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MyWikiBiz, Author Your Legacy — Tuesday October 07, 2008

Author's Note. The initial portion of this essay is the "Interest Statement" that I submitted as a part of my application to graduate school in the Systems Engineering doctoral program at Oakland University, Rochester, Michigan in September 1992.

Systems Engineering : Interest Statement
Jon Awbrey, September 1, 1992

It seemed useful, as a way of sharpening my focus on goals ahead, to write up an extended statement of current research interests and directions. I realize that many features of this sketch are likely to change as details are clarified and as new experience is gained. As an alternative to the longer essay, an abstract is provided as a minimal statement.

Abstract

In briefest terms my project is to develop mutual applications of systems theory and artificial intelligence to each other. In the current phase of investigation I am taking a standpoint in systems theory, working to extend its concepts and methods to cover increasingly interesting classes of intelligent systems. A natural side-project is directed toward improving the economy of effort by unifying a selection of tools being used in these two fields. My instrumental focus is placed on integrating the methods of differential geometry with the techniques of logic programming. I will attempt to embody this project in the form of computer-implemented connections between geometric dynamics and logical AI, and I will measure its success by the extent and usefulness of this realization.

Description of Current and Proposed Work

I intend to focus primarily on the research area of artificial intelligence. In my work of the past few years I have sought to apply the framework of systems theory to the problems of AI. I believe that viewing intelligent systems as dynamic systems can provide a unifying perspective for the array of problems and methods that currently constitutes the field of AI.

The return benefits to systems theory would be equally valuable, enabling the implementation of more intelligent software for the study of complex systems. The engineering of this software could extend work already begun in simulation modeling (Widman, Loparo, & Nielsen, 1989), (Yip, 1991), nonlinear dynamics and chaos (Rietman, 1989), (Tufillaro, Abbott, & Reilly, 1992), and expert systems (Bratko, Mozetic, & Lavrac, 1989), with increasing capabilities for qualitative inference about complex systems and for intelligent navigation of dynamic manifolds (Weld & de Kleer, 1990).

1. Background

In my aim to connect the enterprises of systems theory and artificial intelligence I recognize the following facts. Although the control systems approach was a prevailing one in the early years of cybernetics and important tributaries of AI have sprung from its sources, e.g. (Ashby, 1956), (Arbib, 1964, '72, '87, '89), (Albus, 1981), the two disciplines have been pursuing their separate evolutions for many years now. The intended scope of AI, overly ambitious or otherwise, forced it to break free of early bonds, shifting for itself beyond the orbit of its initial paradigms and the cases that conditioned its origin.

A sample of materials from transition phases of AI's developmental crises may be found in (Shannon & McCarthy, 1956), (Wiener, 1961, 1964), (Sayre & Crosson, 1963), (Young, 1964, 1978), (McCulloch, 1965), (Cherry, 1966), (MacKay, 1969). Any project to resolder the spun-off domains of AI and systems theory will probably resort to a similar flux. In the course of this investigation it was surprising at first to see these old issues rise again, but the shock has turned to recognition. A motion to reinstate thought with action, to amalgamate intelligence with dynamics in the medium of a computational reconstruction, will naturally revert to the neighborhoods of former insights and ride the transits of formative ideas. It is only to be expected that this essay will run across many of the most intersected rights-of-way, if not traveling down and tripping over the very same ruts, then very likely switching onto any number of parallel tracks.

Informed observers may see good reasons for maintaining the separation of perspectives between AI and systems theory. However, the proposition that these two realms share a common fund of theory and practice, not only historically but one that demands and deserves a future development, is an assertion that motivates my efforts here. Consequently, I thought that a justification of my objectives might be warranted. In light of these facts I have written up this extended rationale and informal review of literature, in hopes of making a plausible case for attempting this work.

Rudiments and Horizons

There are harvests of complexity which sprout from the earliest elements and the simplest levels of the discussion that follows. I will try to clarify a few of these issues in the process of fixing terminology. This may create an impression of making much ado about nothing, but it is a good idea in computational modeling to forge connections between the complex, the subtle, and the simple -- even to the point of forcing things a bit. Further, I will use this space to profile the character and the consistency of the grounds being tended by systems theory and AI. Finally, I will let myself be free to mention features of this work that connect with the broader horizons of human cultivation. Although these concerns are properly outside the range of my next few steps, I believe that it is important to be aware of our bearings: to know what our practice depends upon, to think what our activity impacts upon.

1.1. Topos : Rudiments and Immediate Resources

This inquiry is guided by two questions that express themselves in many different guises. In their most laconic and provocative style, self-referent but not purely so, they typically bring a person to ask:

• Why am I asking this question?

• How will I answer this question?

Cast in with a pool of other questions these two often act as efficient catalysts of the inquiry process, precipitating and organizing what results. Expanded into general terms these queries become tantamount to asking:

• What accumulated funds and immediate series of experiences lead up to the moment of surprise that causes the asking of a question?

• What operational resources and planned sequences of actions lead on to the moment of solution that allows the ending of a problem?

Phrased in systematic terms, they ask yet again:

• What capacity enables a system to exist in states of question?

• What competence enables a system to exit from its problem states?

1.1.1. Systematic Inquiry

In their underlying form and tone these questions sound a familiar tune. Their basic tenor was brought to a pitch of perfection by Immanuel Kant, in a canon of inquiry that exceeds my present range. Luckily, my immediate aim is much more limited and concrete. For the present it is only required to ask: How are systematic inquiry and knowledge possible? That is, how are inquiry and knowledge to be understood and implemented as functions of systems and how ought they be investigated by systems theory? In short: How can systems have knowledge as a goal? This effort is constrained to the subject of systems and the frame of systems theory. It will attempt to give system-theoretic analyses of concepts and capacities that can be recognized as primitive archetypes, at least, of those that AI research pursues with avid interest and aspires one day to more fully capture. By limiting questions about the possibility of inquiry and knowledge to the subject and scope of systems theory there may be reason to hope for a measure of practical success.

Kant's challenge is this: To say precisely how it is possible, in procedural terms, for contingent beings and empirical creatures, physically embodied and even engineered systems, to move toward or synthetically acquire forms of knowledge with an a priori character, that is, declarative statements with a global application to all of the situations that these agents might pass through. It is not feasible within the scope of systems theory and engineered systems to deal with the larger question: Whether these forms of knowledge are somehow necessary laws, applying to all conceivable systems and universes. But it does seem reasonable to ask how a system's trajectory might intersect with states whose associated knowledge components have a wider application to the system's manifold as a whole.

1.1.2. Intelligence, Knowledge, Execution

Intelligence, for my purposes, is characterized as a technical ability of choice in a situation as represented. It is the ability to pick out a line on a map, to find a series of middle terms making connections between represented positions. In the situation that commonly calls it out, intelligence is faced with two representations of position. This pair of pointers to points on a map are typically interpreted as indices of current and desired positions. The two images are symbols or analogues of the actual site and the intended goal of a system. They themselves exist in a space that shadows the dynamic reality of the agent involved. But the dynamic reality of the intelligent agent forms a manifold of states that subsists beneath its experience and becomes manifest only gradually and partially in the observations of that agent. It is among the states of this basic manifold that all the real sites and goals of the agent are located.

