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Complex emergent systems, in which interactions among numerous components or “agents” produce patterns or behaviors not obtainable by individual components, are ubiquitous at every scale of the physical universe, for example in neural networks (Deamer and Evans, 2006), turbulent fluids (Frisch, 1995), insect colonies (Camazine et al., 2001), and spiral galaxies (Carlberg, 1992). Complex systems also appear in a range of artificial symbolic contexts, including genetic algorithms (Mitchell, 1996), cellular automata (Wolfram, 2002), artificial life (Adami, 1995), and models of market economies (Holland, 1995).

Life, with its novel collective behaviors at the scale of molecules, genes, cells, and organisms, is the quintessential emergent complex system. Furthermore, the ancient transition from a geochemical world to a living planet may be modeled as a sequence of emergent events, each of which increased the chemical complexity of the prebiotic world (De Duve, 1995; Morowitz, 2002; Hazen, 2005).

Given this ubiquity and diversity, it is desirable to understand the characteristics of emergent complex systems, as well as the factors that might promote complexity in evolving systems. However, complexity has proven difficult to define or measure with precision (Gell-Mann, 1995; Adami, 2003; Shalizi, 2006). A central objective of this study, therefore, is to examine “functional information” (Szostak, 2003) as a quantitative measure of complexity that may be applicable to the analysis and prediction of attributes of a wide range of phenomena in physical and symbolic systems, including evolving biological systems.

An extensive literature explores historical developments and recent advances in the study of complexity and information (Kåhre, 2002; Gell-Mann and Lloyd, 2003; Von Baeyer, 2003; Shalizi, 2006) as well as their application to understanding biological systems (Morowitz, 1978; Bell, 1997; Allen et al., 1998; Solé and Goodwin, 2000; Camazine et al., 2001; Adami, 2003; Avery, 2003; Ricard, 2003). Despite this rich literature, previous discussions of complexity have not generally focused on the relationship between information content and function (Lehman et al., 2000). We propose to measure the complexity of a system in terms of functional information, the information required to encode a specific function.

SYSTEMS AND THEIR FUNCTIONS

In this chapter we consider the functional information of both symbolic systems (letter sequences and Avida artificial life genomes) and biopolymers (RNA aptamers). These systems share several characteristics: first, they consist of numerous individual components or “agents”; second, the agents can combine in a combinatorially large number of different configurations; and third, some configurations display functions that



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