Computational Complexity of Observing Evolution in Artificial-Life Forms

06/24/2018 ∙ by Janardan Misra, et al. ∙ Association for Computing Machinery 0

Observations are an essential component of the simulation based studies on artificial-evolutionary systems (AES) by which entities are identified and their behavior is observed to uncover higher-level "emergent" phenomena. Because of the heterogeneity of AES models and implicit nature of observations, precise characterization of the observation process, independent of the underlying micro-level reaction semantics of the model, is a difficult problem. Building upon the multiset based algebraic framework to characterize state-space trajectory of AES model simulations, we estimate bounds on computational resource requirements of the process of automatically discovering life-like evolutionary behavior in AES models during simulations. For illustration, we consider the case of Langton's Cellular Automata model and characterize the worst case computational complexity bounds for identifying entity and population level reproduction.

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1 Background

Studies on Artificial Evolutionary Systems (AES) are recent attempts to complement real-life theories to study the principles underlying the complex phenomena of life without directly working with the real-life organisms. For example, AES studies can complement theoretical biology by uncovering detailed dynamics of evolution where real life experiments are not possible [50, 37], and by developing generalized formal models for life to determine criterion so that life in any arbitrary model can be observed.

Observations play a fundamental role both in real-life studies as well as in AES research. In case of real-life studies, observations are an integral part of the experimental analysis carried out to uncover the specific dynamics underlying the observed life forms and their properties. In astrobiological studies, though, it is still a challenging problem to detect life in any arbitrary molecular form [17]. The lack of agreement in interpreting the possible presence of life on Mars with Viking Lander labeled release experiments is a case in example [27, 31, 44]. On the other hand, in case of those AES studies, which focus on the problem of the “emergence”’ of life-like behavior, in general there is no known method to even decide beforehand the kind of entities, which might emerge demonstrating such non-trivial life-like behavior, without closely observing the simulations. For example, Nehaniv and Dautenhahn [49] have argued that identification of time varying entities is a difficult problem in the context of formal definitions for self-reproduction and add that in absence of observers it is problematic to decide whether an instance of artificial self-replication be treated at all a life-like one. The exceptions occur only when a study starts with built-in design of the entities e.g., well separated programs, which can self replicate, mutate, and evolve according to the design [1, 53].

Often only informal discussions are presented in AES studies on the mechanisms employed by the researchers to discover the emergent entities and their life-like behavior. These discussions usually remain useful only to the specific models and do not always have the generic perspective. Therefore an important aspect where AES studies demand increasing focus is to study observational processes and mechanisms used in AES studies in their own right resulting into a framework for automated discovery of life-forms and their dynamics in the simulated environments. Whereas in case of real-life studies such automatization might not be feasible in general, with AES studies involving mostly digitized universes and their simulations, it is actually desirable to explore by algorithmic means potentially varied possibilities which these simulations hold yet usually require such detailed observations that it may not always be feasible to carry out for human observers alone. Such an automated discovery of life-forms and the evolving dynamics may bring much promise in AES studies as compared to what could possibly be achieved only with manually controlled observations.

An example of such an automated discovery of life forms is discussed by Samaya in [56]. In order to observe the living loops in his CA model, another “Observer CA” system is designed and embedded within the simulator software. The observer CA is capable of performing the complex image processing operations on the CA configuration given to it as an input by the simulator CA to automatically identify the living loops of different types.

However, because of its implicit nature and the multitude of AES models, a precise characterization of the observation process is generally a difficult problem. Importantly any sufficiently generic framework for studying the observational processes needs to be defined independent of the low-level micro dynamics or the “physical laws” of any specific AES model and should permit the study of higher-level observationally “emergent” phenomena. Initial attempt in this direction appeared in [45, 28]. The central idea proposed in these studies is a semi-formal characterization of the observation process, which leads to abstractions on the model universe, which are consequently used for establishing the necessary elements and the level of evolutionary behavior in that model. In this paper we extend these ideas further and provide some key results including computational complexity theoretic analysis of the problem of algorithmic realization of the observational processes.

A multiset based algebraization is proposed in [47] to achieve an algorithmic characterization of the observational processes for the purpose of entity recognition in artificial-life models. We extend the framework in this work to account for the process of discovery of the life-like evolutionary behavior in AES models. Starting with defining entities and their causal relationships observed during simulations of a model, the extended framework prescribes a series of axioms (conditions) to establish the degree of life-like evolutionary behavior observed in the model. This formal characterization enables to reason about the automated discovery of the life-like entities and their evolutionary behavior, which otherwise is often discovered only manually.

The framework is analyzed in terms of computational resource requirements of the algorithmic implementation of the observation process and consequent automated discovery of the evolutionary behavior in arbitrary AES models. Building upon the earlier results on the computational complexity bounds for the problem of entity recognition, we further characterize the bounds for an automated discovery of various evolutionary components including entity and population level reproduction with epigenetic developments in the child entities involving mutations, heredity, and natural selection. These computational complexity bounds are established distinguishing further between those AES models which allow entities with overlapping structures to coexist in a state and others which do not.

