The “where to look” guidance
of a formal theory is often overlooked.
Developing a formal theory is
akin to learning the rules of a game.
As we have already seen,
signal/boundary systems evolve continually.
So the “rules” must provide for origins of new signals and boundaries
and their selective modification.
Four categories that are
relevant to all signal/boundary systems: diversity, recirculation, niche, and
coevolution.
Resources can even be used
over and over again via recirculation, a mechanism that is important in
sustaining diversity in impoverished circumstances. A money-based system, if I
purchase the services of a carpenter, the money doesn’t “disappear” when I pay
the carpenter. Rather, the carpenter uses it in turn to make purchases of
lumber, food, and other things, perhaps retaining some in savings. The lumber
retailer, in turn, pays a wholesaler, and so on. A dollar of new purchase can
thus have the effect of many dollars when one looks at the system as a whole.
So it is with the nutrients in the rainforest: they pass from organism to
organism.
In the modern world, diversity
is closely related to specialization.
“Niche” is one of the most
important signal/boundary concepts and, at the same time, one of the least
understood. Because there are niches within niches, web-like hierarchies
result. Niches and hierarchies are common to all signal/boundary systems.
Inappropriate levels of
decomposition can make it difficult to answer the question of interest. Every
time we examine a signal/boundary system closely, we see coevolution as a
pervasive feature. Global trade depends directly on local specialization and
global coevolution.
The mechanisms and
interactions falling under the broad categories diversity, recirculation, niche
and hierarchy, and coevolution are central to understanding a wide range of large-scale
and small-scale signal/boundary systems.
The components of a cas are
bounded subsystems (agents) that adapt or learn when they interact. Markets,
languages, and biological cells all fit
the cas framework, the agents being, respectively, buyers and sellers,
speakers, and proteins.
Adaptive agents are defined by
an enclosing boundary that accepts some signals and ignores others, a “program”
inside the boundary for processing and sending signals, and mechanisms for
changing (adapting) this program in response to the agent’s accumulating
experience. In a market, the agents are buyers and sellers and the signals are
bids.
When a question is approached
scientifically, the assumptions (premises) must be made explicit. Moreover, the
assumptions must be formalized so that standard rules of deduction can be used
to derive answers. These rules constrain
the argument to simple obvious manipulations, much like moving pieces according
to the rules of a game. Most cas agents turn out to be persistent patterns
imposed on flows. You cannot learn the rules of chess by keeping only the
statistics of observed moves. The patterns persist because they repair
themselves when disturbed.
Newton’s equations have to be
modified when objects move near the speed of light. Both thought experiments
and computer-based models provide ways of exploring the behaviors of unfamiliar
mechanisms or combinations of mechanisms. Computer-based models have become a
major tool for investigating complex adaptive systems because they can handle
conditional “IF this happens, THEN take this action” interactions. Such
interactions pose difficulties for traditional equation-based models because,
in mathematical terms, they are nonlinear (non-additive). In this respect, the
computer models are akin to experiments. Many runs may be necessary to gain an
overview of the outcomes.
Whether the model is
equation-based or computer-based, there are three distinct approaches to
constructing a model. These approaches are distinguished by their objectives.
1.
The epicycle model and similar
models are data-driven models, often called parametric.
2.
The use of data-driven models
is the most familiar of the three modeling tactics. The objective is to
construct a model that uses data to make predictions. Theories and models for
weather prediction are good examples.
3.
An existence-proof model is
used to show that something is possible.
Typically, exploratory models
start with a designated set of mechanisms, such as the various bonds between amino
acids, with the objective of finding out what can happen when these mechanisms
interact. Exploratory models have a particularly important role when we first
begin to study a complex system, such as a cas. Exploratory models often
develop into existence-proof models. Exploratory models show the adequacy or
the inadequacy of a set of mechanisms for generating a certain range of
signal/boundary observations. An exploratory model can suggest previously
unsuspected connections and interactions. In searching for powerful models,
this temptation to inclusiveness should be resisted. A model’s clarity and
generality depend directly on how much detail has been set aside. Phrasing
questions is an art form, not a deductive process.
Requirement 1 Signals and boundaries,
and all things employing them, should be defined with the help of a formal
grammar that specifies allowable combinations of building blocks. A small set
of generators (e.g., the axioms of Euclidean geometry) is transformed by a
small set of rules (e.g., rules of deduction) into a large array of objects of
interest (e.g., theorems about geometry).
