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On the Generation of Aperiodic and Periodic Necklaces via T-augmentation

Gulliver T.A., Makwakwa I., Speidel U.

Fundamenta Informaticae

2008

Vol. 83, nr 1-2

91-107

In a recent report, Gulliver and Speidel showed that complete sets of aperiodic and - in some cases periodic - necklaces can be generated for arbitrary lengths as fixed-length subsets of variable-length T-codes. The T-codes to which they applied this observation were specifically T-codes constructed by systematic T-augmentation, that is by T-augmentation sequences in which each T-expansion parameter is 1 (simple T-augmentation) and where the T-codes A(p1,p2,1/4,pi) at each T-augmentation level i do not contain any codewords shorter than pi (strictly minimal T-augmentation). This present paper generalizes their result to arbitrary T-codes and formalizes their earlier result as a special case of the general result.

Deterministic Complexity and Entropy

Titchener M. R., Gulliver A., Nicolescu R., Speidel U., Staiger L.

Fundamenta Informaticae

2005

Vol. 64, nr 1-4

443-461

Lempel and Ziv (1976) proposed a computable string production-complexity. In this paper, our emphasis is on providing the rigorous development, where possible, for the theoretical aspects of a more recent and contrasting measure of string complexity. We derive expressions for complexity bounds subject to certain constraints. We derive an analytic approximation to the upper bound to linearize the complexity measure. The linearized measure enables us to propose an entropy measure, observed elsewhere to correspond closely with the Kolmogorov-Sinai entropy in simple dynamical systems.