The Effects of Complex Interaction Rules Between Two Interacting Cellular Automata
TThe Effects of Complex Interaction Rules Between Two Interacting Cellular Automata
Alyssa M Adams
Department of Bacteriology & Computation and Informatics in Biology and Medicine Program,University of Wisconsin-Madison, Madison, WI USAAlgorithmic Nature Group,LABORES for the Natural and Digital SciencesThe Access Network, USA (Dated: August 14, 2020)Biological systems are notorious for their complex behavior within short timescales (e.g. metabolicactivity) and longer time scales (e.g. evolutionary selection), along with their complex spatial orga-nization. Because of their complexity and their ability to innovate with respect to their environment,living systems are considered to be open-ended. Historically, it has been difficult to model open-ended evolution and innovation. As a result, our understanding of the exact mechanisms thatdistinguish open-ended living systems from non-living ones is limited. One of the biggest barriersis understanding how multiple, complex parts within a single system interact and contribute to thecomplex, emergent behavior of the system as a whole. In biology, this is essential for understandingsystems such as the human gut, which contain multiple microbial communities that contribute tothe overall health of a person. How do interactions between parts of a system lead to more complexbehavior of the system as a whole? What types of interactions contribute to open-ended behavior?In this talk, two interacting cellular automata (CA) are used as an abstract model to address theeffects of complex interactions between two individual entities embedded within a larger system.Unlike elementary CA, these CA are state-dependent because they change their update rules as afunction of the systems state as a whole. The resulting behavior of the two-CA system suggeststhat complex interaction rules between the two CA have little to no effect on the complexity of eachcomponent CA. However, having an interaction rule that is random results in open-ended evolu-tion regardless of the specific type of state-dependency. This suggests that randomness does indeedcontribute to open-ended evolution, but not by random perturbations of the states as previouslyspeculated.
Biological systems are notorious for exhibiting complexbehavior within short timescales and within longer evolu-tionary time scales, as well as being complex in spatial or-ganization. One of the biggest barriers to understandingthe exact mechanisms of biological behavior is the lack ofunderstanding how multiple parts within a system inter-act. For example, the human gut microbiome consists ofseveral bacterial and viral communities. Viruses of bacte-ria (phages) are increasingly being recognized as impor-tant components of the human microbiome. Phages mod-ulate microbial communities in the human microbiome bykilling bacteria and driving metabolic activity. However,little is known about the specific roles played by phagesin human systems, particularly how they drive behaviorof bacterial communities.In addition to being complex across time and spatialorganization, biology also evolves open-endedly. How-ever, little is known about the mechanistic relationshipbetween complexity and open-endedness, even within acomputational model. Biological systems have naturalpartitions between subsystems, such as individual organ-isms, individual cells, different cell types, and differentspecies. These partitions evolve in time according to dy-namic and changing laws, whereas computational modelsoften use static laws to evolve a system. This distinctionbetween models and biology prevents several key aspectsof biology from being understood mechanistically. Asa result, there is no universal theory that describes howhaving different levels of organization contributes to com- plexity and open-ended evolution in biology.From a bottom-up approach, it is assumed that a sys-tem’s behavior is entirely determined by the underlyinglaws of the parts that compose it . A top-down ap-proach suggests just the opposite: That the behavior ofindividual parts in a system is determined by the be-havior of the system as a whole . Empirical resultssuggest that biology uses both, and entities within a sys-tem change their laws over time . For example, humansocial systems are composed of individuals and groupsof individuals who change their behavior as a functionof other individuals or groups, who in turn do the same.Biological systems with multiple entities, like communi-ties, change their behavior as a function of other entitieswithin the same system.In online video games, strategies evolve open-endedlyin ways that are reminiscent of biological open-endedevolution . In a popular online video game called Leagueof Legends , several strategies are explored by players.The winning strategies are often the most popular strate-gies. The popular strategies change over time, but theevolution of which strategies are most popular throughtime never settles into an attractor cycle. In fact, thegame developers organized the system of game code,players, and themselves such