The concept of intelligence laid out here has been abstracted from two capacities that it both requires and supports: knowledge and execution. Knowledge is a fund of available representations, a glove-box full of maps. Execution is an array of possible actions and the power of performing them, an executive ability that directs motor responses in accord with the line that is picked out on the map. To continue the metaphor, execution is associated with the driving-gloves, which must be sorted out from the jumble of maps and used to get a grip on the mechanisms of performance and control that are capable of serving in order to actualize choices.

1.1.2.1. Vector Field and Control System

Dynamically, as in a control system, intelligence is a decision process that selects an indicator of a tangent vector to follow at a point or a descriptor of a corresponding operator to apply at a point. The pointwise indicators or descriptors can be any relevant signs or symbolic expressions: names, code numbers, address pointers, or quoted phrases. A "vector field" attaches to each point of phase space a single tangent vector or differential operator. The "control system" is viewed as a ready generalization of a vector field, in which whole sets of tangent vectors or differential operators are attached to each point of phase space. The "strategy" or "policy problem" of a controller is to pick out one of these vectors to actualize at each point in accord with reaching a given target or satisfying a given property. An individual control system is specified by information attached to each dynamic point that defines a subset of the tangent space at that point. This pointwise defined subset is called "the indicatrix of permissible velocities" by (Arnold, 1986, chapt. 11).

In the usage needed for combining AI and control systems to obtain autonomous intelligent systems, it is important to recognize that the pointwise indicators and descriptors must eventually have the character of symbolic expressions existing in a language of non-trivial complexity. Relating to this purpose, it does not really matter if their information is viewed as represented in the states of discrete machines or in the states of physical systems to which real and complex valued measurements are attributed. What makes the system of indications and descriptions into a language is that its elements obey specific sets of axioms that come to be recognized as characterizing interesting classes of symbol systems. Later on I will indicate one very broad definition of signs and symbol systems that I favor. I find that this conception of signs and languages equips the discussion of intelligent systems with an indispensable handle on the levels of complexity that arise in their description, analysis, and clarification.

1.1.2.2. Fields of Information and Knowledge

Successive extensions of the vector field concept can be achieved by generalizing the form of pointwise information defined on a phase space. A subset of a tangent space at a point can be viewed as a boolean-valued function there, and as such can be generalized to a probability distribution that is defined on the tangent space at that point. This type of probabilistic vector field or "information field" founds the subject of stochastic differential geometry and its associated dynamic systems. An alternate development in this spirit might embody pointwise information about tangent vectors in the form of linguistic expressions and ultimately in knowledge bases with the character of empirical summaries or logical theories attached to each point of a phase space.

It is convenient to bring together under the heading of a "knowledge field" any form of pointwise information, symbolic or numerical, concrete or theoretical, that constrains the set of pointwise tangent vectors defined on a phase space. In computational settings this information can be procedural and declarative program code augmented by statistical and qualitative data. In computing applications a knowledge field acquires an aptly suggestive visual image: bits and pieces of code and data elements sprinkled on a dynamic surface, like bread crumbs to be followed through a forest. The rewards and dangers of so literally a "distributed" manner of information storage are extremely well-documented (Hansel & Gretel, n.d.), but there are times when it provides the only means available.

1.1.2.3. The Trees, The Forest

A sticking point of the whole discussion has just been reached. In the idyllic setting of a knowledge field the question of systematic inquiry takes on the following form:

What piece of code should be followed in order to discover that code?

It is a classic catch, whose pattern was traced out long ago in the paradox of Plato's Meno. Discussion of this dialogue and of the task it sets for AI, cognitive science, education, including the design of intelligent tutoring systems, can be found in (H. Gardner, 1985), (Chomsky, 1965, '72, '75, '80, '86), (Fodor, 1975, 1983), (Piattelli-Palmarini, 1980), and in (Collins & Stevens, 1991). Though it appears to mask a legion of diversions, this text will present itself at least twice more in the current engagement, both on the horizon and at the gates of the project to fathom and to build intelligent systems. Therefore, it is worth recalling how this inquiry begins. The interlocutor Meno asks:

Can you tell me, Socrates, whether virtue can be taught, or is acquired by practice, not teaching? Or if neither by practice nor by learning, whether it comes to mankind by nature or in some other way? (Plato, Meno, p. 265).

Whether the word "virtue" (arete) is interpreted to mean virtuosity in some special skill or a more general excellence of conduct, it is evidently easy, in the understandable rush to "knowledge", to forget or to ignore what the primary subject of this dialogue is. Only when the difficulties of the original question, whether virtue is teachable, have been moderated by a tentative analysis does knowledge itself become a topic of the conversation. This hypothetical mediation of the problem takes the following tack: If virtue were a kind of knowledge, and if every kind of knowledge could be taught, would it not follow that virtue could be taught?

For the present purpose, it should be recognized that this "trial factorization" of a problem space or phenomenal field is an important intellectual act in itself, one that deserves attention in the effort to understand the competencies that support intelligent functioning. It is a good question to ask just what sort of reasoning processes might be involved in the ability to find such a middle term, as is served by "knowledge" in the example at hand. Generally speaking, interest will reside in a whole system of middle terms, which might be called a "medium" of the problem domain or the field of phenomena. This usage makes plain the circumstance that the very recognition and expression of a problem or phenomenon is already contingent upon and complicit with a particular set of hypotheses that will inform the direction of its resolution or explanation.

One of the chief theoretical difficulties that obstructs the unification of logic and dynamics in the study of intelligent systems can be seen in relation to this question of how an intelligent agent might generate tentative but plausible analyses of problems that confront it. As described here, this requires a capacity for identifying middle grounds that ameliorate or mollify a problem. This facile ability does not render any kind of demonstrative argument to be trusted in the end and for all time, but is a temporizing measure, a way of locating test media and of trying cases in the media selected. It is easy to criticize such practices, to say that every argument should be finally cast into a deductively canonized form, harder to figure out how to live in the mean time without using such half-measures of reasoning. There is a line of thinking, extending from this reference point in Plato through a glancing remark by Aristotle to the notice of C.S. Peirce, which holds that the form of reasoning required to accomplish this feat is neither inductive nor deductive and reduces to no combination of the two, but is an independent type.

Aristotle called this form of reasoning "apagogy" (Prior Analytics, 2.25) and it was variously translated throughout the Middle Ages as "reduction" or "abduction". The sense of "reduction" here is just that by which one question or problem is said to reduce to another, as in the AI strategy of goal reduction. Abductive reasoning is also involved in the initial creation or apt generation of hypotheses, as in diagnostic reasoning. Thus, it is natural that abductive reasoning has periodically become a topic of interest in AI and cognitive modeling, especially in the effort to build expert systems that simulate and assist diagnosis, whether in human medicine, auto mechanics, or electronic trouble-shooting. Recent explorations in this vein are exemplified by (Peng & Reggia, 1990) and (O'Rorke, 1990).

