As an effort to facilitate discussion, and to educate new members, we are constructing this glossary of  terms and concepts. Anyone is welcome to comment on the terms and concepts as well as define them, but please cite references whenever used. New entries will be added if you make sure you isolate them in the message and indicate that you want to post them.

Press HERE to add an entry.

ComplexReturn to Complexity Home Page

Algebraic Topology
The branch of topology which uses algebraic symbols for encoding spatial relations.   It includes graph theory and therefore is a basis for all of network theory.   It is also the basis for vector calculus and network theory and vector calculus are brought together in a wonderful paper by Franklin H. Branin Jr.: "The algebraic-topological basis for network analogies and the vector calculus." in Proceedings of the Symposium on Generalized Networks, Polytechnic Institute of Brooklyn, 1966.   [Don Mikulecky, Dec 21, 2000]
A process with five attributes [Source: Knuth, D.E., The Art of Computer Programming, 2nd  Edition, Vol. 1, Addison Wesley (Reading), 1973, pp. 4-6.]. (1) It terminates after a finite   number of steps. (2) Each step is unambiguously defined. (3) It has zero or more input data.  (4) It has one or more output data. (5)It must be effective; there must be a Turing-machine   equivalent. For example, there is no algorithm that solves the Busy Beaver problem [Steve    Kercel, Dec. 19, 2000].
Anticipatory System
{From taped interview: July, 1997}
RR: Well, "anticipation", the way I use the term, is a style of control. And it's based, not as cybernetic systems are on a deviation from a desired behavior, anticipation is based on having a predictive model of the system you're trying to control and using the predicted behavior to generate the control which will modify the behavior in a desired way. The combination of the free system that you want to control and a control based on such a model, predictive model... its behavior is what I call an "anticipatory system".
Other definitions:
A system in which change is due, in part, to the systems predictive models of the future.[Tom Staiger, Dec. 21, 2000].  The model used by the system is a model of its world.  It is used to govern present behavior in   "anticipation" of future events based on predictions from its model.  It is a clear example of fourth cause or final cause. [Don Mikulecky, Dec. 21, 2000, Jan. 5, 2001]
I think it is better to say in "prediction" of future events. Prediction takes place in the present and can be right or wrong. It results in anticipation when the model commutes with reality. So a system is anticipatory not only because it has a predictive model, but because it has the ability to refine that model so that it commutes somewhat reliably. Does this help avoid defining in terms of the same word? [John Kinneman, Dec. 21, 2000]
          I like this, since it is short and to the point, but I worry how "predictive models of the future" is cashed out so as to avoid             reference to anticipation. Might be impossible, but best if avoided if possible. Thoughts? [John Collier, Dec. 22, 2000]
Predict: Foretell on the basis of observation, experience, or scientific reason.
          Anticipate: To forsee and deal with in advance.  To act before.
(From Webster) .[Don Mikulecky, Jan. 3 & 5, 2001]
Aristotian Causes
Aristotle formulated a thesis that the task of science is to study the "why of things," and went on to say that there a four different ways of answering the question: Material Cause, Efficient Cause, Formal Cause, and Final Cause.
Aristotelian causality:
Four classes of answers to the question, "Why did the event occur?" The four classes are material,efficient, formal, and final. Aristotle presumed that there were no uncaused effects,   and that all effects are the result of a controlled transformation. Reductionism only admits  material, formal and efficient cause, and keeps them separated. Complex systems are driven by all four classes, and they are inseparable.[Steve Kercel, Dec. 19, 2000].
Autopoietic Unity
Something distinct from its surrounds (a unity) that has the property that its only product is itself.  There is no separation between producer and product. {Note the closeness to closed to efficient cause and the necessity of a boundary, DCM}
From Maturana & Varela, The Tree of Life (1987)  pp 48-49. [Don Mikulecky, Jan. 5, 2001].
Busy Beaver Problem:
The Busy Beaver problem is the problem of finding B(n), the maximum number of ones that a Turing machine with n states and an alphabet of {1, B} will write to an initially blank tape. Classic example of an incomputable probelm. [Source: Rosen, Kenneth, Discrete mathematics and its applications, 4th Edition, WCB/McGraw-Hill, (Boston),1999, pp. 666-674.] [Steve Kercel, Dec. 19, 2000].
Category Theory
"The first presentation of the theory of categories which was at all definitive was by Eilenberg and MacLane in 1954.  Allthough originally developed to treat certain problems in algebraic topology (giving rise to a new branch of mathematics called homological algebra) it has become, after a slow start, one of the major unifying influences in mathematics. It does not seem to be generally appreciated, though, that the theory of categories is a natural tool in any science which involves the use of model systems or abstractions of any type.  Indeed this is precisely what algebraic topology is all about: the  study of a topological or geometric object in terms of a sequence of of abstract algebraic models or images of it.  Thus, algebraic totpology occupies the same position, within mathematics itself, as does the building of mathematical models to understand physical or biological systems outside of mathematics." from Rosen's 1972 paper : "Some Relational Cell Models: The Metabolism-Repair System." Chapter 4 of Foundations of Mathematical Biology Vol. 2,
217-253. N.Y. & London, Academic Press.  [Don Mikulecky, Dec 21, 2000]
Causal Entailment:
A causal linkage. The relationship p causes q, where p and q are events in ontological reality is an instance of causal entailment. The basis for all of our "rational" thought rests on the belief that such relationships exist between events in the world.  It is not a random situation.
Cellular Automata
Church-Turing Hypothesis:
Partial recursive functions are the only computable functions, and these are the functions computable by Turing
Machines. Unproven; rests on the definition of "computable".
Strong Church-Turing: every finitely realizable physical system can be
perfectly simulated by a universal model computing machine operating by
finite means. Unproven, but based on a quantum-mechanical universal
machine, which may or may not be the same thing as a Turing Machine.

