A Divergence Formula for Randomness and Dimension (Short Version)

@inproceedings{Lutz2009ADF,
  title={A Divergence Formula for Randomness and Dimension (Short Version)},
  author={Jack H. Lutz},
  booktitle={CSP},
  year={2009}
}
  • J. H. Lutz
  • Published in CSP 23 June 2009
  • Computer Science, Mathematics
If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $\beta$ is the uniform probability measure. This paper shows that $\dim^\beta(S)$ and its dual $\Dim^\beta(S)$, the {\it strong dimension} of $S$ with respect to $\beta$, can be… 
1 Citations

A divergence formula for randomness and dimension

  • J. H. Lutz
  • Computer Science, Mathematics
    Theor. Comput. Sci.
  • 2008

References

SHOWING 1-10 OF 22 REFERENCES

A divergence formula for randomness and dimension

  • J. H. Lutz
  • Computer Science, Mathematics
    Theor. Comput. Sci.
  • 2008

The dimensions of individual strings and sequences

  • J. H. Lutz
  • Mathematics, Computer Science
    Inf. Comput.
  • 2003

Hausdorff dimension in probability theory II

1. Summary Let (ft, (B, u) be a probability measure space on which is defined a stochastic process {x,} with finite state space. In 1-3 we define a notion of fractional dimension, in terms of and

Dimension in complexity classes

  • J. H. Lutz
  • Computer Science, Physics
    Proceedings 15th Annual IEEE Conference on Computational Complexity
  • 2000
A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martin-gales. When the resource bound /spl Delta/(a parameter of the theory) is unrestricted, the

A Kolmogorov complexity characterization of constructive Hausdorff dimension

The Definition of Random Sequences

Finite-state dimension

The main theorem shows that the finite-state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information-lossless finite- state compressors.

A Survey of the Theory of Random Sequences

By introducing the concept of a universal algorithm Kolmogorov gave a rigorous definition of program complexity which no more depends on the choice of a special machine (or algorithm).

Elements of Information Theory

The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.

A unified approach to the definition of random sequences

  • C. Schnorr
  • Computer Science
    Mathematical systems theory
  • 2005
Using the concept of test functions, a general framework within which many recent approaches to the definition of random sequences can be described is developed, and a thesis on the “true” concept of randomness is formulated.