But there is another reason why the factorization problem presents an especially acute obstacle to progress in the system-theoretic approach to AI. When the states of a system are viewed as a manifold it is usual to imagine that everything factors nicely into a base manifold and a remainder. Smooth surfaces come to mind, a single clear picture of a system that is immanently good for all time. But this is how an outside observer might see it, not how it appears to the inquiring system that is located in a single point and has to discover, starting from there, the most fitting description of its own space. The proper division of a state vector into basic and derivative factors is itself an item of knowledge to be discovered. It constitutes a piece of interpretive knowledge that has a large part in determining exactly how an agent behaves. The tentative hypotheses that an agent spins out with respect to this issue will themselves need to be accommodated in a component of free space that is well under control. Without a stable theater of action for entertaining hypotheses, an agent finds it difficult to sustain interest in the kinds of speculative bets that are required to fund a complex inquiry.

States of information with respect to the placement of this fret or fulcrum can vary with time. Indeed, it is a goal of the knowledge directed system to leverage this chordal node toward optimal possibilities, and this normally requires a continuing interplay of experimental variations with attunement to the results. Therefore it seems necessary to develop a view of manifolds in which the location or depth of the primary division that is effective in explaining behavior can vary from moment to moment. The total phenomenal state of a system is its most fundamental reality, but the way in which these states are connected to make a space, with information that metes out distances, portrays curvatures, and binds fibers into bundles — all this is an illusion projected onto the mist of individual states from items of code in the knowledge component of the current state.

The mathematical and computational tools needed to implement such a perspective goes beyond the understanding of systems and their spaces that I currently have in my command. It is considered bad form for a workman to blame his tools, but in practical terms there continues to be room for better design. The languages and media that are made available do, indeed, make some things easier to see, to say, and to do than others, whether it is English, Pascal (Wirth, 1976), or Hopi (Whorf, 1956) that is being spoken. A persistent attention to this pragmatic factor in epistemology will be necessary to implement the brands of knowledge-directed systems whose intelligence can function in real time. To provide a computational language that can help to clarify these problems is one of the chief theoretical tasks that I see for myself in the work ahead.

A system moving through a knowledge field would ideally be equipped with a strategy for discovering the structure of that field to the greatest extent possible. That ideal strategy is a piece of knowledge, a segment of code existing in the knowledge space of every point that has this option within its potential. Does discovery mark only a different awareness of something that already exists, a changed attitude toward a piece of knowledge already possessed? Or can it be something more substantial? Are genuine invention and proper extensions of the shared code possible? Can intelligent systems acquire pieces of knowledge that are not already in their possession, or in their potential to know?

If a piece of code is near at hand, within a small neighborhood of a system's place in a knowledge field, then it is easy to see a relationship between adherence and discovery. It is possible to picture how crumbs of code could be traced back, accumulated, and gradually reassembled into whole slices of the desired program. But what if the required code is more distant? If a system is observed in fact to drift toward increasing states of knowledge, does its disposition toward knowledge as a goal need to be explained by some inherent attraction of knowledge? Do potential fields and propagating influences have to be imagined in order to explain the apparent action at a distance? Do massive bodies of knowledge then naturally form, and eventually come to dominate whole knowledge fields? Are some bodies of knowledge intrinsically more attractive than others? Can inquiries get so serious that they start to radiate gravity?

Questions like these are only ways of probing the range of possible systems that are implied by the definition of a knowledge field. What abstract possibility best describes a given concrete system is a separate, empirical question. With luck, the human situation will be found among the reasonably learnable universes, but before that hope can be evaluated a lot remains to be discovered about what, in fact, may be learnable and reasonable.

1.1.3. Reality and Representation

A sidelight that arose in the characterization of intelligence is recapitulated here. Beginning with experience described in phenomenal terms, the possibility of objective knowledge appears to depend on a certain factorization or decomposition of the total manifold of experience into a pair of factors: a fundamental, original, objective, or base factor and a representational, derivative, subjective, or free factor. To anticipate language that will be settled on later, the total manifold of phenomenal experience is said to factor into a bundle of fibers. The bundle structure corresponds to the base factor and the fiber structure corresponds to the free factor of the decomposition. Fundamental definitions and theorems with respect to fiber bundles are given in (Auslander & MacKenzie, ch. 9). Discussions of fiber bundles in physical settings are found in (Burke, p. 84-108) and (Schutz, 1980). Concepts of differential geometry directed toward applications in control engineering are treated in (Doolin & Martin, ch. 8). An ongoing project in AI that uses simple aspects of fiber methods to build cognitive models of physics comprehension is described in (Bundy & Byrd, 1983).

An exorbitant number of words has just been wrapped around the seemingly obvious and innocuous distinction between a reality and a representation. Of course, whole books have been written on the subjects of reality and representation, though not necessarily in that order (Putnam, 1988). The topic is especially debated in the philosophy of science, e.g. (Duhem, 1914), (Russell, 1956), (Van Fraassen, 1980), (Hacking, 1983), (Salmon, 1990), and various individual essays in (Quine, 1960, '69, '74, '76, '80, '81). Much of what is said there about the relation of theories to realities has a bearing on the relation of simulation models and AI representations to their underlying realities (Halpern, 1986), (Ginsberg, 1987). A useful historical perspective on the problem of scientific knowledge in relation to the world is supplied by (Losee, 1980). The history of an alternative tradition is treated in (Prasad, 1958).

These questions go back to the beginnings of philosophy. Plato's dialogue The Sophist is one early inquiry that has a special relevance, in its substance and method, for the current context. There is a certain type of recursive and paradigmatic character to the strategy of its analysis. In its quest after the nature of the true philosopher it proceeds in manner that strikingly foreshadows modern debates about the Turing test. What spirit can winnow the grain from the chaff, what screen can sift the fine from the coarse, what threshold can keep the spirit in the letter? These may indeed have been our kind's earliest decision problems. Modern commentary on this dialogue and the context of its times may be found in (Plato/Benardete, 1986), (Kerferd, 1981), (Rosen, 1983), and (Lanigan, 1986).

There is a reason for the seeming excess of labels and packaging invested around this distinction between reality and representation. The razor that would function as advertised and earn its patent to separate sharp realities from fuzzy impressions is not a toy to be wielded lightly. Until it is certain just where to cut, other means may be required to manage, organize, store, and control the fringes of a systematic imagination. It is my hope to turn this measure of redundancy to an informative purpose later on when the distinction begins to seem both more elusive and more vital. An uncertainty in this dimension can become positively noisy in its interference with the observation and communication of static situations and potentially noxious in its undermining of a system's capacity for dynamic control. The difficulty to be faced is this: There can be genuine questions about what actually forms the best factorization of the total manifold into a base space and a remainder.

The most fitting factorization is not necessarily given in advance, though any number of possibilities may be tried out initially. The most suitable distinction between phenomenal reality and epiphenomenal representation can be a matter determined by empirical or pragmatic factors. Of course, with any empirical investigation there can be logical and mathematical features that place strong constraints on what is conceivably possible, but the risk remains that the proper articulation may have to be discovered through empirical inquiry carried on by a systematic agent delving into its own world of states without absolutely dependable lines as guides. The appropriate factorization, ideally the first item of description, may indeed be the chief thing to find out about a system and the principal thing to know about the total space of phenomena it manifests, and yet persist in being the last fact to be fully settled.