that this is guaranteed . Thegame developers change the game’s code every two weeks,which changes important aspects of the game. This, inturn, changes the set of possible strategies that playersare able to choose strategies from. This guarantees that a r X i v : . [ n li n . C G ] A ug the evolution of the most popular strategies never settlesinto an attractor cycle, since the state space of strate-gies changes regularly. In addition, the methods thatthe game developers use to update the game’s code aredetermined largely by empirical analysis of player be-havior. The game developers attempt to strengthen theweaker strategies and weaken the stronger strategies. Inthis sense, League of Legends is a system composed ofmultiple entities and levels of organization. Within thelarger system, the set of popular strategies evolves open-endedly, even under an unchanging interaction rule be-tween game developers and the game’s code . This sug-gests there is indeed a causal relationship between havingmultiple subsystems within a system and that system’sability to exhibit open-ended evolution.The vocal behavior of domestic cats interacting withhumans is an example of two interacting individualswithin a system of communication. Since cats do notvocalize very often in the absence of humans, it is as-sumed that domestic cats adapted to living with humansby vocalizing more often than they do in the wild . Itwas found that cat vocalizations are unique to each cat-human pair, which may indicate the cat’s choice of vocal-ization tones is unique to the individual relationship witha specific human . This is likely due to an interactionrule between both individuals; humans provide feedbackmechanisms to the cat, which provides an evolutionarypressure on the cat to vocalize sounds that result in de-sired outcomes.In this paper, two interacting cellular automata (CA)are used as an abstract model to address the effectof complex interactions between two individual entitieswithin a larger system. Unlike elementary cellular au-tomata (ECA), these CA change their update rules asa function of some part of the system as a whole ateach time step. Because biological evolution is widelyaccepted as being open-ended, CA that are capable ofopen-ended evolution according to a definition tailoredfor discrete, bounded, synchronous models are identified.The relationship between the complexities of various in-teraction functions between these CA, and the qualita-tive characteristics of each CA’s resulting behavior is ex-plored. Due to the algorithmic nature of this model, ap-proximations to algorithmic complexity are used in placeof entropy-based measures of complexity throughout thisanalysis . I. APPARENT OPEN-ENDED EVOLUTION
Even though biological evolution is widely acceptedas being open-ended , there is no scientific consensuson an exact, quantifiable definition of open-endedness .But for bounded, discrete dynamical models with syn-chronous update rules, open-ended evolution (OEE) canbe formally identified if that model’s evolutionary trajec-tory (discrete states over time) is both unbounded andinnovative . This definition heavily relies on the Poincar´e Recurrence Theorem for discrete, deterministic systems.The Poincar´e recurrence time t P of a finite, determinis-tic, bounded, and dynamical model provides a time con-straint on when it will repeat exactly. For 1-dimensionalECA, t P = 2 w , where w is the number of cells in a singleCA state. This is because an ECA can, at most, expressevery possible state in its evolutionary state trajectory,and because the update rule r is fixed, visiting the samestate more than once would cause the state trajectory torepeat itself exactly. unbounded evolution is the ability for a single CAmodel c ∈ C embedded within a larger system u of twointeracting CA ( c and c ) to defy the Poincar´e recur-rence time by repeating its patterns of expressed states s ( t ) , s ( t ) , s ( t ) . . . s ( t r ) in a time greater than t P . Be-cause c and c within u are not isolated and do not evolve under a single, unchanging update rule like ECAs,the t P that c needs to “beat” is determined by the t P ofan ECA with the same size w as c . Definition 1 Unbounded evolution : A finite, deter-ministic, and bounded dynamical system u , which can bedecomposed into subsystems c ∈ C and c ∈ C that in-teract according to a function f , exhibits unbounded evo-lution if there exists a recurrence time t r in c or c suchthat s f ( t ) = s ( t ) , s ( t ) , s ( t ) . . . s ( t r ) is non-repeating for t r > t P , where t P is the Poincar´e recurrence time for anequivalent isolated (non-interacting) system c ∈ C . Here, f is the interaction rule function between c and c , defined in section II. Because the state evolution (tra-jectory) of c is compared to counterfactual state trajecto-ries of ECAs of the same size, this implies that ECAs areinherently incapable of unbounded evolution. Innovation is defined as : Definition 2 Innovation : A finite, deterministic, andbounded dynamical system u , which can be decomposedinto subsystems c ∈ C and c ∈ C that interact accord-ing to a function f , exhibits