Weak Church-Turing: any effectively specifiable processes is computable by
a Turing machine.[Tom Holyrod, Dec. 22, 2000]
Closed under Efficent Cause
This closure is a unique way of ending an infinite progression of causal entailments. The machine is distinct from the organism in just this way.  In simple language, it means that organisms have their "builder" internalized and need not be made by something else.  This idea has been embodied in the cell theory for some time: Living cells come from other living cells.  It is the spirit of Maturana and Varellas "autopoietic unity". [Don Mikulecky, Dec. 20, 2000]
Comparative Compleity, Degrees of Complexity, etc.
The use of some "measure" of complexity to classify and categorize the formal systems used on the right hand side of the modeling relation or models themselves. [Don Mikulecky, Jan. 8, 2001]
{From taped interview: July, 1997}
RR: Well that's a little bit harder to describe. Complexity is really recognized by the failure of all our attempts to deal simply with these systems.  Simplicity is easier to define. I define a system to be simple if it has certain properties and anything else is a system that isn't simple; I call "complex". Simplicity is one of the things we inherited from physics; a philosophy of science: all systems can be broken up in a certain canonical set of ways and all systems are built up out of pieces that arise from such decompositions, again in a certain canonical set of ways.So, a system is simple if you can take it apart in a familiar fashion or put it together from pieces in a familiar fashion. That's what basically itmeans for a system to be simple. The whole idea behind physics was that all systems were simple. And that's the way science progresses, by
finding the right pieces and the right ways of putting the pieces back together. The lesson I bring from biology is that most systems, MOST systems are not even simple. Most systems are more like organisms. There's no one fixed set of parts into which they can all be decomposed...
Other definitions:
Complexity is the property of a real world system that is manifest in the inability of any one formalism being adequate to capture all its properties. It requires that we find distinctly different ways of interacting with systems. Distinctly different in the sense that when we make successful models, the formal systems needed to describe each distinct aspect are NOT derivable from each other.[Don Mikulecky, Dec. 19, 2000]
Computable process:
Can be replaced with an equivalent Turing machine. Congruent with sequentially causal natural systems. [Steve Kercel, Dec. 19, 2000]
I'd be careful with that last sentence. Defining computability in terms of Turing computability is equivalent to Church-Turing (delete the "Congruent with..." sentence.)[Tom Holyrod, Dec. 22, 2000]
Computational Science:
An academic discipline that assumes that processes of life and mind are computable. [Source:Dr.Istvan Berkeley, a Cognitive Scientist] [Steve Kercel, Dec. 19, 2000]
The approach to knowledge based on the idea that there is no passive way to obtain knowledge.  The observer is always an active participant.  Rosen's Modeling Relation captures this very well especially as developed in detail in Anticipatory Systems.  Subjectivity is recognized and incorporated in any knowledge seeking activity.  The "reality" we can achieved is always a construct, no matter how strongly it is grounded in sensory "data".  Data by itself, without interpretation via the modeling relation is useless.  [Don Mikulecky, Dec. 20, 2000]
Church's Thesis
The assertion that causal entailments in the external world must conform to some notion of effectiveness which can be contained within a given formal system.  It equates effectiveness with computability by insisting that everything meaningful can be reduced to syntax and dealt with algorithmicly.  Anything not conforming to this requirement is defined out of the realm of effectiveness.  The Gdel Incompleteness Theorem can be interpreted as a refutation of this thesis.  Controversy arises around this due to the restrictive interpretations placed on  Gdel's theorem by some who believe it is only justifiably applied to mathematics and not the physical world.[Don Mikulecky, Dec. 20, 2000]
The "art of steersmanship," Its principles apply whether the thing being steered is a mechanism or an organism [Source: Ashby, W.R., An Introduction to Cybernetics, Third Impression, John Wiley and Sons (New York), 1958, pp.1-5.]. Ashby's concept of cybernetic complexity requires closed or impredicative loops of causality. [Steve Kercel, Dec. 19, 2000]
In Aristotelian causality, that which is caused. Phenotypical behavior. An event resulting from the interaction of Aristotelian causes. That which is transformed from material cause. [Steve Kercel, Dec. 19, 2000]
Efficient Cause
In Aristotelian causality, the law governing the transformation of a material cause into an effect. [Steve Kercel, Dec. 19, 2000]