Requirement 2 Each generator
used by the signal/boundary grammar should have a location in an underlying
geometry, and combinations of generators should be mobile within that geometry.
Local conditions play an essential role in the adaptation and evolution of
complex adaptive systems.
Requirement 3 The grammar
should be capable of generating programmable agents—bounded conglomerates that
can execute arbitrary signal-processing programs. In more general terms, the
agents must be “programmable,” using the generators and rules provided by the
grammar. In short, we should define boundaries, and the resulting agents, as
signal-processing conglomerates over a fixed set of generators. Identical
agent-conglomerates can have distinct individual histories because of
conditional reactions to different local environments and agents. Behaviors
that enhance the persistence of a particular agent configuration favor the future
influence of that configuration, as well as adaptations based on that
configuration. As an agent-conglomerate becomes more complicated, its structure
may actually develop as the agent matures, as in the case of multi-celled
organisms.
Requirement 4 The
signal/boundary grammar must provide for reproduction by collecting resources,
whereby an agent-conglomerate reproduces by collecting copies of the generators
that define its structure. Agents acquire generators from the strings of elements
that pass through their outer boundary. The elements so acquired can be
recombined to form building blocks for a copy of the agent. To provide for this
process, the grammar discussed in the previous three requirements must be
extended to emphasize using acquired elements to create new boundaries and
conglomerates. When the new conglomerates combine to make a new version of the
agent, reproduction is implemented. An agent-conglomerate that does not produce
such a copy eventually dissolves (dies).
Requirement 4, in combination
with the other three requirements, characterizes a signal/boundary theory in
which signals, boundaries, and rules are represented by strings generated from
a small alphabet. Agents are defined by specifying a boundary hierarchy that
contains signals and rules at various levels. The agents are situated in an
underlying geometry, and agents are formed out of a spatially distributed
population.
The generating procedure must
provide for the generation of agents with any possible programmable behavior,
so that an agent’s behavior is not arbitrarily limited by the generating
procedure. Then the generating procedure must provide for Darwinian selection
in which agents that collect resources more rapidly than others contribute more
of their characteristics to future generations. In particular, “subroutines”
that appear in rapidly reproducing agents should appear in new combinations in
the future.
The agents are diverse rather
than standardized, and both their behavior and their structure change as they
interact.
There is no universal
competitor or global optimum in a cas. Because of these ever-increasing
possibilities for interaction, improvements are always possible. For individual
agents, the mechanisms of change aim at improvement rather than optimization. Innovation
is a regular feature of cas.
In a complex adaptive system,
equilibria are rare and temporary. Adaptation by recombination of “building
blocks” is a continuing process at all levels, and coevolution clearly has a
central role in this continuing adaptation. Because the agents incorporate
adaptive mechanisms, the systems continue to innovate. In other words, the
mechanisms of change are primarily mechanisms of exploration rather than
exploitation. Lack of a universal competitor and innovation also imply that cas
agents change their behavior on two different time scales.
Fast The immediate reaction of an agent to the signals from other agents is
conditional, taking a condition/action form.
Slow Agent adaptation in the evolutionary sense requires the appearance of new
agents in the cas.
In a cas, anticipations change
the course of the system. In a cas, the environment includes other agents, so
the internal model usually includes models of other agents.
These three characteristics—no
universal competitor, innovation, and anticipation—pose different barriers to
understanding, but they do help in setting requirements that cas agents must
meet.
Classifier systems (Lanzi
2000), provide a class of formally defined, rule-based signal-processing
systems useful for defining agents. Individual rules in the system, called
classifiers, are of the form IF (a required set of signals is present) THEN
(send an outgoing signal based on these signals). Genetic algorithms, when
applied to classifier systems, provide a computer-executable model of
populations undergoing natural selection. A fitness function assigns to each
string the number of offspring it will produce. In this “string of genes”
format, each gene on the string is drawn from a fixed set of alternatives,
called alleles.
When the genetic algorithm is
used with classifier systems, the population consists of strings representing
the rules of the system. Then the genetic algorithm, generation by generation,
searches for improvement within the set of possible rules. When rules of
above-average fitness share well-linked clusters of alleles, crossover rapidly
uncovers and exploits those possibilities. An agent is situated when it is
placed in an environment that consists of other agents and “inanimate” objects
such as obstacles and patches of resources. For cas signal-processing agents,
we will invoke a set of detectors for determining the changing features of the
surrounding environment.