The paper is organized as follows: In Section 2, we will formally elaborate the framework followed by computational complexity theoretic analysis of it in Section 3. Section 4 presents a discussion of related work, and is followed by concluding remarks in Section 5, including a discussion on the limitations of the framework and pointers for future research.

2 The Framework

2.1 Preliminaries

Next we will briefly review the basics of multiset theory and the notions from the theory of computational complexity, which would be used in the formal exposition the framework.

2.1.1 Multiset Theory

A multiset is a collection of objects, which may contain multiple copies of its elements, e.g., . Formally, a multiset on a set can be considered as a mapping associating nonnegative integers () (representing multiplicity) with each element of , , where . denotes that appears in with multiplicity . Some of the operations on the multiset are defined as follows: For multisets and

Multiset Comparison: Subset: and Equality:

Join: For multisets on ,

Intersection:

Power Set: For a multiset on ,

Cartesian Product: . So, Cartesian product of a multiset on with classical set is defined as

Size: For a multiset on , its size

Functions: The mapping based definition of a multiset given above does not directly support the notion of a function on a multiset. Recall that for classical sets, a function is a special subset of the Cartesian product of the domain and the range sets such that each element of the domain set is associated with only one of the elements of the range set. This conditions when applied over the multisets defined as above cannot be easily generalized since for that we need to distinguish all the multiple copies of the individual elements. Therefore, we restrict our attention to defining a special class of functions, which will be used in the paper. The functions we consider are those which restrict the range to a classical set. Let us consider such a function , where is a multiset on and is the range set. satisfies the following constraint:

Informally, this constraint demands all the multiple copies of each should be associated with different elements of by the function . Notice that such a function itself is a classical set - this is an important property which will be used later. Consider for example a multiset as the domain and the set as the range of the function defined as , , which satisfies the above constraint with and . A special subclass of these functions called ‘one-to-one’ or ‘injective’

functions further restrict that no two ordered pairs can have identical second elements in the pairs: If

is ‘one-to-one’ then for all we have . For example, a one-to-one function may be defined as .

For further details on multiset theory, reader is referred to [62, 11, 58].

2.1.2 Computational Complexity

The field of computational complexity studies issues related to the computational resources (e.g., CPU time, memory) required for the execution of algorithms and the inherent difficulty in designing algorithms for specific problems [51, 15]. For example, time complexity of a problem is the (minimum) number of time steps that would be required by any algorithm to solve an instance of the problem as a function of the size of the input. When computational complexity for a specific problem is considered, necessary resource requirements for any algorithm solving that problem are measured indicating the inherent computational difficulty of that problem. On the other hand, when a specific algorithm for a problem is considered, computational complexity measures are only limited to that algorithm.

Computational complexity measures are in general specified without explicitly considering the details of the actual underlying machine architecture and rather by assuming some abstract representation of it, e.g., Turing Machine model and RAM (Random Access Machine) model. These abstract models are known as the models of computation and computational complexity measures are specified only in reference to a model of computation. We will consider the RAM model as the underlying model of computation for the computational complexity theoretic discussions to be presented in this paper. RAM model is an example of the traditional von Neumann architecture of a sequential machine which can execute sequential computer programs and is computationally equivalent to the Turing machine model.

Computational complexity theory often classifies problems into various complexity classes based upon their inherent resource requirements. A

computational complexity class w.r.t. a specific computational resource is the set of all those problems which minimally require at least the amount of resource specified for that class. For example, the time complexity class NP is the set of all those problems for which a solution can be verified in polynomial number of time steps on a deterministic Turing machine model. Many important problems fall in this class including the Boolean formula satisfiability problem [51], the protein folding problem [10], clustering problem, timetable design problem, staff scheduling problem etc. (see [25] or the wiki page [64] for a detailed list of NP Complete problems.) An important technique used for proving that a problem is hard w.r.t. a complexity class is known as the reduction method, in which a relatively simpler translation (as compared to the computational requirements of the class ) is defined, which reduces an input instance of some problem , which is already known to be belonging to the class , into an instance of the problem .

Asymptotic order notation known as (big Oh) is often used to measure the bounds on computational complexity for algorithms and problems [15]. If , then is said to be upper bounded by for for all the positive values of the input of size after certain point. Formally,

Informally, it means, grows no faster than . Also we have the following useful asymptotic properties [38, page 80]: If and ,

(1)
(2)

A specific type of computational complexity analysis often considered is the worst case analysis, which analyses the resource requirements for solving the worst possible input instance of the problem. Such analysis sets an upper bound on the resource requirements for any input instance of the problem. For example, for the problem of searching for a given item in an input list of size , the worst case time complexity would be for the case where the item to be searched is not present in the list. Since with arbitrary amount of resources any input instance for a computable problem can be solved, worst case analysis defines the lowest possible bounds for solving the worst case input instance. Sometimes, however, it is not possible to give exact (tightest) bounds for a problem under consideration, in that case, some relatively relaxed upper bound is given instead. The worst case analysis of a specific algorithm solving the problem provides one such upper bound on the worst case computational complexity of that problem itself since it demonstrates that those computational resources would be sufficient to solve any instance of the problem, though another better algorithm may exist requiring less resources.

