Abstract
In this paper we define and study finite state complexity of finite strings and infinite sequences and study connections of these complexity notions to randomness and normality. We show that the finite state complexity does not only depend on the codes for finite transducers, but also on how the codes are mapped to transducers. As a consequence we relate the finite state complexity to the plain (Kolmogorov) complexity, to the process complexity and to prefix-free complexity. Working with prefix-free sets of codes we characterise Martin-Löf random sequences in terms of finite state complexity: the weak power of finite transducers is compensated by the high complexity of enumeration of finite transducers. We also prove that every finite state incompressible sequence is normal, but the converse implication is not true. These results also show that our definition of finite state incompressibility is stronger than all other known forms of finite automata based incompressibility, in particular the notion related to finite automaton based betting systems introduced by Schnorr and Stimm [28]. The paper concludes with a discussion of open questions.
This work was done in part during C. S. Calude’s visits to the Martin-Luther-Universität Halle-Wittenberg in October 2012 and the National University of Singapore in November 2013, and L. Staiger’s visits to the CDMTCS, University of Auckland and the National University of Singapore in March 2013. The work was supported in part by NUS grant R146-000-181-112 (PI F. Stephan).
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Calude, C.S., Staiger, L., Stephan, F. (2014). Finite State Incompressible Infinite Sequences. In: Gopal, T.V., Agrawal, M., Li, A., Cooper, S.B. (eds) Theory and Applications of Models of Computation. TAMC 2014. Lecture Notes in Computer Science, vol 8402. Springer, Cham. https://doi.org/10.1007/978-3-319-06089-7_5
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