Title Kolmogorov Complexity Estimation and Analysis

Methods for discerning and measuring Kolmogorov Complexity are discussed and their relationships explored. A computationally efficient method of using Lempel Ziv 78 Universal compression algorithm to estimate complexity is introduced. 1 Abstract—Methods for discerning and measuring Kolmogorov Complexity are discussed and their relationships explored. A computationally efficient method of using Lempel Ziv 78 Universal compression algorithm to estimate complexity is introduced. I. INTRODUCTION olmogorov Complexity is a fundamental measure of information with growing applications and importance [2], [4]. Estimation of Kolmogorov Complexity is key to objective information system monitoring and analysis. References [7], [2]-[4] contain many applications of Kolmogorov Complexity; also see [1] for background on this subject. All applications of Kolmogorov Complexity are limited due to its incomputable nature and are impacted by improvements or innovations in the ability to estimate Kolmogorov Complexity well. In this paper we review a generic method for estimating complexity – the Lempel-Ziv 78 (LZ78) [11] universal compression algorithm, discuss its limitations in estimating complexity, and derive a computationally efficient method of using this algorithm to estimate complexity. We then develop two measures for the estimation of complexity – power spectral density based estimation and expected time of sequence production. We discuss the relationships between these methods of estimation and other estimators. Additionally we introduce a third parameter related to complexity – SPAN. Relationships between these measures of complexity are compared and discussed, and their relationships to compression-based estimates of Kolmogorov Complexity are explored.