“Complex Adaptive Systems’ Use of Fuzzy Logic”
Complex Adaptive Systems’
Use of Fuzzy Logic
A complex adaptive system is defined as “…an adaptive network exhibiting aggregate properties that emerge from the local interaction among many agents mutually constituting their own environment.”[1] This entails a number of concepts. The first is emergent properties—the development of operations on higher levels which are not immediately apparent from the given properties of the component parts of the system.[2] The second is local action—the component parts only deal with their immediate neighbors rather than calculating behavior on the basis of the system as a whole.[3] The third is a number of agents making up the component parts of the overall Complex Adaptive System—the interactions of numerous agents giving rise to the unexpected (“emergent”) properties displayed by the CAS overall behavior.[4] The fourth is adaptive behavior---the component agents’ change their behavior in response to pressures from the micro-level environment and these actions may in turn change the behavior of the entire system.[5]
The operation of a CAS is adapted to dealing with problems in real world situations, in which decisions must be made on “local, imperfect, delayed and conflicting information” under time constraints.[6] Actions under such imperfect conditions are usually undertaken through the application of heuristic devices prescribing actions .[7]
The mechanism employed by agents in solving real-world problems is called a “schema”. A schema is a formula for interpreting the agent’s environment and determining actions (if any) to be taken. The process of schema formation and application is described by Gell-Mann[8]
The schema is a dynamic mechanism for interpreting the environment of the agent. It is open to change in response to new information received from the environment—if an action taken by the agent is perceived to have had less than optimal results, then the schema is altered in order to improve the response of the agent and the consequent result. As a means of interpreting the world and actions to be taken within it, the schema necessarily exhibits certain common characteristics in all contexts.[9]
In order for the description-prescription of the schema to be usable, it must necessarily be compact—shorter than the underlying stimuli and the action(s) to be taken in response to the stimuli. Since a degree (often a great degree) of compression is required, detail must be sacrificed in the construction of the schema. Some information (often a great deal of information) must be left out in order that the schema be of a manageable and useful length. Such compression through the omission of unnecessary (“random”) information is called coarse graining, “…a level of detail up to which the system is described, with finer details being ignored.”[10] The coarse graining involved in the construction and modification of schemata necessarily (because of the localized nature of agent experience and activity) renders the form of each schema to depend upon the context within which the agent exists.[11]
The imprecision of the coarse-grained schema will thus, in most cases, necessitate an adjustment to improve its response to stimuli. Gell-Mann has described the “seasons” of a schema’s adjustment, what he calls a “CAS cycle”:
- coarse graining,
- identification of the perceived regularities,
- compression into a schema,
- variation of schemata,
- application of schemata to real world,
- consequences in the real world exerting selection pressures that affect the competition among schemata.[12]
Such imprecision in schemata renders the descriptions rendered by a schema uncertain. Because of the uncertainty inherent in a schema, the information and propositions contained within such a schema can be represented by fuzzy sets. Fuzzy logic in general and fuzzy sets specifically are a form of logic which acknowledges uncertainty and attempts to reach the best possible conclusions and courses of action within the uncertain environment. The more an object x belongs to A, the closer to 1 is its grade of membership….[13]
Fuzzy sets thus operate to deal with uncertainties within a universe of certain knowledge. Fuzzy logic does not abandon certainty by declaring all things subjective and incapable of proof. It does, however, take note of and make use of the uncertainties that lies within realms wherein there is no doubt about the outer limits of the situation.
Fuzzy logic and fuzzy sets thus attempt to reach an interim conclusion based upon acknowledged uncertainty recognizes the imperfection of human perception and reasoning. It is thus useful in projecting into the future by deliberately accepting the inherent uncertainty and making provision for “gray language to describe a gray world.”[14]
The process of reasoning using such “gray language” has been explored by fuzzy logic theorists for more than thirty years. Fuzzy logic results in usable if admittedly uncertain conclusions, a condition which is necessarily a consequence of the uncertain “gray” world with “unreliability of the information that is resident in the data.”[15]
The process of accomodating coarse-grained (uncertain) schemata to the real world through repetitive trial and error is duplicated by the fuzzy logic machine known as a fuzzy system, also called “an adaptive fuzzy system. “[16]
A fuzzy systems senses conditions at appropriate intervals (what may be an appropriate interval is context dependent) and then takes action to adjust the sensed state of affairs to an ideal state. At the next appropriate interval it senses conditions again and makes such subsequent adjustments as may be required to align that condition with the optimal situation. The level duration of corrective activity is dictated by certain principles which govern the fuzzy system. The governing principles and even the optimal situation are subject to revision on the basis of experience.[17]
The process of adjusting the principles governing the application of corrective action to produce the most desirable result is a process of competition among different principles (or schemata, to return to our original terminology.) The choice of which of several competing schemata will be tried to decide a course of action is not necessarily an all-or-nothing bivalent decision of this one but not that one.