Individual rules in a classifier
system, like the individual instructions for a computer program, are generally
simple and easy to understand. However, a system with many interacting rules is
as difficult to analyze as a computer program with many subroutines. Interestingly,
most reactions between proteins are determined by relatively small
subsequences, called active sites, within the string defining the protein. Uniform
distribution corresponds to a classifier-system signal list making all signals
available to all rules.
The state of a game at any
time is the placement of game pieces on the board. Just as the play of the game
amounts to moving from board configuration to board configuration according to
the rules of the game, so the dynamics of a system amounts to moving from state
to state according to the laws of the system. In most games, the play from any
board configuration B onward doesn’t depend on the moves leading to
configuration B. What is “legal” from configuration B onward depends only on B,
not on how you got to B. The same is true in physics: knowing the state of a
system is sufficient for determining future possibilities.
Network theory, more formally
called graph theory, offers a general way of describing arbitrary cas
interactions by extracting the pattern from the details. Thus, by operating on
tags in signals and conditions used by established signal/boundary
interactions, cross-breeding and other adaptive mechanisms can produce new,
redirected signals and conditions. In order to understand signal/boundary
systems we must consider the formation of boundaries and signals, not just
their existence.
In complex adaptive systems in
general, and in signal/boundary systems in particular, adaptive mechanisms
mediate the formation of new structures, often adding specialized versions of
existing components to an agent. As a result, flows of resources and signals in
the agent pass through increasing numbers of stages, each of which encompasses
some special activity of the original flow.
Nevertheless, Smith’s pin
factory illustrates three major characteristics of signal/boundary
interactions: that new boundaries distinguish the specialists, that resources
flow from specialist to specialist, and that signals synchronize interactions.
Building block is a generator
if it is immutable over the span of its existence. It may be copied and it may
be combined with other generators, but it has a fixed set of “connectors”
subject to a fixed set of connection rules, as in the case of chemical elements.
Removal of a generator is often disastrous, as in the removal of a single
instruction from a computer program. A building block is a conglomerate if it
can grow and “fission” into additional conglomerates related to the “parent”
conglomerate. In most cases, a conglomerate can be defined in terms of an
interconnected set of generators. However, a conglomerate isn’t of much
interest unless it persists long enough to serve as a building block for more
complex structures.
Conglomerates can usually be
described in terms of changing combinations of building blocks drawn from the
next lower level of description—neurons in the case of the brain, reactions in
the case of the cell, and so on. In complex adaptive systems, emergent
properties often occur when coevolving signals and boundaries generate new
levels of organization. Newer signals and boundaries can then emerge from
combinations of building blocks at this new level of organization.
In particular, Darwinian
selection, acting on a signal/boundary system, can ensure that only building
blocks residing in fit structures survive to be tested in other structures. When
a building block is tested and survives in a variety of contexts, its
usefulness becomes well confirmed in a statistical sense. The procedures examined
here, rather than treating individual building blocks independently of one
another, place strong emphasis on context provided by other building blocks. Ignoring
context would be tantamount to ignoring the conditional interactions that
highlight signal/boundary interactions.
Recombination occurs because
the chromosomes that control development pair up, and one chromosome within a
pair physically crosses over the other, exchanging segments on one side of the
crossover point. Recombination is a strong method for introducing plausible new
rules, but those rules are hypotheses that must be tested and selected.
Darwinian selection has been suggested, but how do we implement it in this
rule-based approach? There are three requirements:
1.
There must be a population so
that the rules can compete.
2.
There must be a comparison of
rules that results in a rating that indicates the rule’s performance within the
population.
3.
There must be a selection
process, so that rules that are successful in the competition can be favored.
Rules that are stronger
(better confirmed as hypotheses) should contribute more to the evolving
organization of the signal/boundary system. Recombination and reproduction are
then applied to the rules to explore further possibilities (hypotheses). A
boundary can be copied only if sufficient elements are collected to make a copy
of the corresponding entry and exit tags, and an agent’s rate of reproduction
is determined by its ability to collect relevant elements. The agent’s internal
compartments will be treated as a collection of tag-based urns. This treatment
yields the constrained diffusion typical of coupled reactions mediated by
semi-permeable membranes.
