Automata , 1968, Inf. Control..

[19]  Fernando Pereira,et al.  Weighted Automata in Text and Speech Processing , 2005, ArXiv.

[20]  Richard J. Lipton,et al.  A Probabilistic Remark on Algebraic Program Testing , 1978, Inf. Process. Lett..

[21]  Mikel L. Forcada,et al.  An Implementation of Deterministic Tree Automata Minimization , 2007, CIAA.

[22]  Joël Ouaknine,et al.  On the Complexity of Equivalence and Minimisation for Q-weighted Automata , 2013, Log. Methods Comput. Sci..

[23]  Mehryar Mohri,et al.  On the Computation of Some Standard Distances Between Probabilistic Automata , 2006, CIAA.

[24]  Tao Jiang,et al.  Minimal NFA Problems are Hard , 1991, SIAM J. Comput..

[25]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..