1.1.3.1. Levels of Analysis

The primary factorization is typically only the first in a series of analytic decompositions that are needed to fully describe a complex domain of phenomena. The question about proper factorization that this discussion has been at pains to point out becomes compounded into a question about the reality of all the various distinctions of analytic order. Do the postulated levels really exist in nature, or do they arise only as the artifacts of our attempts to mine the ore of nature? An early appreciation of the hypothetical character of these distinctions and the post hoc manner of their validation is recorded in (Chomsky, 1975, p. 100).

In linguistic theory, we face the problem of constructing this system of levels in an abstract manner, in such a way that a simple grammar will result when this complex of abstract structures is given an interpretation in actual linguistic material.

Since higher levels are not literally constructed out of lower ones, in this view, we are quite free to construct levels of a high degree of interdependence, i.e., with heavy conditions of compatibility between them, without the fear of circularity that has been so widely stressed in recent theoretical work in linguistics.

To summarize the main points: A system of analytic levels is a theoretical unity, to be judged as a whole for the insight it provides into a whole body of empirical data mediately gathered. A level within such a system is really a perspective taken up by the beholder, not a cross-section slicing through the phenomenon itself. Although there remains an ideal of locating natural articulations, the theory is an artificial device in relation to the nature it explains. Facts are made, not born, and already a bit factitious in being grasped as facts.

The language of category theory preserves a certain idiom to express this aspect of facticity in phenomena (MacLane, 1971), which incidentally has impacted the applied world in the notions of a database view (Kerschberg, 1986) and a simulation viewpoint (Widman, Loparo, & Nielsen, 1989). In this usage a level constitutes a functor, that is, a particular way of viewing a whole category of objects under study. For direct applications of category theory to abstract data structures, computable functions, and machine dynamics see (Arbib & Manes, 1975), (Barr & Wells, 1985, 1990), (Ehrig, et al., 1985), (Lambek & Scott, 1986), and (Manes & Arbib, 1986). A proposal to extend the machinery of category theory from functional to relational calculi is developed in (Freyd & Scedrov, 1990).

1.1.3.2. Base Space and Free Space

The base space is intended to capture the fundamental dynamic properties of a system, those aspects to which the other dynamic properties may be related as derivative quantities, free parameters, or secondary perturbances. The remainder consists of tangential features. For simple physical systems this second component contains derivative properties, like velocity and momentum, that are represented as elements of pointwise tangent spaces. In an empirical sense these features do not properly belong to a single point but are attributed to a point on account of measurements made over several points. Of course, from the dual perspective it is momentum that is real and position that is illusion, but that does not affect the point in question, which concerns the uncertainty of their discernment, not the fact of their complementarity.

1.1.3.3. Unabridgements

Part of my task in the projected work is to make a bridge, in theory and practice, from simple physical systems to the more complex systems, also physical but in which new orders of features have become salient, that begin to exhibit what is recognized as intelligence. At the moment it seems that a good way to do this is to anchor the knowledge component of intelligent systems in the tangent and co-tangent spaces that are founded on the base space of a dynamic manifold. This means finding a place for knowledge representations in the residual part of the initial factorization. This leads to a consideration of the questions: What makes the difference between these supposedly different factors of the total manifold? What properties mark the distinction as commonly intended?

From a naturalistic perspective everything falls equally under the prospective heading of physis, signifying nothing more than the first inklings of natural process, though not everything is necessarily best explained in detail by those fragments of natural law which are currently known to us. So it falls to any science that pretends to draw a distinction between the more and the less basic physics to describe it within nature and without trying to get around nature. In this context the question may now be rephrased: What natural terms distinguish every system's basic processes from the kinds of coping processes that support and crown the intelligent system's personal copy of the world? What protocols attach to the sorting and binding of these two different books of nature? What colophon can impress the reader with a need to read them? What instinct can motivate a basis for needing to know?

1.1.4. Components of Intelligence

In a complex intelligent system a number of relatively independent modules will emerge as utilities to subserve the purpose of knowledge acquisition. Chief among these are the faculties of memory and imagination, which operate in closely coordinated representation spaces of the manifold, and may be no more than complementary ways of managing the same turf. These capacities amplify the sensitivity and selectivity of intelligence in the system. They support the transcription of momentary experience into records of its passing. Finally, they collate the fragmentary notes and diverse notations of dynamic experience and catalyze their conversion into unified forms and organizations of rational knowledge.

1.1.4.1. Imagination

The intellectual factor or knowledge component of a system is usually expected to have a certain quality of mercy, that is, to involve actions which are Reversible, Assuredly, Immediately, Nearly. Even though every action obeys physical and thermodynamic constraints, processes that suit themselves to being used for knowledge representation must exhibit a certain forgiveness. It must be possible to move pointers around on a map without irretrievably committing forces on the plain of battle. Actions carried out in the image space should not incur too great a pain or price in terms of the time and energy they dissipate. In sum, a virtue of symbolic operations is that they be as nearly and assuredly reversible as possible. This "virtual" construction, as usual, declares a positively oriented proportion: operations are useful as symbolic transformations in proportion to their exact and certain reversibility.

Imagination's development of elaborate and seemingly superabundant resources of imagery is actually governed by strict obedience to the cybernetic law of requisite variety, which determines that only variety in the responses of a regulator can counter the flow of variety from disturbances to essential variables, the qualities the system must act to keep nearly constant in order to survive in its current and preferred form of being (Ashby, ch. 10 & 11). Aristotle, thinking that the human brain was too flimsy and spongy a material to embody the human intellect, thought it might be useful as a kind of radiator to cool the blood. This is actually a pretty good theory, I think, if it is recognized that the specialty of the brain is to regulate essential variables of human existence on a global scale through the discovery of natural laws. To view the brain as a theorem-o-stat is then fairly close to the mark.

1.1.4.2. Remembrance

The purpose of memory, on the other hand, requires states that can be duly constituted in fashions that are diligently preserved by the normal functioning of the system. The expectation must be faithfully met that such states will be maintained until deliberately reformed by due processes. Intelligent systems cannot afford to indiscriminately confound the imperatives to forgive and forget. Reversibility applies to exploratory operations taking place interior to the dynamic image. An irreversible recording of events is generally the best routine strategy to keep in play between outer and inner dynamics. But reversibility and its opposite interact in subtle ways even to maintain the stability of stored representations. After all, when physical records are disturbed by extraneous noise without the mediation of attention's due process, the ideal system would work to immediately reverse these unintentional distortions and ungraceful degradations of its memories. To abide their time, memories should lie in state spaces with stable equilibria, resting at the bottoms of deep enough potential wells to avoid being tossed out by minor quakes.

