Efficient Learning of Typical Finite Automata from Random Walks

Abstract This paper describes new and efficient algorithms for learning deterministic finite automata. Our approach is primarily distinguished by two features: (1) the adoption of an average-case setting to model the “typical” labeling of a finite automaton, while retaining a worst-case model for the underlying graph of the automaton, along with (2) a learning model in which the learner is not provided with the means to experiment with the machine, but rather must learn solely by observing the automaton's output behavior on a random input sequence. The main contribution of this paper is in presenting the first efficient algorithms for learning non-trivial classes of automata in an entirely passive learning model. We adopt an on-line learning model in which the learner is asked to predict the output of the next state, given the next symbol of the random input sequence; the goal of the learner is to make as few prediction mistakes as possible. Assuming the learner has a means of resetting the target machine to a fixed start state, we first present an efficient algorithm that makes an expected polynomial number of mistakes in this model. Next, we show how this first algorithm can be used as a subroutine by a second algorithm that also makes a polynomial number of mistakes even in the absence of a reset. Along the way, we prove a number of combinatorial results for randomly labeled automata. We also show that the labeling of the states and the bits of the input sequence need not be truly random, but merely semi - random . Finally, we discuss an extension of our results to a model in which automata are used to represent distributions over binary strings.

[1]  Boris A. Trakhtenbrot,et al.  Finite automata : behavior and synthesis , 1973 .

[2]  Umesh V. Vazirani,et al.  Strong communication complexity or generating quasi-random sequences from two communicating semi-random sources , 1987, Comb..

[3]  Ronald L. Rivest,et al.  Inference of finite automata using homing sequences , 1989, STOC '89.

[4]  Ronald L. Rivest,et al.  Diversity-based inference of finite automata , 1994, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[5]  Dana Ron,et al.  On the learnability and usage of acyclic probabilistic finite automata , 1995, COLT '95.

[6]  Dana Angluin,et al.  Learning Regular Sets from Queries and Counterexamples , 1987, Inf. Comput..

[7]  Ya. M. Barzdin.  Deciphering of Sequential Networks in the Absence of an Upper Limit on the Number of States , 1970 .

[8]  Leonard Pitt,et al.  The minimum consistent DFA problem cannot be approximated within any polynomial , 1993, JACM.

[9]  Kevin J. Lang.  Random DFA's can be approximately learned from sparse uniform examples , 1992, COLT '92.

[10]  N. Littlestone.  Learning Quickly When Irrelevant Attributes Abound: A New Linear-Threshold Algorithm , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[11]  Ronitt Rubinfeld,et al.  On the learnability of discrete distributions , 1994, STOC '94.

[12]  DANA ANGLUIN,et al.  On the Complexity of Minimum Inference of Regular Sets , 1978, Inf. Control..

[13]  E. Mark Gold,et al.  Complexity of Automaton Identification from Given Data , 1978, Inf. Control..

[14]  Neri Merhav,et al.  Universal prediction of individual sequences , 1992, IEEE Trans. Inf. Theory.

[15]  Leslie G. Valiant,et al.  Computational limitations on learning from examples , 1988, JACM.

[16]  Miklos Santha,et al.  Generating Quasi-random Sequences from Semi-random Sources , 1986, J. Comput. Syst. Sci..

[17]  David Haussler,et al.  Predicting {0,1}-functions on randomly drawn points , 1988, COLT '88.

[18]  Leslie G. Valiant,et al.  Cryptographic limitations on learning Boolean formulae and finite automata , 1994, JACM.

[19]  Avrim Blum.  Some tools for approximate 3-coloring , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[20]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, STOC '84.

[21]  Yossi Azar,et al.  Biased random walks , 1992, STOC '92.

[22]  Vijay V. Vazirani,et al.  Random polynomial time is equal to slightly-random polynomial time , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[23]  Leonard Pitt,et al.  Prediction-Preserving Reducibility , 1990, J. Comput. Syst. Sci..

[24]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[25]  E. Mark Gold,et al.  System identification via state characterization , 1972 .

[26]  Ronald L. Rivest,et al.  Learning complicated concepts reliably and usefully , 1988, Annual Conference Computational Learning Theory.