Not to be TOO picky, but has anybody compared the modern definitions of the words "effective" and "efficient"? 'Efficient' used to mean what "effective" still means, and "effective" doesn't carry the 'least amount of work' baggage. I think "effective cause" is clearer. [Tom Holyrod, Dec. 22, 2000]
The answer to a "why?" question about some event or entity in nature.   Answered in terms of the Aristotelian "becauses".
The study of the causal underpinings of system behavior, that is , the reason why it changes state the way it does.  In living systems, their "physiology". [Don Mikulecky, Jan. 9, 2001 {based on Rosen, Essays on Life Itself, pp 313}]
Fabricated vs. Physiology
Final Cause
In Aristotelian causality, the function for which an event occurs. Not admissable in reductionist science. [Steve Kercel, Dec. 19, 2000]
Formal Cause
In Aristotelian causality, constrains the form of the event. Typically parametric or genotypic description of phenotypic effect.  [Steve Kercel, Dec. 19, 2000]
Frozen Cell Problem
Functional Component
Functional component: [Source: Rosen, R.] The difference between the two complex systems defines the "functional component." The difference between the behaviors of the two complex systems defines the function. In a complex system, a component with a function is the unit of organization. A functional component is context dependent. It has inputs, both from the larger system of which it is a component, and the environment of the larger system. It also has outputs, both to the larger system, and the environment. If the environment, A, changes, then the function of the component, B, changes. A can typically be described by a family of mappings that carries a set (the range X, where xX) to another set (the domain Y, where yY), such that, y = a(x), or more formally, A: X Y. B can typically be described by another family of mappings that carries a set (the range U, where uU) to another set (the domain V, where vV), such that, v = b(u), or more formally, B: U V. The functionality, F, of the functional component can be described as a mapping that maps a domain set of mappings (A, where mapping aA) to a range set of mappings, (B, where mapping bB), such that b = f(a), or F: A B. [Steve Kercel, Dec. 19, 2000]
A property of a mathematical object is often called generic if a sufficiently small but otherwise arbitrary perturbation of the object produces another object with the same property.  In other words, if something is generic with respect to a given property, it is to that extent indistinguishable from any of its immediate neighbors.   When we use mathematical language to image the material word through the modeling relation we tend to deal with the nongeneric.  Conservation laws, symmetry conditions, and the like dominate mathematical physics, for example and they are strongly nongeneric.  So too are the material systems which these languages describe.
1)    It is generic for a real number to be irrational; it is nongeneric for a number to be rational, or even to be computable in the usual sense.
2)    It is generic for two lines in a plane to intersect; it is nongeneric for them to coincide or be parallel.
3)    It is generic for a differential form to be nonexact or nonintegrable.
4)    It is generic for sets to be infinite.
Generic properties are what we expect to see when we approach something in an objective, unbiased way. Nevertheless, by any objective criterion, it is the rational numbers that are rare and our predalection for them tells us more than about numbers. {from: Essays on Life itself, pp 148,175, and 326}.
Gdel's Theorm
No matter how one tries to formalize a particular part of math, syntactic truth in the formalization is narrower than the set of truths about numbers. (Rosen, 7) It shows that there are truths that mathematics cannot prove.
Halting Problem
A set of objects X is impredicative if and only if there exists a property, P(x), of an object xX, where X is the set of objects possessing property P(x).[Source: Kleene, S.C., Introduction to Metamathematics, vanNostrand , Princeton, 1950, p. 42.] In other   words, an impredicative object participates in its own definition. Impredicativity is not an          appeal to circular logic. Circular logic is an attempt in formal logic to use a proposition to prove itself. In contrast, impredicative definitions (such as the definition of the least upper bound of a bounded set of real numbers), use closed loops of inferential entailment to create an implicit definition. Impredicativities are despised by mathematicians but are tolerated because they are  indispensable in mathematics. [Steve Kercel, Dec. 19, 2000]
Incomputable process:
No Turing machine can be found that will replace the process. Congruent with closed-loop non-material causality in natural systems. [Steve Kercel, Dec. 19, 2000]
No proof of congruence is known.  Could be just computationally irreduceable.[Tom Holyrod, Dec. 22, 2000]
Inferential entailment:
Logical implication. The relationship p implies q, where p and q are propositions is an instance of inferential entailment. [Steve Kercel, Dec. 19, 2000]
The dependence of the dynamics of one system on the dynamics of another. 