A collection of classic and recent papers on the significance of reversibility questions for information acquisition and computational intelligence is gathered together in (Leff & Rex, 1990). The bearing of irreversible processes on the complex dynamics of physical systems is treated in (Prigogine, 1980). Monographs on the physics of time asymmetry and the time direction of information are found in (Davies, 1977) and (Reichenbach, 1991). Relationships between periodicity properties of formal languages and ultimately periodic behavior of finite automata are discussed in (Denning, Dennis, & Qualitz, sec. 6.4) and (Lallement, sec. 7.1). Existential and cultural reflections on the themes of return, repetition, and reconstruction are presented in Kierkegaard, 1843) and (Eliade, 1954). The topographic, potential-surface, or landscape metaphor for system memories, e.g. as elaborated in the self-organizing "memory surface" model of (de Bono, 1969), was influential in the early history of AI and continues to have periodic reincarnations, e.g. (Amit, sec. 2.3). Distributed models of information storage emphasizing sequential memory and reconstructive retrieval are investigated in (Albus, 1981) and (Kanerva, 1988).

Work in cognitive science and AI, in the history of its ongoing revolutions and partial resolutions, has shown a recurring tendency to lose sight of the breadth and power that originally drew it to examine such faculties as memory and imagination. The fact that this form of forgetfulness happens so often is already an argument that there may be some reason for it, in the sociology and psychology of science if not in the nature of the subject matter. No matter what the cause the pattern is seen again and again. The spirit of the original quest that imparts a certain verve to the revolutionary stages of a field's development repeatedly devolves into a kind of volleyball game, an exercise engaged in by opposing parties to settle, by rhetorical hook or strategic crook, which side of a conceptual net the whole globe in contention shall be judged to rest. But most of the purportedly world-shattering distinctions are rendered ineffective by the lack of any operational, much less consensual, definitions. The most heated border disputes arise over concepts for which no clear agreement exists even as to the proper direction of inquiry, whether the form of argument demanded ought to be working from a definition or groping toward a definition of the concept at issue.

It may be inevitable as a natural part of the annealing process of any specialized instrument of science to periodically enter phases of chafing over indeterminate trifles. But it remains a good idea to preserve a few landmarks sighting on the initial aims and the original goals of the inquiry. With respect to imagination, memory, and their interaction within the media of description and expression, a wide field of illumination on the expanses rolling out from under their natural scope is cast by the following sources: (Sartre, 1948), (Yates, 1966), and (Krell, 1990). The critique of pragmatism for "differences that don't make a difference" is legend, e.g. (James, 1907). The form of reasoning that argues toward a definition is bound up with the question of abductive reasoning as described by C.S. Peirce (CE, CP, NE). An interesting, recent discussion of the problem of definition appears in (Eco, 1983).

1.2. Hodos : Methods, Media, and Middle Courses

To every thing there is a season. To every concept there are minimal contexts of sensible application, the most reduced situations in which the concept can be used to make sense. Every concept is an instrument of thought, and like every method has its bounds of useful application. In directions of simplicity, a concept is bounded by the minimum levels of complexity out of which it is, initially, recurrently, transiently, ultimately, able to arise. There is one form of rhetorical question that people often use to address this issue, if somewhat indirectly. It presents itself initially as a genuine question but precipitates the answer in enthymeme, dashing headlong to break off inquiry in the form of a blank. Ostensibly, the space extends the question, but it is only a pretext. The pretense of an open sentence is already filled in by the unexpressed beliefs of the questioner.

"What could be simpler than ___ ?" this question asks, and the automatic completions that different customs have usually borne in mind are these: sets, functions, relations. My present purpose is to address the concept of information, and specifically the kind that results from observation. In answer to the question of its foundation, I have not found that the concept of information can make much sense in anything short of the following framework.

Three-place relations among systems are a minimum requirement. Information is a property of a sign system by virtue of which it can reduce the uncertainty of an interpreting system about the state of an object system. Thus information is a feature that a state in a system has in relation to two other systems. The fundamental reality underlying the concept of information is the persistence of individual systems of relation, each of which exhibits a certain kind of relation among three domains and satisfies a definable set of definitive properties. Each domain in the relation is the state space of an entire system: sign system, interpreting system, object system, respectively. When a set of properties is identified that captures what all such sign systems have in common, a definition of the concept of a sign system will have been discovered. But what form of argument will serve to bring us to a definition, in this case or in its more general setting? Certainly, it cannot be that form of thought, unaided, that requires a definition to start.

More carefully said, information is a property that can be attributed to signs in a system by virtue of their relation to two other systems. This attribution projects a relation among three domains into a simpler order of relation. There are various ways of carrying out this reduction. Not all of them can be expected to preserve the information of the original sign relation. An attribution may create a logical property of elements in the sign domain or it may construct functions from the sign domain to ranges of qualitative meaning or quantitative measure.

1.2.1. Functions of Observation

An observation preserved in a permanent record marks the transience of a certain compound event, the one that an observational account creates in conjunction with the events leading up to it. If an observation succeeds in making an indelible record of an event, then a certain transient of the total system has been passed. To the extent that the record is a lasting memory there is a property of the system that has become permanent. The system has crossed a line in state space that it will not cross again. The state space becomes strictly divided into regions the system may possibly visit again and regions it never will. Of course, an equivalent type of event may happen again, but it will be indexed with a different count. The same juxtaposition of events in the observed system and accounts by the observing system will never again be repeated, if memory faithfully serves.

But perfect observation and permanent recordings are seldom encountered in real life. Therefore, informational content must be found in the distribution of a system's behavior across the whole state space. A system must be recognized as informed by events whenever this distribution comes to be anything other than uniform, or in relative terms deviating from a known baseline. As to what events caused the information there is no indication yet. That kind of decoding requires another cycle of hypotheses about reliable connections with object systems and experiments that lay odds on the systematic validation of these bets. The impressions that must be worked with have the shape of probability distributions. The impression that an event makes on a system lies in the difference between the total distribution of its behavior and the distribution generated on the hypothesis that the event did not happen.

A system of observation constitutes a projection of the object system's total behavior onto a space of observations, which may be called the space of observing or observant states. The object system's total state space is not necessarily a well-defined entity. It can only be imagined to lie in some unknown extension of the observing space. How much information a system may have is defined only relative to a particular system of observation. It is often convenient to personify all the various specifications of observational systems and spaces under a single name, the observer. Every bit of information that a system maintains with respect to an observer constrains the system's behavior to half the observed state space it would otherwise have. When designing systems it is preferred that this bit of information reside in a well-defined register, a localized component of anatomical structure in a designed-to-be-known decomposition of the intelligible object system.

However, the kind of direct product decomposition that would make this feasible is not always forthcoming. When investigating a system of unknown design, it cannot be certain that all its information is embodied in localized components. It is not even certain that a given observation system is detecting the level, mode, or site in which the majority of its information is stored. Even when it is found that a system occupies a small selection or a narrow distribution of its possible states and increases its level of informedness with time, this may yield a quantitative measure of its determination and progress but it does not offer a motive, neither a reference to the objects nor a sense of the objectives that may be driving the process.

In order to assess the purpose and object of an information process, it is important to examine and distinguish several applications that the common measure of information might receive. A first employment scales the information that an object system possesses by virtue of being in a certain state, as among the possibilities envisioned by an observer. A second grades the information that a state in a sign system provides to reduce the uncertainty of an interpreter about the state of an object system. A third weighs the information that a self-informed intelligent system can exercise with respect to the control of its own state transformations.