Increased throughput, then,
lets the agent collect the resources needed to copy its structures more
rapidly. Accordingly, increased throughput is linked to the agent’s Darwinian
fitness—the agent’s contribution to future generations. Cash plays a similar
role in an economic niche, and its passage through a chain of buyers and
sellers gives rise to the multiplier effect.
More generally, in a network,
a niche is a community having a great number of internal connections but
relatively fewer external connections, making it possible for the community to
exhibit partially autonomous behavior. In each case, recirculation of resources
means that activity in the niche cannot be determined by simply adding up the
activities of the different agents occupying the niche.
Language acquisition, looked
at in this way, is a system that adds generative rules as the learning agent’s
experience refines its levels of consciousness. The result is an acquired set
of generators (vocabulary) and a progressively refined set of generating rules
(a grammar).
The objective of a dgs
(dynamic generated system) framework for studying signal/boundary systems is
similar. Find dgs generators and generating rules that open the way to modeling
arbitrary signal/boundary systems, with particular emphasis on the coevolution
of signals and boundaries.
It is helpful to re-examine the
four interlocking models of signal/boundary interactions keeping the points
just made in mind. The italicized words in the following four descriptions
refer to the five requirements—building blocks, spatial distribution,
programmability, implicit fitness, and internal models:
• Classifier-system models specify the performance/ credit
assignment/ rule discovery activities of an adaptive agent in a complex
adaptive system. Because a classifier system (cfs) uses a population of rules
(strings), rule discovery can be implemented with a genetic algorithm, using
recombination to generate new tags. Because the rules can be sequenced, cfs
models can implement any computer-executable model, supplying programmability. In
particular, sets of rules can act as internal models—subroutines that allow an
agent to look ahead and anticipate consequences of current actions.
• Reaction nets model the interactions of adaptive agents in
collecting and processing resources. The flows of resources in a population of
agents are handily modeled with reaction nets. When tags are used to define the
net, progressive changes in the net can also be generated.
• Tagged urns specify the effect of boundaries on the throughput of
coupled reactions. Boundaries can be modified by changing the entry and exit
tags, thus making it possible to study the coevolution of signals and
boundaries.
• Echo models situate agents in a geometry and specify an agent’s
structure in terms of elements that can be obtained by collecting resources
available at sites in the geometry. By requiring an agent to collect the
resources necessary for replication, an implicit fitness is introduced—a
fitness that can change according to the context provided by other agents
(e.g., resource exchanges).
With this provision, the
concept of niche is explicitly defined in terms of communities of agents.
This model of ontogeny uses
the dgs format to reformulate five biological mechanisms discussed in earlier
chapters: gene (de-)repression, semi-permeable membranes, cell adhesion,
catalyzed reactions, and replication through reassembly.
All complex adaptive systems
exhibit obvious internal boundaries that divide the cas into a diverse array of
semi-autonomous subsystems called agents. Each agent has a “program” that
guides its interactions with other agents and other parts of its environment. To
provide a common framework for the programs used by different kinds of cas
agents, the book concentrates on classifier systems for the following reasons:
• Classifier systems use
conditional (If/Then) signal-processing rules, thus capturing the conditional
interactions characteristic of signal/boundary systems.
• A classifier system executes
many signal-processing rules simultaneously without incurring problems of consistency,
thus capturing the simultaneous interactions typical of complex adaptive
systems.
• Classifier systems are
computationally complete, so any computational procedure (e.g., any agent
strategy) can be presented by a set of classifier-system signal-processing
rules
• Tags in a classifier system
play the role of active sites, motifs, and the like in different complex
adaptive systems, allowing direct comparisons of tag-mediated signal/boundary
networks in different complex adaptive systems.
• Classifier systems are
specifically designed for use with genetic algorithms, so that the coevolution
of signals and boundaries can be generated and studied.
• Compartmented populations
Using probabilities to designate the concentrations of different
colors of balls in an urn allows calculations relevant to the multiplicity of
signals in a typical cas compartment; urns arrayed hierarchically mimic a
hierarchical arrangement of cas compartments.
• Diffusion through
semi-permeable membranes Moving balls from urn to urn, subject to
entry and exit conditions, models the effect of diffusion through
semi-permeable membranes and the counterparts of constrained diffusion in other
signal/boundary systems.
• Conditional action
Using conditional probabilities to represent reaction rules paves the
road for using Markov processes to study signal processing in hierarchical
arrays of semi-permeable membranes.