2.2 The Formal Structure of the Framework

To illustrate the framework, we will use the examples of Cellular automata (CA) based Langton Loops [35] and calculus based Algorithmic chemistry[18]. Brief background on these models is given next.

John von Neumann defined CA [63] to explain the generic logic of self reproduction in mechanical terms. His synchronous CA model was a two dimensional grid divided into cells, where each cell would change in parallel its state based upon the states of its neighborhood cells, its own state and its transition rule. For this CA model, von Neumann defined a virtual configuration space where he demonstrated analytically that there exists some universal replicator configuration which could replicate other configurations as well as itself. Though universal replicators are not found in nature and such self replicator was extremely large in its size, the underlying logic of treating states of cells both as ‘data’ as well as ‘instruction’ was a fundamental contribution of this model since it was also discovered later in case of real-life where DNA sequences specify both transcription as well as translation for their own replication in a cell. Another strength of von Neumann’s formulation was its ability to give rise to unlimited variety of self replicators [41, 42]. Over the years this model was simplified and reduced in size considerably [14, 81-105].

Finally Langton introduced loop like self replicating structures in [35], which retained the ‘transcription - translation’ property of von Neumann’s model excluding the capability of universal replication and symbolic computation. Langton’s original self-replicating structure is a 86-cell loop constructed in two-dimensional, 8-state, 5-neighborhood cellular space consisting of a string of core cells in state , surrounded by sheath cells in state . These loops have since then, been extended into several interesting directions including evolving Evoloops in [56].

Algorithmic Chemistry (AlChemy) was introduced in [18] and further discussed in [19, 22, 20, 21]. The main focus of AlChemy is to study the principles behind the emergence of molecular organizations through an approximate abstraction of real chemistry as calculus with finite reductions. Starting with a random population of terms (molecules), using different filtering conditions on reactions, authors describe the emergence of different kinds of organizations: Level organization consisting of a set of self copying term(s) and hypercycles with mutually copying terms, Level self maintaining organizations consisting of terms such that every term is effectively produced as a result of reaction between some other terms in the same organization and lastly Level meta-organization consisting of two or more Level sub organizations such that molecules migrate between these self maintaining sub-organizations. Authors also discuss an algebraic characterization of Level and Level organizations without referring to the underlying syntactical structure of the terms (molecules) or the micro dynamics (reduction semantics and filtering conditions) governing the output of reactions. Further details on these models can be found in the above references.

In the ensuing discussion, we will use “AES model” and “model”, “Observation process” and “Observer” interchangeably to add convenience in presentation. Similarly “real-life” is used in the paper to refer to organic life on earth in contrast with the “artificial-life”. Also, Observer Abstractions will refer to specific observations and corresponding abstractions made upon the AES model during its simulations. Axioms are used to specify conditions which need to be satisfied in order to draw valid inferences e.g., recognition of entities and their causal relationships. The aim is to define these Axioms such that only valid claims for the presence of life-like phenomena in a model can be entertained. Auxiliary formal structures are also used in the intermediate stages of analysis. E.g., distance measure for determining dissimilarity between entities with respect to their specific characteristics.

2.2.1 Observation Process and the Model Universe

We first briefly summarize the framework presented in [47] for the purpose of entity recognition in artificial-life models. Next, we will extend it for the purpose of observing evolutionary components.

Observation Process. : An observation process is defined as a computable transformation from the underlying model structure to observer abstractions , where is the set of process independent abstractions and is the set of process dependent abstractions.

States. : set of observed states111We will use to denote individual states. of the model across simulations.

In general a state could be considered as a collection of all observable atomic structural elements and their (observationally relevant) characteristics in the model at any given time point during the simulation. A multiset xan be used to represent state of a model at any instance during its simulation.

Observed Run. : set of observed sequence of states ordered with respect to the temporal progression of the model during its simulation. Each such sequence represents one observed run of the model. A sequence of states is formally represented as a partial mapping , where is the set of non negative integers acting as indexes for the states in the sequence.

Let denote the set of states appearing in a specific run . An important consideration while defining a run is to decide the temporal granularity. The granularity would determine which states an observation process is observing (or is able to observe) and that would in turn determine to what extent life-like behavior could be potentially observed by the observation process.

and a set of observed runs thus define the observed dynamic structure of the model as a state machine with respect to a given observation process. Where

  • is the set of starting states, i.e., .

  • is the transition relation between states, i.e., .

2.2.2 Entities and Their Characteristics

Observer Abstraction 1 (Entity Set).

: Multiset of entities observed and uniquely identified by the observer in a state of the model for a given run .

is the multiset of entities observed and uniquely identified by the observer across the states in the given run .

The criterion to select the set of uniquely identifiable entities in a given state of the AES model is entirely dependent on the observation process as specified by the AES researcher.