These distinctions can be traced back through the ideas of pragmatism to a couple of distinctions made by Aristotle in the first textbook of psychology. In Aristotle's Peri Psyche or On the Soul he discerns in the essential nature of things the factors of form and matter. In regard to animate creatures Aristotle divines that the actuality of their intelligence is found in their form while it is the potentiality of mind that is embodied in matter. The form and actuality of mentality is like the edge of an implement that makes it effective in its intended purpose. The formal aspect is an organic shape impressed upon and infused within the material substrate of life. The matter of the mind merely supplies a medium for the potentiality of mental functioning. Subsequently Aristotle divides the form of actuality into two senses, exemplified in turns by the possession and the exercise of knowledge. Can such distinctions, devices of ancient pedigree on fields of patent verdigris, bring a significant bearing to the conduct of modern inquiries in applied AI? This question is considered among the topics of (Awbrey & Awbrey, 1990).

At this point the notion of observation put forward above would seem identical to the notion of representation that is usual in AI in cognitive science. But mathematicians and physicists reserve the status of representation to maps that are homomorphisms, in which some measure of structure preservation is present. And if these two notions are confounded, what sort of observation would enable the detection of whether maps preserve structure or not?Therefore it seems necessary to preserve a more general notion of observation which permits arbitrary transformations, not just the linear mappings or morphisms that properly constitute representations.

It has been appreciated in mathematics and physics for at least a century that an isomorphism is almost totally useless for the purposes that motivate representation and that a single representation is hardly ever enough. Representations are exactly analogous to coordinate projections or spectral coefficients in a fourier analysis. It is a necessary part of their function to severely reduce the data, and this engenders the complementary necessity of having a complete set of projections in order to reconstitute the data to the extent possible.

The extent to which a representation found embodied in a system is an isomorphic representation of its object system is the extent to which that information has not really been processed yet. Only a piecemeal reductive, jointly analytic form of representation can supply grist for the mill that applies rational knowledge to making incisive judgments about action. To object that the reality itself does not exist in the analyzed form created by a system of representation is like objecting to changing the form of bread in the process of digesting it. It is only necessary to remember that representations are supposed to be different from the realities they address, and that the nature of one need not existentially conflict with the nature of the other.

In exception to the general rule, some work in AI and cognitive science has reached the verge of applying the homomorphic idea of representation, although in some cases the arrows may be reversed. Notable in this connection is the concept of structure-mapping exploited in (Gentner & Gentner, 1983) and (Prieditis, 1988) and the notion of quasi-morphism introduced in (Holland, et al., 1986). One of the software engineering challenges implicit in this work is to provide the kind of standardized category-theoretic computational support that would be needed to routinely set up and test whole parametric families of such models. An off-the-shelf facility for categorical computing would of course have many other uses in theoretical and applied mathematics.

1.2.1.1. Observation and Action

It seems clear that observations are a special type of action, and that actions are a special type of observable event. At least, actions are events that may come to be observed, if only in the way that outcomes of eventual effects are recognized to confirm the hypotheses of specific causes. Is every action in some sense an observation? Is every observable event in some sense an observation, a commemoration, an event whose occasion serves to observe something else? If this were so, then the concepts of observation and action would be special cases of each other. Computer scientists will have no trouble accepting the mutual recursion of complex notions, so long as the conceptual instrument as a whole does its job, and so long as the recursion bottoms out somewhere. The mutual definition can find its limit in two ways. It can ground out centrally, with a single category of primitive element that has all the relevant aspects being analyzed, here both perception and action. It can scatter peripherally, resolving into simple elements that distinctively belong to one category or another.

1.2.1.2. Observation and Observables

Independently of their distinctness as categories, what is the relation of the observing and the observable as roles played out in the theater of observation? Observation may be the noting of internal or external events, but more than contemplation it requires the possibility of leaving a record. Nothing serves as observation unless notches can be made in a medium that retains the indenture through time. By this analysis, observation is found to be involved in the very same relation that signs have to their objects. The observation is a sign of its observed object, event, or action. In spite of the active character of concrete observation, it still seems convenient in theoretical models (like turing machines) to divide observation across two abstract components: an active, empirical part that arranges apparatus for a complex test and goes looking for what's happening (on unforeseen segments of tape), and a passive, logical part that represents the elementary reception and pure contingency of simply noting without altering what's under one's nose (or read head).

1.2.1.3. Observation and Interpretation

The foregoing discussion of observation and observables seems like such a useless exercise in hair-splitting that a forward declaration of its eventual purpose is probably called for at this point. Section 2 will introduce a notation for propositional calculus, and Section 3 will describe a proposal for its differential extension. To anticipate that development a bit schematically, suppose that a symbol "x" stands for a proposition (true-false sentence) or a property (qualitative feature). Then a symbol "dx" will be introduced to stand for a primitive property of "change in x". Differential features like "dx", depending on the circumstances of interpretation, may be interpreted in several ways. Some of these interpretations are fairly simple and intuitive, other ways of assigning them meaning in the subject matter of systems observations are more subtle. In all of these senses the proposition "dx" has properties analogous to assignment statements like "x := x+1" and "x := not x". In spite of the fact that its operational interpretation entails difficulties similar to that of assignment statements, I think this notation may provide an alternate way of relating the declarative and procedural semantics of computational state change.

In one of its fuller senses the differential feature "dx" can mean something like this: The system under consideration will next be observed to have a different value for the property "x" than the value it has just been observed to have. As such, "dx" involves a three-place relationship among an observed object, a signified property, and a specified observer. Note that the truth of "dx" depends on the relative behavior of the system and the observer, in a way that cannot be analyzed into absolute properties of either without introducing another observer. If "dx" is interpreted as the expectation of a certain observer, then its realization can be imagined to depend on both the orbit of the system and the sampling scheme or threshold level of the observer. In general, differential features can involve the dynamic behavior of an observed system, decisions about a designated property, and the attention of a specified observer in ways that are irreducibly triadic in their level of complexity.

For example, the system may "actually" have crossed the line between "x" and "not x" several times while the observer was not looking, but without additional oversight this is only an imaginary or virtual possibility. And it is well understood that oversight committees, though they may serve the purpose of a larger objectivity by converging in time on broadly warranted results, in the mean time only compound the complexity of the question at issue. Therefore, it should be clear that the relational concept indicated by "dx" is a primitive notion, in the general case irreducible to concepts of lower order. The relational fact asserted by "dx" is a more primary reality than the manifold ways of parceling out responsibility for it to the interaction of separate agents that are subsystems of the whole. The question of irreducibility in this three-place relation is formally equivalent to that prevailing in the so-called sign relation that exists among objects, signs, and interpreting signs or systems.