innovation if there exists astate trajectory s f ( t ) = s ( t ) , s ( t ) , s ( t ) . . . s ( t r ) that isnot contained in the set of all possible state trajectories { s I } for an equivalent isolated (non-interacting) system c ∈ C . That is, a subsystem CA c exhibits innovation by Def-inition 2 if its state trajectory cannot be produced by anECA of the same size. Both Definitions 1 and 2 reflect theintuitive notions of “on-going production of novelty” and“unbounded evolution” but do not necessarily meanthe complexity of individual states increases with time.Furthermore, OEE is apparent on the scale of a single CAembedded within a larger system. This is in agreementwith our intuition of OEE within biology— the evolutionof life as a whole appears to evolve open-endedly, but it isembedded within a larger system that is not necessarilyopen-ended. II. MODEL
FIG. 1: Each of these CA evolve over time (downwards)and change their update rule r ∈ R = [0 , f type as described in the text. Becauseresults are exhaustive, only CA with w = 3 areconsidered.The model system explored here is composed of twofinite, deterministic, and spatially bounded interactingCA with fixed widths w and periodic boundary condi-tions. Each CA starts exactly like an ECA with a fixedupdate rule r ∈ R = [0 , s . Both CA use one of the followingtypes of functions f type ,t at time t to change their rule r ,which will then determine the next state s t +1 :1. r x,t = f this state ,t ( s x,t ): r x,t is determined only bythe current state s x,t of that CA.2. r x,t = f other state ,t ( s y,t ): r x,t is determined only bythe current state s y,t of the opposing CA.3. r ,t = r ,t = f both states ,t ( s ,t , s ,t ): Both r ,t and r ,t are determined by the current states s t of bothCA.4. r x,t = f mixed ,t (random choice( s ,t , s ,t ∨ s x,t ∨ s y,t )): r x,t depends on a random choice of both CA states,the state of that CA, or the state of the opposingCA.Because all possible initial states for a given w and ini-tial rules r ∈ R = [0 , w = 3 were explored. For each of all possiblecombinations of s ,w , r , state trajectories were recordedfor 2 ∗ w time steps. An illustration of this model isshown in Figure 1.For each interaction function type f type , the exactmappings f type ,i between states s and rules r was gen-erated randomly. 5000 random mappings were createdfor each f type , and only six of these 5000 mappings wereused. The six mappings were chosen based on their rel-ative approximate complexity values (described in III)— three mappings with relatively high complexity and threemappings with relatively low complexity. Because the in-teraction function for f mixed depends on a random choicemade at each time step t , the complexity could not bemeasured for f mixed ,i mappings, since a static mappingdoes not exist. However, the exact random choice in map-pings were based on the other three interaction types. Foreach of the interaction function types, the six individualmappings are denoted as i ∈ [0 , III. METHODS
Both CA state trajectories were checked for apparentOEE according to Definitions 1 and 2. Algorithmic com-plexity (Kolmogorov-Chaitin-Solmonoff complexity) can-not be computed exactly due to the Halting Problem,but can be approximated using the Block DecompositionMethod (BDM) . This is an upper-bound approximationof the algorithmic complexity, which, in short, measuresthe size of the smallest computer program that can pro-duce the string of symbols being measured .The BDM can be used to approximate the algorithmiccomplexity of 1-dimensional or 2-dimensional objects. Byrepresenting each interaction function mapping f type ,i asan adjacency matrix, it is possible to approximate thecomplexity for any non-changing f type ,i . But since eachinteraction rule mapping f type ,i was generated randomly,this would affect the expected range of BDM values forany f type ,i . BDM is largely known for quantifying therandomness of mechanisms capable of producing an ob-ject. This was mitigated by selecting mappings withhigh and low BDM values relative to a batch of 5000randomly-generated mappings.The CA state trajectory of both CAs can be repre-sented in two ways for the purposes of measuring theBDM. The first is to measure the BDM of each statein the state trajectory. Then the mean of the BDMvalues for each individual state s are calculated perCA state trajectory. The second is by enumeratingall possible states for a CA of size w and measuringthe BDM of the sequence of enumerated states. For w = 3, there are 2 w = 8 possible states, making itcomputationally tractable to measure the BDM for theentire state trajectory. For computational tractabilityreasons, the Python 3 package pybdm (https://pybdm-docs.readthedocs.io/en/latest/) does not support se-quences with an alphabet size over 9, thus making itpossible to use pybdm to calculate the BDM for each ofthese measurements. IV. RESULTS
Figure 2 shows the number of open-ended CA state tra-jectories (% OEE) for both CAs as a function of the dif-ferent interaction function types f type , as defined by Defi-nitions 1 and 2. There were no OEE state trajectories for 2 ( ( &