The ontological relationship between function and realization that makes something complex.[John Kinneman, Jan. 8, 2001]  {Ed.'s note:  John has picked up on a point about which Rosen seems ambiguous .  This assertion is based on Rosen's identification of "complex" with the "real world" (Personal communication, 1997)  and the nesting of "organism" into the class of complex things.  "All living things are complex but all complex things are not living".  However, he also says the following in the taped interview from 1997: (JR is his daughter Judith)
"JR: Well, your goal all along has been to understand why living things are alive.

RR: Yeah. What makes something live? That is really the great mystery in all of science, in all of nature, and in all of thought.

JR: So can you sum that up by saying that the reason they are alive is because they are complex enough to be alive?

RR: I feel that complexity is almost another way of saying these systems are alive.

JR: They achieve a certain level of complexity and life is an emergent property of that complexity?

RR: Yeah. I would... That's a fair statement."

The following is from Howard H. Pattee

Questioner: What do you consider the necessary conditions for life?

Rosen: What you have to have, at least in so far as we formalize our intuitions about organisms, are modes of coupling with the world which can be regarded as metabolic; we must have inputs from the world, typical material inputs that supply energy and which provide the capacity for renewing the structure of the organism, whatever it might be. . . And you also have to have a kind of genetic apparatus, something which carries information, which tells how the parts which the metabolic part of the system produces shall be assembled, both to renew the substance of the organism, and also as a separate function, to reproduce it. I think anything that we would want to call alive would have to have at least these two basic functions: the function of metabolism and what I call the genetic function. ["A Question of Physics: Conversations in Physics and Biology" P. Buckley and D. Peat, eds., Univ. of Toronto Press, 1979, p. 89]