If a particular observer is taken as a standard, then discussion reduces to a universe of discourse about various two-place relations, that is, the relations of a system's state to several designated properties. Relative to this frame, a system can be said to have a variety of objective properties. An observer may be taken as a standard for no good reason, but usually a system of observation becomes standardized by exhibiting properties that make it suitable for use as such, like the fabled daily walks of Kant through the streets of Konigsberg by which the people of that city were able to set their watches (Osborne, p. 101). This reduction is similar to the way that a pragmatic discussion of signs may reduce to semantic and even syntactic accounts if the context of usage is sufficiently constant or if a constant interpreter is assumed. Close analogies between observation and interpretation will no doubt continue to arise in the synthesis of physical and intelligent dynamics.

1.2.2. Symbolic Media

A critical transition in the development of a system is reached when components of state are set aside internally or assimilated from the environment to make relatively irreversible changes, indelible marks to record experiences and note intentions. Where in the dynamics of a system do these signs reside? In what nutations of equilibrium does the system insinuate its libraries of notation, the tokens of passed, pressing, and prospective moments of experience? What parameters are concretely set as memorials to the results of observations performed, the outcomes of actions observed, and the plans of action contemplated to provide the experience of desired effects? What bank accumulates all the words coined and spent on sights and deeds? What mint guarantees the content and determines the form of their first impressions?

1.2.2.1. Papyrus, Parchment, Palimpsest

Starting from the standpoint of systems theory a sizable handicap must be overcome in the quest to figure out: "What's in the brain that ink may character?" and "Where is fancy bred?" (McCulloch, 1965). If localized deposits of historical records and promissory notes are all that can be found, a considerable amount of reconstruction may be necessary to grasp the living reality of experience and purpose that underlies them still. A distinction must be made between the analytic or functional structure of the phase space of a system and the anatomical structure of a hypothetical agent to whom these states are attributed. The separation of a system into environment and organism and the further detection of anatomical structure within the organism depend on a direct product decomposition of the space into relatively independent components whose interactions can be treated secondarily. But the direct product is a comparatively advanced stage of decomposition and not to be expected in every case.

This point draws the chase back through the briar patch of that earlier complexity theory, the prime decomposition or group complexity theory of finite automata and their associated formal languages or transformation semigroups (Lallement, ch. 4). This more general study requires the use of semi-direct products (Rotman, 1984) and their ultimate extension into wreath products or cascade products, along with the corresponding notions of divisibility, factorization, or decomposition (Barr & Wells, 1990, ch. 11). This theory seems to have reached a baroque stage of development, either too difficult to pursue with vigor, too lacking in applications, or falling short of some essential insight. It looks like another one of those problem areas that will need to be revisited on the way to integrating AI and systems theory.

1.2.2.2. Statements, Questions, Commands

When signs are created that can be placed in reliable association with the results of observations and the onsets of actions, these signs are said to denote or evoke the corresponding events and actions. This is the beginning of declarative, imperative, and interrogative uses of symbolic expressions.

The interrogative mode is associated with residual properties of the state occupied by a system. The question marks a difference between states denoted by declarative expressions, a divergence between expectation and actuality. The inquisitive use of a sign notes a surprise to be explained, usually by adducing the signs of less obvious facts to the account. A surprise incites the system to an effort whose end is to bring the system's habits of expectation in line with what actually happens on a recurring basis.

The imperative mode is associated with convergent possibilities of the states in which a system may come to reside. The command calls attention to a discrepancy between actuality and intention, a difference between the states independently declared to be observed and desired. The injunctive use of a sign sets a problem to be resolved, usually by executing the actions enjoined by a series of signs. A problem incites the system to an effort whose end is to bring what actually happens on a recurring basis in line with the system's hopeful anticipations. If this problem turns out to be intractable, then the expectation that these intentions can be fulfilled may have to be changed. In this way the different modes of inquiry are often embroiled in intricate patterns of interaction.

In proceeding from surprise and problem states to arrive at explanations and plans of action that are suited to resolving these states, the system's aim is expedited by certain resources, all of which involve massive and complex systems of signs and symbolic expressions. It helps to have a library, an access to the records of individual and collective past efforts and experiences. To be used for clear and present indications this library must have a ready index of its contents, a form of afterthought that is not too thoughtless in design. It helps to a have laboratory, a workshop or a study, any facility where imagination reigns for composing and testing improvised programs and theories, for prototyping on-the-spot inventions. To be used for free and unbiased evaluation this factory of imagination must be a mechanism of forethought without malice, where symbolic expressions extempore are not confused with actions and do not exact the same price in energy spent and pain risked.

But how can all this information and flexibility, constraint vying with freedom of interpretation, be accorded a place in the present state of a system? Can Epimetheus and Prometheus find a way to "get along" in the current state of things? Is the phase space of a system really big enough for both of them?

If signs and symbols are to receive a place in systems theory it must be possible to construct them from materials available on that site. But the only thing a system has to work with is its own present state. How do states of a system come to serve the role of signs? How can it make sense to say that system regards one of its own states as a sign of something else? How do certain states of a system come to be taken by that system, as evidenced by its interpretive behavior, as signs of something else, some object or objective? A good start toward answering these questions would be made by defining the words used in asking them. In looking at the concepts that remain to be given system-theoretic definitions it appears that all of these questions boil down to one: What character in the dynamics of a system would cause it to be called a sign-using system, one that acts as an interpreter in a non-trivial sense?

1.2.2.3. Pragmatic Theory of Signs

The theory of signs that I find most useful was developed over several decades in the last century by C.S. Peirce, the founder of modern American pragmatism. Signs are defined pragmatically, not by any essential substance, but by the role they play within a three-part relationship of signs, interpreting signs, and referent objects. It is a tenet of pragmatism that all thought takes place in signs. Thought is not placed under any preconceived limitation or prior restriction to symbolic domains. It is merely noted that a certain analysis of the processes of perception and reasoning finds them to resolve into formal elements which possess the characters and participate in the relations that a definition will identify as distinctive of signs.

One version of Peirce's sign definition is especially useful for the present purpose. It establishes for signs a fundamental role in logic and is stated in terms of abstract relational properties that are flexible enough to be interpreted in the materials of dynamic systems. Peirce gave this definition of signs in his 1902 "Application to the Carnegie Institution":

Logic is formal semiotic. A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C. This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time. (Peirce, NEM 4, 54).

It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized. (Peirce, NEM 4, 21).

A placement and appreciation of this theory in a philosophical context that extends from Aristotle's early treatise On Interpretation through John Dewey's later elaborations and applications (Dewey, 1910, 1929, 1938) is the topic of (Awbrey & Awbrey, 1992). Here, only a few features of this definition will be noted that are especially relevant to the goal of implementing intelligent interpreters.

One characteristic of Peirce's definition is crucial in supplying a flexible infrastructure that makes the formal and mathematical treatment of sign relations possible. Namely, this definition allows objects to be characterized in two alternative ways that are substantially different in the domains they involve but roughly equivalent in their information content. Namely, objects of signs, that may exist in a reality exterior to the sign domain, insofar as they fall under this definition, allow themselves to be reconstituted nominally or reconstructed rationally as equivalence classes of signs. This transforms the actual relation of signs to objects, the relation or correspondence that is preserved in passing from initial signs to interpreting signs, into the membership relation that signs bear to their semantic equivalence classes. This transformation of a relation between signs and the world into a relation interior to the world of signs may be regarded as a kind of representational reduction in dimensions, like the foreshortening and planar projections that are used in perspective drawing.

