Investigating Emergence by Coarse Graining Elementary Cellular Automata Andrew Weeks 1 , Fiona Polack 1 and Susan Stepney 1 1 Department of Computer Science, University of York, UK, YO10 5DD Abstract We extend coarse graining of cellular automata to investi- gate aspects of emergence. From the total coarse grain- ing approach introduced by Israeli and Goldenfeld, Coarse- graining of cellular automata, emergence, and the pre- dictability of complex systems, Phys. Rev. E, 2006, we devise partial coarse graining, and show qualitative differences in the results of total and partial coarse graining. Mutual infor- mation is used to show objectively how coarse grainings are related to the identification of emergent structure. We show that some valid coarse grainings have high mutual informa- tion, and are thus good at identifying and predicting emer- gent structures. We also show that the mapping from lower to emergent levels crucially affects the quality emergence. Introduction We are interested in observing and modelling complex emer- gent systems, with the goal of understanding how we could begin to specify and implement engineered emergent sys- tems. Emergence is variously characterised; we start from Ronald et al’s definition of emergence: “The language of design L1 and the language of observation L2 are distinct, and the causal link between the elementary interactions pro- grammed in L1 and the behaviors observed in L2 is non- obvious to the observer...” (Ronald et al., 1999). Here, we refer to the local level, of the implementation substrate, as L. The language of observation represents a global, or coarse- grain, level where emergent behaviour is observable that we refer to as the specification, S. After Shalizi (2001), we de- fine emergence in information-theoretic terms, as the greater predictive efficiency of descriptions in S over those in L. In natural complex systems, it is hard to define languages L and S, and to determine accurate mappings between them. Here, the complex emergent systems are elementary cellular automata (ECAs); their language is simple and well-defined, and thus mappings can be identified and analysed. One perception of an emergent system is that its high level behaviour is independent of the low level behaviour. How- ever, the emergent properties are actually a carefully chosen extract of the low-level behaviour. The observational dis- continuity allows us to identify emergent behaviour. Else- where (Weeks et al., 2007), we show that coarse graining is R M Figure 1: Rules and mappings. Shaded cells represent value 1. R shows how the “name” of an ECA rule is derived: each of the 8 possible ECA initial state is shown in bigen- dian order; below it, the next state of the central cell is shown; the rule is “named” by reading off the values of the next state. Here, R is the transition table for ECA rule 150 (10010110 2 ). M represents the coarse grain mapping 0110, with grain g =2. a simple form of emergence. If we can state coarse-grained rules, then we can use the coarse level to predict behaviour. Because information is lost in the higher level we cannot predict behaviour correctly in all cases, but the rules should be able to predict some common futures. Here, we explore emergence through coarse graining ECAs and measurement of mutual information between levels. Coarse Graining ECAs An ECA is a one-dimensional cellular automaton, with two states and a neighbourhood of three. There are 256 ECA rules, of which 88 are distinct (not just spatial reflections or 0-1 inversions). Rule sets are named by taking the decimal representation of the binary string that represents the outputs of the transition rules from all neighbourhood states taken in bigendian order (figure 1). The coarse graining of ECAs was investigated by Israeli and Goldenfeld (2006). In a coarse graining at grain g, the values of a contiguous block of g cells at the fine level are projected, or mapped, to the value of a single cell at a coarse level (figure 1). Israeli and Goldenfeld (2006) require their coarse grain- ings to be total, that is, to satisfy the commutativity con- dition that running the fine ECA for n × g time-steps then Artificial Life XI 2008 686