• Simultaneous processing
Repeated random draws of pairs of colored balls model the effects
of simultaneous pairwise reactions occurring in production lines, recycling,
niches, and the like.
•
Coevolution Using a genetic algorithm to modify signal tags
and entry and exit conditions offers a direct approach to adaptive,
coevolutionary changes that modify boundary permeability
Because complex adaptive
systems are organized around populations, both offspring options can be
explored without abandoning advantages already attained by the parents. As a
result, succeeding generations make greater use of tags that offer additional
advantages. In other words, progressive cross-breeding explores the levels
between “craftsman” generalists and “production-line” specialists in the milieu
of signals and resources available to the agents. A well-defined, widely
applicable concept of niche should encompass common features of the different
versions:
Multiplier effects
Resources don’t disappear on consumption; they are transformed in passing
from agent to agent. As a result, an injection of an additional resource (say,
from outside the niche) propagates from agent to agent, multiplying the effects
of the injection. (The economic version of this effect was proved by
Samuelson.)
Recycling Agents
are distributed spatially, and they interact in tangled loops of conditional
responses and resource-passing (e.g., phenotypic plasticity in ecosystems
(Levin 1999)).
Persistence Often
patterns of interaction, imposed on a “flow” of ever-changing agents, persist
well beyond the lifetimes of individual agents. These persistent patterns act
as niches within which diverse agents can co-exist. The introduction of a new
kind of agent to a niche—say, from some other niche (e.g., an invasive species)
or by recombination of tags (e.g., a new flu virus)—can cause a torrent of
further innovations exploiting the new possibilities.
A niche is a diverse array of
agents that regularly exchange resources and depend on that exchange for
continued existence. The conditional actions of the agents lead to non-additive
effects that cannot be usefully averaged, so a purely statistical approach
isn’t likely to provide answers to this question. Statistical approaches wind
up in the same cul-de-sac as statistical approaches to understanding computer
programming.
The “trends” suggested by a
series of “snapshots” based on an average over agents’ activities rarely give
reliable predictions or opportunities for control—instead of “clearing” of a
market, we get “bubbles” and “crashes.” A more fruitful approach to questions
of this kind requires an examination of the coevolution of situated
agents—agents that have locations in both a geographic sense and a network
sense. The “bandits” offer a boundary-oriented setting for examining situated
agents.
With this provision, a
“generalist” agent accesses a broad array of urns but often winds up in queues
with low payoff per agent because of crowding, whereas a “specialist” agent
accesses a queue that is less crowded but more difficult to locate. As we have
seen, widely different agent-based signal/boundary systems, ranging from
biological cells to governments, exhibit the same general features, notably the
following:
·
semi-autonomous subsystems
(agents)
·
hierarchical organization
(agents composed of agents)
·
sustained diversity (agents exploiting
different strategies)
·
extensive recycling of
resources (agents converting resources and passing them on
However, most complex adaptive
systems don’t settle down. In considering mathematical approaches to
signal/boundary systems, it is important to recognize that agents situated in a
physical environment do encounter regularities. That is, the environments
encountered are a highly constrained subset of the set of all conceivable
environments. It is a bad mistake to assume that nearly all conceivable
environments occur with non-zero probability. In realistic environments,
regularities offer exploitable opportunities for the formation of niches
involving persistent, interdependent interactions of diverse agents.
Complex adaptive
systems—cells, rainforests, markets, language, and the Internet, to name a
few—are characterized by complex, ever-changing interactions of signals and
boundaries.
This book provides exploratory
models for examining the adaptations generated by signal/boundary coevolution. The
exploration centers on two kinds of models: adaptive agents and tagged urns.
Adaptive agents (defined by
signal-processing rules that provide for parallel processing and adaptation
under a genetic algorithm) model the evolution of hierarchical systems that
employ many signals (resources) interacting simultaneously
Tagged urns (modifications of
the urns used in probability theory) use entry and exit conditions to control
the flow of balls (signals) between urns, thus providing explicit formal models
of semi-permeable boundaries.
In these models, both signals
and boundaries are constructed from building blocks provided by tags and parts
of tags—the counterparts of active sites, motifs, and message headers. The
framework of dynamic generated systems combines adaptive-agent models and
tagged-urn models to provide a precise formalism for exploring the evolution of
signal/boundary systems.
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