As an example, consider the case of two dimensional CA lattice based model. A cell in the lattice is represented as , where is the coordinate and is the state of the cell. When a cell is in state , it is also known as a quiescent cell. Let denote the set of all cells. For a given cell , let and , which can be extended to the set of cells: , , . Also let output the coordinate wise non quiescent cells in the surrounding neighborhood of a cell. In terms of these, each entity in a state can be characterized by two values - - the connected set of non-quiescent cells and an associated pivot. The (function) gives the coordinates for a cell uniquely associated with an entity in CA lattice in a particular state. Pivot works as an state invariant for each such identified entity (loop).

In general an observation process may recognize an entity in a state as a subset of the observable atomic structures, that is, (structure of) an entity can be defined as a subset of the state itself. Formally,

In case of two entities being identical i.e., consisting of identical multisets of atomic structures, additional tagging may also be required. Notice that such a subset based structural identification of entities also allows entities with overlapping sets of atomic elements. In such scenarios also application of additional tagging is essential. Tagging, in general, can be defined as a total function:

Where is the set of unique tags to be associated with the entities. In the following discussion, we would implicitly assume that whenever necessary, identified entities in each state are also tagged such that with tagging, each identified entity would naturally be distinguishable from others. If two entities are considered different, that would mean either the these entities have distinct structures or they are given different tags by the observation process.

Also we have the following two axioms imposing consistency requirements on the entity identification. First axiom states that every entity in a state is uniquely identified. Second axiom states that the set of entities identified in identical states should be the same.

Axiom 1 (Axiom of Unique Identification of Entities).

An entity must be uniquely identified in a given observed run . Formally, Tagging is a one-to-one function on states, that is, .

Axiom 2 (Axiom of Unique Identification in States).

Entity set must be uniquely identified for each state in a given observed run . Formally,

This axiom may also be considered as the soundness axiom for entity recognition. Finally we have the following important axiom of non-ignorance, which can be considered as a complementary completeness axiom for entity recognition.

Axiom 3 (Axiom of non-Ignorance).

Let and denote the multisets of entities observed in states respectively. Then

In other words, this axiom forbids that an observer omit identification of an entity in a state but in a different state identifies it as consisting of the same atomic elements which were also available in .

Having defined the sequence of states with temporal ordering and the entities identified by their tags, we will now proceed to discuss how an observer might define the detailed observable characteristics for such entities. Using these characteristics it can infer relevant relationships among entities e.g., descendant relationship, heredity, and variation. To this aim, we will define ‘character space’ as a set of values for the observed characteristics. These values might be purely symbolic without any relative ordering or can be ordered using suitable ordering relation.

Observer Abstraction 2 (Character Space).

The observer should define the set of all possible mutually independent (or orthogonal) and measurable characteristics for possible entities in the model as a multi dimensional character space , where each of is the set of values for characteristic. Each of make one orthogonal dimension in the space .

Corresponding to each entity there is a point in , say , where

. For a vector

element () will be denoted as . For some of the characteristics observer might define a ‘partial ordering’ ( for ), which can be used to compare values for those characteristics. The absence of a characteristics in an entity is represented by a special zero element such that if is (partially) ordered then . .

Notice that, observable characteristics need not to be limited to syntactic level or structural properties and can also include semantic properties - observable patterns of behaviors - though semantic properties are much more difficult to observe and measure since they require abstracting the patterns of reactions over a range of states and also semantic equivalence between entities might be computationally undecidable, in general, e.g., program equivalence [30, Ch. 8].

In case of Langton’s CA model, an obvious characterization for is a two dimensional space with being the set of all non-quiescent connected set of cells and being the set of corresponding pivots. We will next extend the framework to infer various components of the life-like evolutionary behavior.

2.2.3 Distance Measure

A “dissimilarity measure” () defines the “observable differences” () between the characteristics of the entities in a population. The distance measure defined below can be used by the observer to distribute entities into separate clusters such that entities in the same cluster are sufficiently similar while entities from different clusters are distinguishably different in their characteristics. Exact definition of distance function is also model dependent.

Observer Abstraction 3 (Distance Measure).

An observer defines a decidable clustering distance measure , where is the set of values to characterize the observable “differences” between entities in .

Examples include the Hamming distance to define distance between genomic strings in the Eigen’s model of molecular evolution [57], set of points where two computable functions differ in their function graphs, or the set of instructions where two programs may differ. One of the known criterion to define the concept of species is “phenotype similarity” [54], which can also be seen as another example for distance measure.

In case of Langton’s CA model, is defined such that where is if both entities have the same number of cells arranged identically or else it is and is when the pivots for both the entities are the same and otherwise.

