Self-referencing cellular automata: A model of the evolution of information control in biological systems

Cellular automata have been useful artificial models for exploring how relatively simple rules combined with spatial memory can give rise to complex emergent patterns. Moreover, studying the dynamics of how rules emerge under artificial selection for function has recently become a powerful tool for understanding how evolution can innovate within its genetic rule space. However, conventional cellular automata lack the kind of state feedback that is surely present in natural evolving systems. Each new generation of a population leaves an indelible mark on its environment and thus affects the selective pressures that shape future generations of that population. To model this phenomenon, we have augmented traditional cellular automata with state-dependent feedback. Rather than generating automata executions from an initial condition and a static rule, we introduce mappings which generate iteration rules from the cellular automaton itself. We show that these new automata contain disconnected regions which locally act like conventional automata, thus encapsulating multiple functions into one structure. Consequently, we have provided a new model for processes like cell differentiation. Finally, by studying the size of these regions, we provide additional evidence that the dynamics of self-reference may be critical to understanding the evolution of natural language. In particular, the rules of elementary cellular automata appear to be distributed in the same way as words in the corpus of a natural language. Introduction Cellular automata (CA) are model complex systems that combine spatial memory with relatively simple update rules to produce rich dynamic patterns (Wolfram, 2002). In this regard, they can be viewed as models for life. “Rules” in DNA encode policies that iteratively react to the environment by modifying it. Thus, over evolutionary time scales, natural selection can explore the nucleic-acid rule space and amplify those rules which provide useful functions. With this narrative in mind, researchers have used in silico artificial selection to explore the CA rule space and amplify rules with certain computational abilities (Breukelaar and Bäck, 2005; Das et al., 1994; Hordijk, 2013; Mitchell et al., 1994). Moreover, there has been much interest in understanding the demographic dynamics of these CA rule populations over their evolutionary history (Hordijk, 2013; Mitchell et al., 1994). That is, artificial selection of these evolving CA’s has itself become a model for the innovation intrinsic to natural selection. One major difference between evolving cellular automata (EvCA) and evolving natural organisms is the lack of feedback in the fitness channel of the former. In EvCA, each new generation faces the same selective pressures as prior generations. However, with new generations of natural organisms, there is feedback between the current demographics and the selective pressures shaping future demographics. Goldenfeld and Woese (2011) point out that these self-referential dynamics are a unique characteristic of life – making life distinctly different from any other physical system. Two of us (SIW and PCWD) have proposed that self-referential dynamics are one of the hallmarks of life, emerging with its origin (Walker and Davies, 2013). Where EvCA’s will only innovate in the presence of external “abiotic” pressures, natural organisms put pressure on themselves to re-organize even without an external fitness driver. Similarly, at everyday and ontogenetic time scales, conventional CA’s do not easily embed the regulatory mechanisms that permeate throughout life. A single genome gives rise to a wide variety of differentiated cell phenotypes which, locally, appear to follow a consistent set of operational rules but globally appear to have no shared program. By modeling feedback explicitly, these phenomena can be explained using gene regulatory network (GRN) frameworks (Davidson, 2010; Schlitt and Brazma, 2007), where the expression level of one gene promotes or inhibits the expression level of another and thus “latches” cells into different types. Thus, by making feedback between state and dynamics explicit in CA’s, it may be possible to enrich their ability to model how organisms emerge, evolve, develop, and react to both their external environment and their internal state. Here, we introduce self-referencing cellular-automaton framework we call PICARD where PICARD Implements CA Rules Differently (PICARD). Like a traditional one-dimensional CA, PICARD executions move from one iteration to another by some rule. However, whereas traditional CA’s require the rule to be static and externally specified, PICARD infers the iteration rule from the current state of the CA itself. As we will show, executions from multiple static CA’s can be embedded within a single PICARD – a PICARD can be identical to one static CA from certain initial conditions and another static CA from other initial conditions. Thus, a PICARD can combine the computational abilities of different CA’s within one entity. Moreover, whereas CA’s differ by their rule, PICARD’s differ by their state-torule mapping. Because there are many more state-to-rule mappings than there are CA rules, the PICARD parameter space is potentially much richer for later evolutionary investigations. To some extent, PICARD is a simple attempt to add dynamical feedback that is missing in traditional evolutionary cellular automata. However, because iterations are generated by rules that are encoded in previous iterations, PICARD feedback is the kind of self-reference that is thought to be a characteristic feature of life (Goldenfeld and Woese, 2011; Hofstadter, 1979; Kataoka and Kaneko, 2000a,b; Walker and Davies, 2013). Our CA approach shares many similarities with self-referencing functional-dynamics developed by Kataoka and Kaneko to model the evolution of rules (Kataoka and Kaneko, 2000a,b). Attempting to avoid the biochemical complexities of the evolution of nucleic acids, they turn their focus on the evolution of natural language. Furthermore, they draw connections between attractors in their coupled-logistic-map landscape and words that accumulate in language. Although their framework is very different than the automata we study here, we too have results that appear to be strongly connected to the evolution of natural language. Thus, augmenting cellular automata with self-reference widens our ability to model to evolution of language in unanticipated ways. Cellular Automata with PICARD Mappings Although PICARD implements CA rules differently, once each rule is defined, a PICARD iteration is identical to an iteration of a conventional elementary CA. As shown in Fig. 1(a), a traditional elementary CA generates each row based on the pattern in the preceding row. The iteration rule is a lookup table that maps each triplet of bits in the preceding row to a single bit in the following row. Thus, with the right initial conditions, some rules can produce intricate patterns over many generations, as shown in Fig. 1(b). Where PICARD differs from a traditional CA is that no static rule is specified. Instead, a map is provided from each row to the rule that will operate on it. We are primarily interested in CA’s with more than eight cells; consequently, this mapping is a coarse graining of the system – some rules will necessarily correspond to multiple different row patterns. With this multiple realizability of rules in mind, each PICARD row and rule can be viewed as a microstate– macrostate pair. Consequently, in the following, we will use b