This is here to invite comment.  Some of the group strongly recommend that we recognize living as a property broader than those possesed by organisms.   For starters, I go to Maturana and Varela "The Tree of life": "Ontogeny is the history of structural changes in a particular living being. ... it is born in a particular place, in a medium that constitutes the ambience in which it emerges and in which it interacts.  This ambiance appears to have a structural dynamics of its own, *operationally distinct* from the living being....Between them there is a necesary structural congruence (or the unity disappears)."
Material Cause
In Aristotelian causality, the object that serves as the source of the transformation. The thing that was transformed into an effect.[Steve Kercel, Dec. 19, 2000]
Mapping of an ontologiical event to an epistemological proposition. Independent of either formal or natural system. Must be discovered independently of either. Cannot typically be determined by inverting the prediction mapping. Together with prediction, constitutes World 2 in Popper's 3-world philosophy of science. Together with prediction, the connection between the formal and natural systems in Rosen's modeling relation.[Steve Kercel, Dec. 19, 2000]
Measurement Problem
Here are a series of definitions  classified according to Richard Brodie's book: Virus of the Mind
Biological definition:
"A unit of cultural inheritance, hypothesized as analogous to the particulate gene, and as naturally selected by virtue of its 'phenotypic' consequences on its own survival and replication in the cultural environment"  Dawkins, The Extended Phenotype (1982) [Andrew Gonzalez, Jan. 4, 2001]
Dawkins, "The selfish gene" pp192-201 "The new soup is the soup of human culture. We need a name for a new replicator, a noun that conveys the idea of a unit of cultural transmission, or a unit of *immitation*"[Don Mikulecky, Jan. 4, 2001]
Psychological definition: (from Plotkin)
The unit of cultural heredity analogous to the gene.  It is the internal representation of knowledge.[Don Mikulecky, Jan. 4, 2001]
A model is a particular case of the Modeling Relation which commutes.  That is it is a successful encoding  of a percept (our mind's "image" of a causal event in the real world) into a formal system, the use of that formal system to "explain" the causal event in the real world, and a decoding back to the real world to compare and confirm the congruence between the implication in the formal system and the causal event in nature. [Don Mikulecky, Dec.21, 2000]
The study of events occurring in reality. World 1 in Popper's 3-world philosophy of science. Presumes that there is a reality in which events might occur. Comparable to natural system in Rosen modeling relation. [Steve Kercel, Dec.19, 2000]  The reason why a system has come to have the identity it does.  In simple systems this can be reduced to the systems epistemology, but not in complex systems.   In complex systems it is generally necessary to go to larger systems for the reasons for a given system's existence.[Don Mikulecky, Jan. 9, 2001 {based on Rosen, Essays on Life Itself, pp 313}]
It is a bounded process, it includes subprocesses of metabolism and repair, and is closed to efficient cause. There is no presumption of specific chemical makeup, and no limits on behavior other than the necessity of including those just listed.[Tom Staiger, Dec. 19, 2000]

A question: Is something more also needed? Does this definition of organism handle the necessary role of environment as context for organism? Or does efficient closure, by itself, still leave an organism separate from its environment? Perhaps some other entailments are necessary to capture the environmental context and meanings conferred thereby. I think closure to final cause would do that, because purpose, meaning, function (as an organism)are all context-derived, I think. I would guess that without such an organizing relationship with the larger system (environment/context) it can't exist as an organism.[John J Kineman, Dec. 19, 2000]

A comment: The efficient cause closure is a unique reqirement.The environment does not provide this.The environment provides material cause as it would for any machine.This may be, in part, contextual, but it also is true that the organism is an autopoietic unity at the same time. [Maturana & Varella]. There are all four causes involved here in complex ways. Efficient cause , by being closed, is what makes the distinction. The need for interaction with the ecosystem, etc. is a entwining of final causes which I would hardly describe as closure. [Don Mikulecky, Dec. 19,2000]

Mapping of an epistemological proposition to an ontological event. Independent of either formal or natural system. Must be discovered independently of either. Cannot typically be determined by inverting the measurement mapping. Together with measurement, constitutes World 2 in Popper's 3-world philosophy of science. Together with measurement , the connection between the formal and natural systems in Rosen's modeling relation. [Steve Kercel, Dec.  19, 2000]
Relational Biology
A term introduced by Nicholas Rashevsky in his 1954 paper : "Topology and life: In search of general mathematical principles in biology and sociology", Bull.Math. Biophys. 16: 317-348.  Later adopted by his student, Robert Rosen, who developed it into an approach to some of the most important problems in biology.  Rosen used category theory as his way of creating models of a kind never done before.  His use of M,R systems as relational models paved the way to a dichotomous distinction between machines and organisms.  [Don Mikulecky, Dec 21, 2000]
The branch of mathematics concerned with geometrical features that remain unchanged after twisting, stretching, and other deformations which do not actually cut the space.   Sometimes called "rubber sheet geometry".  It is the study of how systems are connected together and in that way it is the means for studying relationships between things.[Don Mikulecky, Dec 21, 2000]
Turing Machine
Turing Test
Universal Turing Machine

Glossary  reposted for new additions, comments, etc.  December 18, 2000.

Don Mikulecky

Complexity Research Group