Even among schemata, competition leavened with cooperation is sometimes both possible and advantageous. In the realm of theories, for instance, competing notions are not always mutually exclusive; sometimes a synthesis of several ideas comes much closer to the truth than any of them does individually.[18]
The competition among schemata does not take place in a vacuum. The environment within which each agent functions is made up of other CASs, each following the pattern of learning and adjusting to its environment. Accordingly, refinement of schemata by an agent must take into account the on-going refinements of the schemata of agents within its locality. Often the process can work to the mutual benefit of agents.
. It is often beneficial for complex adaptive systems to join together to form a collective entity that also functions as a complex adaptive system, for instance when individuals and firms in an economy operate under a government that regulates their behavior in order to promote values important to the community as a whole.[19]
The functions of groups of people interacting with other groups of people on whatever basis (or, for that matter individuals within a group, or Complex Adaptive Systems interacting on any level) is pictured by a concept called a “fitness landscape.” A fitness landscapes enables the comparison of one CAS with others of its kind in terms of how well in deals with its environment (including the other agents on its own and other
“…metaphorically, gaze enviously at a nearby peak, but be unable to reach it[20] because that would require crossing a valley of lower fitness.”[21]
Although the peaks and valleys (in whatever combination) are useful mental images for the purpose of picturing the movement of an agent from one level of fitness to another (and an excellent reminder of the potential dangers inherent in the abandonment of any safe haven for an uncertain future), Stuart Kauffman has given us a more formal definition:
In considering program space I defined a fitnessI shall choose to define the fitness of a Boolean network in terms of a steady target pattern of activity and inactivity among the N elements of the network. This target is the (arbitrary) goal of adaptation. Any network has a finite number of state cycle attractors. I shall define the fitness of any specific network by the match of the target pattern to the closest state on any of the net’s state cycles.[22] A perfect match yields a normalized fitness of 1.0. More generally, the fitness is the fraction of the N which match the target pattern.[23]
Construction and subsequent operation of a schema necessarily must take cognizance of the similar activity of other Complex Adaptive Systems in the agent’s locality. Agents must, if they are to be successful, make provision to adjust to the adjustments of other agents in the environment, particularly when those other adjustments will have a direct and relatively immediate impact on the original actor. Such mutually impacting activity, changing the environment for all participants is called a “coupled fitness landscape,” one in which the actions of each agent will have consequences for its neighbors. As has already been noted, such joint activity may be either good or bad for the individual, but will always have an effect.[24]”Populations climb a very complex fitness landscape which is changed and distorted in complex ways by the very act of climbing.”[25]
The reciprocal development of schemata and subsequent actions by participants in such a coupled fitness landscape has been captured in a hypothesis by Kauffman.[26]
Thus the interaction of states involves a number of players each interacting with other players (not necessarily all, however) in a number of ways. The various methods of interaction can (and probably will) involve different levels of power between the actors involved. The differing levels of power are relative to the designated actors and the levels are subject to constant change. The states and societies engage in this activity to meet some need, to respond to a perceived challenge offered by the environment. The environment will include the plans and actions of other states or societies. The problem becomes one of representing these many dynamic factors.
A neural net is a method that can be used for such representation. A neural net[27] is a diagram in which each factor is graphically represented by a figure (e.g. an ellipse) with arrows connecting all factors that exert influence on one another. Relative strengths of influence can also be indicated on the connecting arrows. Changes in the environment can be graphically introduced and the effects of the change in one aspect traced throughout the entire system. [28]
The difficulty in such a diagram is the fact that many actions and their consequences may be unreflected in the diagram (which after all suffers from the same sorts of coarse-graining limitations as any other sort of schema) and thus wholly unexpected.[29] The contingency of actions taken by human beings makes such a device useful for understanding the effects of possible actions.[30]
The unexpected result of the actions taken by individual entities within a system is the consequence of change by the individual to adapt to the stresses of its environment,[31] which changes then impact other individuals within the system. Lewin describes this ripple effect of an individual’s action moving throughout a system[32] .[33]
The process of percolation of change through an entire fitness landscape may require several looks at the FCM representing its state in order to observe the sequential changes caused by an individual actor’s change.
In the study of a human organization, such as a tribal society or a business firm, one may encounter at least three different levels of adaptation, on three different time scales. 1) On a short time scale, we may see a prevailing schema prescribing that the organization react to particular external changes in specified ways; as long as the schema is fixed, we are dealing with direct adaptation. 2) On a longer time scale, the real world consequences of a prevailing schema (in the presence of events that occur) exert selection pressures on the competition of schemata and may result in the replacement of one schema by another. 3) On a still longer time scale, we may witness the disappearance of some organizations and the survival of others, in a Darwinian process.[34]
1
[1] Cederman, Lars-Erik. Emergent Actors in World Politics. Princeton University Press: Princeton, New Jersey (1997) [hereinafter “Cederman”] p. 50
[2]Cederman p. 51
[3]Ibid.
[4]Ibid. Cederman characterizes the third property as involving a “large number of agents” which is relevant to his basic concern of the definitions of a nation and a state. The relatively small number of actors in international relations removes the operation of the adjective in this paper.
