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In the quest for a mathematical measure able to capture and shed light on the dual notions of information and complexity in biosequences, Hazen et al. have introduced the notion of Functional Information (FI for short). It is also the result of earlier considerations and findings by Szostak and Carothers et al. Based on the experiments by Charoters et al., regarding FI in RNA binding activities, we decided to study the relation existing between FI and classic measures of complexity applied on protein-DNA interactions on a genome-wide scale. Using classic complexity measures, i.e, Shannon entropy and Kolmogorov Complexity as both estimated by data compression, we found that FI applied to protein-DNA interactions is genuinely different from them. Such a fact, together with the non-triviality of the biological function considered, contributes to the establishment of FI as a novel and useful measure of biocomplexity. Remarkably, we also found a relationship, on a genome-wide scale, between the redundancy of a genomic region and its ability to interact with a protein. This latter finding justifies even more some principles for the design of motif discovery algorithms. Finally, our experiments bring to light methodological limitations of Linguistic Complexity measures, i.e., a class of measures that is a function of the vocabulary richness of a sequence. Indeed, due to the technology and associated statistical preprocessing procedures used to conduct our studies, i.e., genome-wide ChIP-chip experiments, that class of measures cannot give any statistically significant indication about complexity and function. A serious limitation due to the widespread use of the technology. References J.M. Carothers, S.C. Oestreich, J.H. Davis, and J.W. Szostack. Informational complexity and functional activity of RNA structures. J. AM. CHEM. SOC., 126 (2004), pp. 5130-5137. R.M. Hazen, P.L. Griffin, J.M. Carothers, and J.W. Szostak. Functional Information and the emergence of biocomplexity. Proc. of Nat. Acad. Sci, 104 (2007), pp. 8574-8581. J.W. Szostak. Functional Information: molecular messages, Nature, 423 (2003).