Automatic functions, linear time and learning

The present work determines the exact nature of {\em linear time computable} notions which characterise automatic functions (those whose graphs are recognised by a finite automaton). The paper also determines which type of linear time notions permit full learnability for learning in the limit of automatic classes (families of languages which are uniformly recognised by a finite automaton). In particular it is shown that a function is automatic iff there is a one-tape Turing machine with a left end which computes the function in linear time where the input before the computation and the output after the computation both start at the left end. It is known that learners realised as automatic update functions are restrictive for learning. In the present work it is shown that one can overcome the problem by providing work tapes additional to a resource-bounded base tape while keeping the update-time to be linear in the length of the largest datum seen so far. In this model, one additional such work tape provides additional learning power over the automatic learner model and two additional work tapes give full learning power. Furthermore, one can also consider additional queues or additional stacks in place of additional work tapes and for these devices, one queue or two stacks are sufficient for full learning power while one stack is insufficient.

[1]  Dana Angluin,et al.  Inductive Inference of Formal Languages from Positive Data , 1980, Inf. Control..

[2]  Frank Stephan,et al.  Language Learning from Texts: Mind Changes, Limited Memory and Monotonicity (Extended Abstract). , 1995, COLT 1995.

[3]  Anil Nerode,et al.  Automatic Presentations of Structures , 1994, LCC.

[4]  Sanjay Jain,et al.  ON AUTOMATIC FAMILIES , 2011 .

[5]  F. C. Hennie Crossing sequences and off-line Turing machine computations , 1965, SWCT.

[6]  F. C. Hennie,et al.  One-Tape, Off-Line Turing Machine Computations , 1965, Inf. Control..

[7]  Carl H. Smith,et al.  On the impact of forgetting on learning machines , 1995, JACM.

[8]  Manuel Blum,et al.  Toward a Mathematical Theory of Inductive Inference , 1975, Inf. Control..

[9]  Sasha Rubin,et al.  Automata Presenting Structures: A Survey of the Finite String Case , 2008, Bulletin of Symbolic Logic.

[10]  Juris Hartmanis,et al.  Computational Complexity of One-Tape Turing Machine Computations , 1968, JACM.

[11]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[12]  Sanjay Jain,et al.  Learnability of Automatic Classes , 2010, LATA.

[13]  A. Nies Computability and randomness , 2009 .

[14]  John Case,et al.  Automatic Learning of Subclasses of Pattern Languages , 2011, LATA.

[15]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[16]  Achim Blumensath,et al.  Automatic structures , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[17]  Thomas Zeugmann,et al.  Learning indexed families of recursive languages from positive data: A survey , 2008, Theor. Comput. Sci..

[18]  Cristian S. Calude Information and Randomness: An Algorithmic Perspective , 1994 .

[19]  D UllmanJeffrey,et al.  Introduction to automata theory, languages, and computation, 2nd edition , 2001 .

[20]  Leonard Pitt,et al.  Inductive Inference, DFAs, and Computational Complexity , 1989, AII.

[21]  Daniel N. Osherson,et al.  Systems That Learn: An Introduction to Learning Theory for Cognitive and Computer Scientists , 1990 .

[22]  John Case,et al.  Automatic Functions, Linear Time and Learning , 2012, CiE.

[23]  E. Mark Gold,et al.  Language Identification in the Limit , 1967, Inf. Control..

[24]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1997, Texts in Computer Science.