# On the Complexity of Finite Sequences

@article{Lempel1976OnTC, title={On the Complexity of Finite Sequences}, author={Abraham Lempel and Jacob Ziv}, journal={IEEE Trans. Inf. Theory}, year={1976}, volume={22}, pages={75-81} }

A new approach to the problem of evaluating the complexity ("randomness") of finite sequences is presented. The proposed complexity measure is related to the number of steps in a self-delimiting production process by which a given sequence is presumed to be generated. It is further related to the number of distinct substrings and the rate of their occurrence along the sequence. The derived properties of the proposed measure are discussed and motivated in conjunction with other well-established…

## 2,238 Citations

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- 1993

A possible definition of complexity of finite pattern is proposed, under which the complexity of random strings is very small, while the aperiodic string with strong random correlation between symbols has maximum complexity.

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A function, B(x) is introduced which assigns a real number to a string, x, which is intended to be a measure of the randomness of x, the Kolmogorov complexity of x.

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This paper will focus on complexity measures that are not only of theoretical interest, but that can also be obtained in an algorithmic manner, such as the linear complexity, thelinear complexity profile, and the tree complexity.

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The results show that the test is sensitive to a variety of alternatives to randomness and suggest that the proposed test statistic is a reasonable measure of the stochastic complexity of a finite sequence of discrete random variables.

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DNA sequences are the subject of special attention, due to the importance of the object, manifestations of repetition of different types, and numerous examples of using the concept of complexity in solving problems of classification and evolution of various biological objects.

Complexity of strings in the class of Markov sources

- Computer ScienceIEEE Trans. Inf. Theory
- 1986

Shannon's self-information of a string is generalized to its complexity relative to the class of finite-state-machine (FSM) defined sources by a theorem stating that, asymptotically, the mean complexity provides a tight lower bound for the mean length of all so-called regular codes.

## References

SHOWING 1-5 OF 5 REFERENCES

Information-Theoretic Limitations of Formal Systems

- Computer ScienceJACM
- 1974

An attempt is made to apply information-theoretic computational complexity to meta-mathematics by measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theoremic proofs.