2.2.4 Observable Limits on Mutational Changes

For most of the non trivial AES models entities may also change (mutate) over the course of their interaction with the environment (or other entities.) These changes in the structure as well as the characteristics of the entities might in turn make it difficult for an observer to recognize the entities across states. Therefore it is essential to specify the limits under which an observer can recognize entities across states even in the presence of such mutational changes. This is an inherent limiting property on the part of the observer and could vary among observers. Based upon the limit referred here as , an observer can establish whether two entities in different successive states are indeed the same with differences owning to mutations or not. The smaller the limit, the harder it will be for an observer to keep recognizing entities across states and a mutated entity may get classified as a new one. As entities are observed in more and more refined levels of details, their apparent similarities melt away and differences become sharply noticeable. Another alternative method which could be used for establishing the persistence of entities across states is the identification of temporally invariant properties of the entities of interest. In the current framework such temporally invariant properties of entities could also be defined as a separate characteristic in .

Observer Abstraction 4 (Mutation Bound).

Based upon the choice of clustering distance measure , the observer selects some suitable , which will be used to bound mutational changes for proper recognition. is a vector such that each element specifies an observer-defined threshold on the recognizable mutational changes for corresponding characteristic.

It is important to note that the choice of critically affects further inferences. For example, a choice of very large values would result in the lack of identification of variability in characteristics among entities. This in turn might make an observer to recognize several (otherwise non identical) entities as identical and thus the dynamics of model which could have been possible to uncover with fine grained identification of the differences would not be possible. On the other hand if an observer decides to select very small values for then it cannot recognize persistence of an entity across states under changes. Therefore it is important to define the bounds optimally. Next, we define Recognition relation to establish the persistence of entities even in the presence of mutational changes:

In case of Langton’s CA model, let us select , which means the observer can recognize an entity in future states even with mutations (changes in the states, number, or the arrangement of cells comprising the entity) provided that the pivot remains the same.

Definition 1 (Recognition Relation).

The observer establishes recognition of entities across states of the model with (or without) mutations by defining the function : , which is a partial function and satisfies the following axioms:

Axiom 4.

.

Informally, the axiom states that entities to be recognized as the same have be observed in successive states. Note that is anti symmetric (and therefore partial) to ensure that entities are recognized based upon the temporal progression of the model and not in any other arbitrary order.

Axiom 5.

is an injective function, that is,

Informally, the axiom states that no two different entities in one state can be recognized as the same in the next state.

Axiom 6.

. .

Informally is that , which is recognized in the next state by the observer as with possible mutations bounded by . In other words if entity mutates and changes in the next state and identified as , then observer might be able to recognize and as the same if these changes (between and ) are bounded by .

In case of our example of Langton’s CA model, a recognition relation satisfying above axioms would imply that two entities in consecutive states are recognized same only if they have the same pivots. This also means the observer can recognize an entity even with change in the number, state, and geometrical arrangement in the cells across states provided that entity does not shift in CA lattice altogether (which would result in the change of the pivot.) Axiom , which states that is an injective function would also hold because no two entities in the same state share the same pivot. This is because pivot as defined before is connected to all other cells of the entity and all the non-quiescent cells which are connected in any state are taken together as one entity. Thus two different entities in the same state always consist of cells such that cells in one entity are not connected with the cells of second entity, and hence always have different pivots.

2.2.5 Observed Causality

In order to infer meaningful relationship between entities, which could later be used as a basis for inferring interesting macro level properties in the model, an observer needs to identify “causal” relationships among entities independent of the underlying ‘physical laws’ of the model.

Observer Abstraction 5 (Causality).

. establishes the observed causality among the entities appearing in the successive states of a run .

Notice that in order to establish causal relation between entities, observer need not necessarily know the underlying reaction semantics or the micro level dynamics of the model. Only requirement is that the observer’s claimed causality conforms with the stated axioms. Since causality is largely a observer and model dependent property, we will illustrate it further in Section 2.3.1 by defining an additional axiom of reproductive causality, which will be used in turn to infer reproductive relationships among entities.

For example, in case of Langton’s CA model, the relation between entities in consecutive states is defined as follows: such that where and we require

Intuitively what we demand with above definition of causal relation is that (child) entity was part of the (parent) entity and at certain stage it “breaks off” from the (parent) entity , as can be seen in Figure 2 at time step .

2.3 Observing Evolution

Having defined the observation process as a computable transformation from the underlying sequence of observed states of the model to the set of components involving entities and their observable characteristics with measurable differences as well as observable limits on such differences, we will now consider a specific observation process for observing evolutionary behavior in the AES studies. The following discussion will define components in for observing the fundamental evolutionary components: reproduction with mutations and epigenetic developments, heredity, and natural selection. can also be defined considering other criterion of recognizing life-like phenomena, for example, metabolism [5], complexity [2], self organization [32], autonomy and autopoisis [65].

2.3.1 Reproduction

Reproduction is one of the fundamental components of evolution. Through reproduction, entities pass on their characteristics to the next generation and increase the population size. Reproduction is possibly the only way by which (abstract) entity features can persist across generations in case of those AES models, where entities do not persist forever. In an observed framework, the way an observer can establish reproduction is by providing observed evidence for it. This is done by defining causal descendance relationships among the entities across states, whereby parent and the child entities are recognized by the observer as being sufficiently similar and “causally” connected across the states. Formally, we add a new Axiom for the causal relation defined before:

Axiom 7 (Reproductive Causality).

