Finite-State Complexity and the Size of Transducers

Finite-state complexity is a variant of algorithmic information theory obtained by replacing Turing machines with finite transducers. We consider the state-size of transducers needed for minimal descriptions of arbitrary strings and, as our main result, we show that the state-size hierarchy with respect to a standard encoding is infinite. We consider also hierarchies yielded by more general computable encodings.

[1]  L. Goddard.  Information Theory , 1962, Nature.

[2]  N. V. Vinodchandran,et al.  Entropy rates and finite-state dimension , 2005, Theor. Comput. Sci..

[3]  Jeffrey Shallit.  What this country needs is an 18c piece , 2003 .

[4]  Jeffrey Shallit,et al.  Automaticity: Properties of a Measure of Descriptional Complexity , 1994, STACS.

[5]  Jack H. Lutz,et al.  Finite-state dimension and real arithmetic , 2007, Inf. Comput..

[6]  Shou-Feng Wang,et al.  𝒫𝒮-regular languages , 2011, Int. J. Comput. Math..

[7]  Abhi Shelat,et al.  Approximating the smallest grammar: Kolmogorov complexity in natural models , 2002, STOC '02.

[8]  R. Guy.  Unsolved Problems in Number Theory , 1981 .

[9]  Abhi Shelat,et al.  Approximation algorithms for grammar-based compression , 2002, SODA '02.

[10]  Lance Fortnow,et al.  Resource-Bounded Kolmogorov Complexity Revisited , 1997, STACS.

[11]  Jeffrey Shallit,et al.  Automaticity I: Properties of a Measure of Descriptional Complexity , 1996, J. Comput. Syst. Sci..

[12]  Wojciech Rytter,et al.  Grammar Compression, LZ-Encodings, and String Algorithms with Implicit Input , 2004, ICALP.

[13]  Jeffrey Shallit,et al.  Automatic Complexity of Strings , 2001, J. Autom. Lang. Comb..

[14]  Jean Berstel,et al.  Transductions and context-free languages , 1979, Teubner Studienbücher : Informatik.

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

[16]  Jeffrey Shallit.  The computational complexity of the local postage stamp problem , 2002, SIGA.

[17]  Cristian S. Calude,et al.  Finite-State Complexity and Randomness , 2009 .

[18]  Gregory J. Chaitin,et al.  Algorithmic Information Theory , 1987, IBM J. Res. Dev..