[5]Cederman p.52
[6]Tad Hogg, “The Dynamics of Complex Computational Systems” in Complexity, Entropy and the Physics of Information, Wojciech H. Zurek, ed. Addison-Wesley Publishing Company: Redwood, California (1990) [hereinafter “Physics of Information”] p.208
[7]Ibid.
- [8][E]xperience can be thought of as a set of data, usually inputoutput data, with the inputs often including system behavior and the outputs often including effects on the system.
- The system identifies perceived regularities of certain kinds in the experience, even though sometimes regularities of those kinds are overlooked or random features misidentified as regularities. The remaining information is treated as random [i.e. will not serve as a basis for decision making and/or action], and much of it often is.
- [E]xperience can be thought of as a set of data, usually inputoutput data, with the inputs often including system behavior and the outputs often including effects on the system.
- The system identifies perceived regularities of certain kinds in the experience, even though sometimes regularities of those kinds are overlooked or random features misidentified as regularities. The remaining information is treated as random [i.e. will not serve as a basis for decision making and/or action], and much of it often is. Experience is not merely recorded in a lookup table; instead, the perceived regularities are compressed into a schema. Mutation processes of various sorts give rise to rival schemata. Each schema provides, in its own way, some combination of description, prediction, and (where behavior is concerned) prescriptions for action. Those may be provided even in cases that have not been encountered before, and then not only by interpolation and extrapolation, but often by much more sophisticated extensions of experience.
- The results obtained by a schema in the real world then feed back to affect its standing with respect to the other schemata with which it is in competition
Murray Gell-Mann, “Complex Adaptive Systems” in Complexity: Metaphors, Models and Reality. George A. Cowan, David Pines and David Meltzer, eds. Addison-Wesley Publishing Company : Reading, Massachusetts (1994) [hereinafter “Gell-Mann—1”] pp. 18-19
[9]Summarization: schemata summarily represent important information about objects or events.
Internalization: schemata internalize information from the world by representing it.
Assimilation: schemata inform what they represent; they assimilate states of the environment in a manner consistent with their organization.
Accommodation: information from the environment can alter schemata if there exist adaptive pressures from different patterns of assimilation.
Inclusiveness: schemata are inclusive; they potentially represent all states of the environment that can be experienced. [NOTE: the localized nature of agents limits this inclusive characteristic to the individual experiences of the agent locally—Newman]
Diagnosticity: schemata are diagnostic; they convey information about the history that can be used to predict future states of the environment.
Summarization: schemata summarily represent important information about objects or events.
Internalization: schemata internalize information from the world by representing it.
Assimilation: schemata inform what they represent; they assimilate states of the environment in a manner consistent with their organization.
Accommodation: information from the environment can alter schemata if there exist adaptive pressures from different patterns of assimilation.
Inclusiveness: schemata are inclusive; they potentially represent all states of the environment that can be experienced. [NOTE: the localized nature of agents limits this inclusive characteristic to the individual experiences of the agent locally—Newman]
Diagnosticity: schemata are diagnostic; they convey information about the history that can be used to predict future states of the environment.Recursiveness: schemata can contain other schemata.
Generativity: schemata can be created from other schemata by modification of existing structures
Ben Martin, “The Schema” in Complexity: Metaphors, Models and Reality. George A. Cowan, David Pines and David Meltzer, eds. Addison-Wesley Publishing Company: Reading, Massachusetts (1994) [hereonafter “Metaphors and Models”] pp.276-277.
[10]Gell-Mann, Murray. The Quark and the Jaguar. W. H. Freeman and Company: New York (1994) [hereinafter “Quark”] p. 29.
[11]Quark p. 33
[12]Gell-Mann—1, p. 25
[13]Kandel, Abraham. Fuzzy Mathematical Techniques With Applications. Addison-Wesley Publishing Company: Reading, Massachusetts (1986) [hereinafter “Kandel”] p.2
[14]Kosko, Bart. Fuzzy Thinking. Hyperion Press: New York (1993) [hereinafter “Kosko”] p.96
[15]Kandel p. 36.
[16]Kosko p. 155
[17]Kosko discusses the operation of fuzzy systems in the context of electrical engineering and Buddhism at some length at Kosko pp. 156-180
[18]Quark p. 242
[19]Ibid.
[20]It is just such an inability to reach an increased level of fitness necessary to deal with increased environmental demands, for different reasons, that causes the end of Copan, the T’ang Empire, the Byzantine Empire and the Zulu Empire. See below.
[21]Lewin, Roger. Complexity: Life At The Edge of Chaos. Macmillan Publishing Company: New York. (1992) [hereinafter “Lewin”] p.57.
[22]Clearly this definition is highly context dependent. Should an environmental challenge be presented to the agent, arguably the relevant level of fitness is that obtaining at the moment of challenge rather than any average fitness or fittest state. This reality is recognized among athletes in the process of “peaking” for a competition. For an example of context-dependent failure of readiness, despite average superiority, see Dan Gable’s loss to Larry Owings in the finals of the 142 pound class at the 1970 NCAA Wrestling Championships, Gable’s only loss after 180 consecutive wins while going undefeated in high school and college.