Informally, the axiom of reproductive causality states that if an entity in state is causally connected to entity in the next state, then there must not be any other entity in state , which is also recognized by the observer as in the next state. This is to ensure that mutations are not confused by the observer with reproductions. In essence, this formulation of causality is an abstract specification which demands observers to identify the entities which have been observed to be causal sources for the appearance of a new entity. Only then proper descendance relation for the new entity can be established.

Lemma 2.1.

Causal relation defined for the Langton’s CA model satisfies the Causality Axioms and Reproductive Causality Axiom .

Proof.

Condition insures that and are observed only in the consecutive states as demanded by the axiom . To establish that is not the result of mutations in some other entity observed in past (i.e., ) we note that because of the definition of , and would otherwise have the same pivots, which means pivot of will be included in the set of cells in (since ), which is not possible because and being different entities in the same state cannot have cells in common including pivot as argued above in the proof of previous lemma. ∎

Reproductive Mutations

For evolution to be effective there should be observable differences between the child and the parent entities arising out of reproductive processes. These changes in the characteristics of the entities may or may not be inheritable based upon the design of the model and the simulation instance.

Observable Limits on Reproductive Mutational Changes: Similar to the above discussed , it is important to specify the limits under which an observer can identify whether an entity is an descendant of another entity even though they might not be identical. This necessitates us to introduce another bound on observable reproductive mutations as . This limit on observable reproductive mutations is indeed crucial while working with models where epigenetic development in the entities can be observed [40]. This is because in such models including examples from real life, the “child” entity and the “parent” entities do not resemble with each other at the beginning and observer has to wait until whole epigenetic developmental process gets unfolded and then compare the entities for similarities in their characteristics. assists an observer to establish whether a particular entity could be treated as a “descendant” of another entity or not.

Another reason for introducing the limit is that from the view point of an high level observation process not recording every micro level details, it is quite essential to distinguish between parent entities and other secondary entities involved in the reproductive process. Consider, for example, a model where entity reproduces according to reaction , where is mutant child entity of , which can be determined by an observation process only when it can establish that and are sufficiently similar with respect to their characteristics, while and are not.

Observer Abstraction 6 (Reproductive Mutation Bound).

Based upon the choice of clustering distance measure , the observer selects some suitable , which will be used to bound reproductive mutational changes for proper recognition. is a vector such that each element specifies an observer-defined threshold on the recognizable mutational changes for corresponding characteristics.

In case of Langton’s CA model, let , which means for reproduction observer strictly demands identical geometrical structure of the parent and child entities, though they may have different pivots - this is essential to capture exact replication of the loops.

It is important to note that the choice of critically affects further inferences. For example, small values for might make it harder to establish reproductive relationships among entities and for such an observer every new entity would seem to be appearing de novo in the model. On the other hand choice of very large values would result in the lack of identification of variability in characteristics and thus make it difficult to infer natural selection (discussed later).

We next need an auxiliary relation to determine that the differences due to the reproductive mutations are also bounded by .

Definition 2.

s.t.

Informally for ) to be in , their differences for each single characteristic must be bounded by and should not be recognized as mutating to .

Based on the thus established notion of “causal” relationships between entities and , we will define relation, which connects entities for which an observer can establish descendance relationship across generations.

Definition 3.

=

In this definition the transitive closure of captures the observed causality () across multiple states even in cases when “parent” entities might undergo mutational changes () before “child” entities complete their “epigenetic” maturation with possible reproductive mutations. Intersection with ensures that causally related parent and child entities are not too different from each other, that is, reproductive mutational changes are under observable limit. For , when is observed as an ancestor of .

Figure 1: Graphical view of the relationships between entities in successive states. Recognition relation , Causal relation , and .

Figure 1 depicts graphically the relationships between entities in successive states. Vertical lines represent the states (). Various kinds of arrows represent different relationships: recognition relation , causal relation , and . The end points of the arrows on state lines represent entities.

To illustrate the point further let us also consider the case of Reflexive Autocatalysis: In the simplest form, a reflexive autocatalytic cycle is represented as a system of reaction equations:

where copies of entity are produced at the end and that entity is a variation of entity , i.e., . Such autocatalytic cycles are supposed to be the chemical basis of biological growth and reproduction. Examples include the Calvin cycle, reductive citric acid cycle, and the formose system. Competing cycles of this sort can even undergo limited evolution, though they are supposed to have very limited heredity [60].

In the current framework if an observer could determine the causal relations - , , , and if the entity does not undergo any changes before is produced, that is, . Then would contain so also would establishing the reproduction of through reflexive autocatalytic cycle and with variation.

Theorem 2.2.

The definition of effectively formalizes the recognition of reproductive relationships under parental mutations together with reproductive mutations and epigenetic developments in the child entities.

Proof.

An observation process need to observe child entities so long that their epigenetic development unfolds completely. Since, in general, a priori limits cannot be assumed on the number of states required for such epigenetic development, this requirement of observations across states is captured using transitive closure - , where ensures that (mutational) changes in the parent entities and also the changes in the child entities during epigenetic development are accounted for. To see this,

Lets us assume that in a state , a child entity was observed for the first time and (parent) entity present in the state was observed to be casually connected to it. Suppose that for the entity its epigenetic development unfolds through states such that with changes owing to the development was observed as in these states with . Similarly suppose that parent entity has been undergoing mutations in these states and also in the states before observed as such that . It is clear that would contain , , among other tuples implying that the intersection of with would result in those tuples , where and are sufficiently similar in their characteristic. Therefore if the resultant set is not empty, the observer can establish the reproductive relationship between entities and even under parental mutational changes and the epigenetic changes and reproductive mutations in the child entity. ∎

Using relation, we now can consider the cases of entity level reproduction and Fecundity:

Case 1: Entity Level Reproduction

We consider the case where instances of individual entities can be observed as reproducing even though there might not be any observable increase in the size of the whole population. For a given simulation of the model, an observer defines the following relation:

Definition 4.

The condition in defining is used to ensure that is the immediate parent of and thus there is no intermediate ancestor between and . Notice that . Using relation, in order for the observer to establish reproduction in the model, the following axiom should be satisfied:

Axiom 8 (Reproduction).

This means, if there is reproduction in the model, then there should exists a simulation of the model, where at least one instance of reproduction is observed.

Notice that, for every , some other where has been observed to change to and to through a sequence of states would also be included in the . Therefore, in the following, we limit the attention to temporally least such parent-child pairs in :

Case 2: Population Level Reproduction - Fecundity

Though entity level reproduction is essential to be observed, for natural selection it is the population level collective reproductive behavior (fecundity), which is significant owing to the carrying capacity of the environment. Since carrying capacity is an limiting constraint on the maximum possible size of population, an observer needs to establish that there is no perpetual decline in the size of the population. In other terms for all generations, there exists a future generation that is of the same size or larger. This allows cyclic population sizes where the cycle mean grows (or stays steady) over time. Also in case of fecundity, an observer need not to observe all the parents in the same state, nor do children need to be observed in the same states of the model. Formally we require the observer to establish Fecundity by satisfying the following axiom:

Axiom 9 (Fecundity).

There exist statistically significant number of different generations of reproducing entities in temporal ordering such that where

Informally, the axiom states that for every generation of reproducing entities (), in future there exist generation of its descendant entities () such that the size of descendant generation must be equal or more than current generation. Note that the granularity of the time for determining generations is entirely dependent on the design of the model and the observation process.

Let us next prove that, above formulations successfully yield the following claim in case of our example of Langton’s CA model:

Figure 2: Self-Reproduction in Langton loops; screen shots from [4]
Lemma 2.3.

Axiom of Reproduction and the Axiom of Fecundity are satisfied by the entities and abstractions on Langton Loops defined above.

Proof.

These two axioms can be established by the observer in a specific state sequence as exemplified in Figure 2 and Figure 3 by repeatedly applying the recognition relation when entities are changing in number and states of cells (retaining the pivots) and applying the causal relation when a parent entity splits (e.g. at Time=). The relation connects the initial parent entity and the child entity at Time=.

With respect to Figure 2, an entity is identified at Time= with associated pivot. Between time steps entity changes in number and states of its cells but the pivot remains the same, hence as per the definition of , the observer can recognize the entity in these successive states. At Time=, the (parent) entity is observed to be splitting into two identical copies. One of these is again recognized as the original parent entity because of its pivot and the second entity would be claimed to be causally related with the parent entity as per the definition of . To see this, notice that the parent entity at Time= contains all the cells of the child entity appearing at Time=, which satisfies the definition of . Between time steps and both parent and child entities undergo changes in the number and states of their cells but their pivots remain fixed. Hence they can again be recognized. Finally at Time= the child entity becomes identical to the original parent entity, therefore the parent entity at Time= and the child entity at Time= are related using . The transitive closure finally give us the final descendance relationship between the parent and the child entity. ∎

Figure 3: Fecundity across generation in a population of Self Replicating Langton Loops; screen shots from [4]

2.3.2 Heredity

Heredity, yet another precondition for evolution, can in general be observed in two different levels: Syntactic level and Semantic level. On syntactic level, entity level inheritance is implied by the structural proximity between parents and their progenies ranging over several generations - though in case of continuous structural changes in the parental entities and epigenetic development in progenies, this would require an observer to establish structural similarities over a range of states as discussed earlier with the definition of relation. Also for syntactic inheritance to persist, design of the model needs to ensure that environment, which controls the reaction semantics of entities, remains approximately constant over a course of time so that structural similarities also result into continued reproductive behavior.

Difficulty arises primarily on the level of multi parental reproduction - in this situation an observer might have to stipulate some kind of gender types and might have to relax the mechanism of recognizing the parent-child relationship in a way as happens for example in case of organic life, where male-female reproductive process (often) gives birth to a progeny belonging to “only” one gender type. In such a case, for heredity, an observer need to ensure that, over a course of time all the gender types are sufficiently produced in the population.

On the other hand it is also possible to observe inheritance on the semantic level (ignoring structural differences) in terms of semantic relatedness between entities, whereby an observer can observe that progenies and their parental entities exhibit similarities in their behaviors (e.g., reproductive) under near identical set of environments. This in turn would require an observer to identify the possible sequences of observable reactions between existing entities, which appear to be yielding new set of entities (children) and in the child generation as well there exist a similar observable reproductive process, which enables the (re)production of entities. Such an observation would enable the observer to abstract the reproductive processes currently operational in the model. The inherent difficulties in this view are obvious - in essence an observer needs to abstract the reproductive semantics from the observable reactions in the model, which in turn might require non trivial inferences in absence of the knowledge of the actual design of the model.

Considering the case of real-life from an observation view point, semantic view is in fact an abstraction over all the reproductive processes existing across various species and levels including the case of bacterial organisms, where next generation of bacteria may contain a mix of genetic material from various parental bacterium of previous generation through the process of horizontal transmission. So while in case of syntactic inheritance an observer would only be able to establish inheritance across organisms belonging to same species, using semantic view, he could expand his horizon to the all organic life as a whole.

However, heredity as a mechanism of preservation of syntactic structures, appears to be crucial for those AES models where entities have very limited set of reproductive variations possible, that is, where environment supports only rare forms of entities to reproduce and any changes in the syntactic structure of these reproductive entities may result in the elimination of the reproductive capability. Real-life on earth as well as the model of the Langton loops are definitive examples where most of the variations in the genetic structure, or the loops geometry/transition rules result in the loss of reproductive/replicative capabilities.

Also heredity usually requires further mechanisms to reduce possible undoing of current mutations in future generations owing to new mutations. Therefore, in order to establish inheritance in AES models, sufficiently many generations of reproducing entities need to be observed to determine that the number of parent-child pairs where certain characteristics (both syntactic and semantic) were inherited by child entities without further mutations is significantly larger than those cases where mutations altered the characteristics in the child entities. We can express it as the following axiom:

Axiom 10 (Heredity).

Let a statistically large observed subsequence of a run :

Consider

to be the set of all parent-child pairs observed in . Again let

to be the set of those cases of reproduction where characteristics were inherited without (further) mutations. High degree of inheritance for characteristics implies that

For syntactic inheritance to be observed in a population of entities, we should have some such characteristics which satisfy this condition.

The axiom of heredity together with the axiom of preservation of reproduction under mutation is to ensure that reproductive variation is maintained and propagated across generations.

2.3.3 Natural Selection

There are several existing notions of selection in the literature on evolutionary theory [24, 54, 55, 61, 40, 33]. In the current framework natural selection is considered as a statistical inference of average reproductive success, which should be established by an observer on the population of self reproducing entities over an evolutionary time scale i.e., over statistically large number of states in a state sequence. Other notions of selection using fitness, adaptedness, or traits etc. are rather intricate in nature because these concepts are relative to the specific abstraction of “common environment” shared by entities and “the environment-entity interactions”, which are the most basic processes of selection. Nonetheless selecting appropriate generic abstraction for these from the point of view of an observation process is not so simple. Therefore following the idea from [9, page 19] that on evolutionary scale rate of reproduction is the the only attribute selected directly and characteristics affecting the rate of reproduction are selected only indirectly, a more straightforward approach is considered here whereby reproductive success is assumed to be an indicator of better adaptedness or fitness. We thus define the following (necessary) axioms for the natural selection as generally discussed in the literature [61]:

Axiom 11 (Observation on Evolutionary Time Scale).

An Observer must observe statistically significant population of different reproducing entities for statistically large number of states in a run .

Formally, for a statistically large subsequence of , , the observer determines set of reproducing entities as follows:

Axiom 12 (Sorting).

Entities in should be different with respect to characteristics in and there should exist differential rate of reproduction among these reproducing entities. Rate of reproduction for an entity is the number of child entities it reproduces before undergoing any mutations beyond observable limit. In other words, is defined as following:

Variance in the characteristics of entities in is estimated as:

Similarly variance in the rates of reproduction is estimated as:

The above two axioms though necessary are not sufficient to establish natural selection since these cannot be used as such to distinguish between natural selection with neutral selection [61]. The following axioms are therefore needed to sufficiently establish natural selection.

Axiom 13 (Heritable Variation).

There must exist variation in the inherited mutations in the population of . Formally, let

be the set of child entities carrying reproductive mutations. Let be the largest subset of consisting of those child entities which are unique with respect to at least one characteristic in , that is,

Then axiom of heritable mutation demands that

Informally, which means, a significant fraction of the population of all reproducing entities is unique in its characteristics.

Axiom 14 (Correlation).

There must be non zero correlation between heritable variation and differential rate of reproduction. Formally, a linear correlation coefficient

(also known as the Pearson s product moment correlation coefficient 

[16]) between the characteristic and the rates of reproductions for the entities in is defined as