This definition takes as its subject a certain three-place relation, the sign relation proper, envisioned to consist of a certain set of three-tuples. The pattern of the data in this set of three-tuples, the extension of the sign relation, is expressed here in the form: ‹Object, Sign, Interpretant›. As a schematic notation for various sign relations, the letters "s", "o", "i" serve as typical variables ranging over the relational domains of signs, objects, interpretants, respectively. There are two customary ways of understanding this abstract sign relation as its structure affects concrete systems.

In the first version the agency of a particular interpreter is taken into account as an implicit parameter of the relation. As used here, the concept of interpreter includes everything about the context of a sign's interpretation that affects its determination. In this view a specification of the two elements of sign and interpreter is considered to be equivalent information to knowing the interpreting or the interpretant sign, that is, the affect that is produced in or the effect that is produced on the interpreting system. Reference to an object or to an objective, whether it is successful or not, involves an orientation of the interpreting system and is therefore mediated by affects in and effects on the interpreter. Schematically, a lower case "j" can be used to represent the role of a particular interpreter. Thus, in this first view of the sign relation the fundamental pattern of data that determines the relation can be represented in the form ‹o, s, j› or ‹s, o, j›, as one chooses.

In the second version of the sign relation the interpreter is considered to be a hypostatic abstraction from the actual process of sign transformation. In other words, the interpreter is regarded as a convenient construct that helps to personify the action but adds nothing informative to what is more simply observed as a process involving successive signs. An interpretant sign is merely the sign that succeeds another in a continuing sequence. What keeps this view from falling into sheer nominalism is the relation with objects that is preserved throughout the process of transformation. Thus, in this view of the sign relation the fundamental pattern of data that constitutes the relationship can be indicated by the optional forms ‹o, s, i› or ‹s, i, o›.

Viewed as a totality, a complete sign relation would have to consist of all of those conceivable moments — past, present, prospective, or in whatever variety of parallel universes that one may care to admit — when something means something to somebody, in the pattern ‹s, o, j›, or when something means something about something, in the pattern ‹s, i, o›. But this ultimate sign relation is not often explicitly needed, and it could easily turn out to be logically and set-theoretically ill-defined. In physics, it is important for theoretical completeness to regard the whole universe as a single physical system, but more common to work with "isolated" subsystems. Likewise in the theory of signs, only particular and well-bounded subsystems of the ultimate sign relation are likely to be the subjects of sensible discussion.

It is helpful to view the definition of individual sign relations on analogy with another important class of three-place relations of broad significance in mathematics and far-reaching application in physics: namely, the binary operations or ternary relations that fall under the definition of abstract groups. Viewed as a definition of individual groups, the axioms defining a group are what logicians would call highly non-categorical, that is, not every two models are isomorphic (Wilder, p. 36). But viewing the category of groups as a whole, if indeed it can be said to form a whole (MacLane, 1971), the definition allows a vast number of non-isomorphic objects, namely, the individual groups.

In mathematical inquiry the closure property of abstract groups mitigates most of the difficulties that might otherwise attach to the precision of their individual definition. But in physics the application of mathematical structures to the unknown nature of the enveloping world is always tentative. Starting from the most elemental levels of instrumental trial and error, this kind of application is fraught with intellectual difficulty and even the risk of physical pain. The act of abstracting a particular structure from a concrete situation is no longer merely abstract. It becomes, in effect, a hypothesis, a guess, a bet on what is thought to be the most relevant aspect of a current, potentially dangerous, and always ever-insistently pressing reality. And this hypothesis is not a paper belief but determines action in accord with its character. Consequently, due to the abyss of ignorance that always remains to our kind and the chaos that can result from acting on what little is actually known, risk and pain accompany the extraction of particular structures, attempts to isolate particular forms, or guesses at viable factorizations of phenomena.

Likewise in semiotics, it is hard to find any examples of autonomous sign relations and to isolate them from their ulterior entanglements. This kind of extraction is often more painful because the full analysis of each element in a particular sign relation may involve references to other object-, sign-, or interpretant-systems outside of its ostensible, initially secure bounds. As a result, it is even more difficult with sign systems than with the simpler physical systems to find coherent subassemblies that can be studied in isolation from the rest of the enveloping universe.

These remarks should be enough to convey the plan of this work. Progress can be made toward new resettlements of ancient regions where only turmoil has reigned to date. Existing structures can be rehabilitated by continuing to unify the terms licensing AI representations with the terms leasing free space over dynamic manifolds. A large section of habitable space for dynamically intelligent systems could be extended in the following fashion: The images of state and the agents of change that are customary in symbolic AI could be related to the elements and the operators which form familiar planks in the tangent spaces of dynamic systems. The higher order concepts that fill out AI could be connected with the more complex constructions that are accessible from the moving platforms of these tangent spaces.

1.2.3. Architecture of Inquiry

The outlines of one important landmark can already be seen from this station. It is the architecture of inquiry, in the style traced out by C.S. Peirce and John Dewey on the foundations poured by Aristotle. I envision being able to characterize the simplest drifts of its dynamics in terms of certain differential operators.

It is important to remember that knowledge is a different sort of goal from the run-of-the-mill setpoints that a system might have. The typical goal is a state that a system has actually experienced many times before, like normal body temperature for a human being. But a particular state of knowledge that an intelligent system moves toward may be a state it has never been through before. The fundamental equivocation on this point expressed in Plato's Meno, whether learning is functionally equivalent to remembering, was discussed above. In spite of this quibble, it still seems necessary to regard states of knowledge as a distinctive class. The reasons for this may lie in the fact that a useful definition of inquiry for human beings necessarily involves a whole community of inquiry.

On account of this social character of inquiry, even those states of knowledge which might be arrived at through accidental, gratuitous, idiosyncratic, transcendental, or otherwise inexplicable means are useless for most human purposes unless they can be communicated, that is, reliably reproduced in the social system as a whole. In order to do this it seems necessary as a practical matter, whatever may have been the original process of construction, that such states of knowledge be obtainable through the option of a rational reconstruction. Hence the familiar requirement of proof for mathematical results, no matter how inspired their first glimmerings. Hence the discipline of programming that challenges workers in AI to represent intelligent processes in terms of computable functions, however differently intelligence may have evolved in the frame of biological time.

Aristotle long ago pointed out that there can be no genuine science of the purely idiosyncratic subject, no systematic knowledge of the totally isolated event. Science does not have as its domain all experience but only that subset which is indefinitely repeatable. Likewise on the negative branch, concerning the lack of knowledge that occasions a problem, a state that never recurs does not present a problem for a system. This limitation of scientific problems and knowledge to recurrent phenomena yields an important clue. The placement of intelligence and knowledge in analogy with system attributes like momentum and frequency may turn out to be based on deeply common principles.

1.2.3.1. Inquiry Driven Systems

One goal of this work is to develop a formalism adequate to the description of knowledge-oriented inquiry-driven systems in logical and differential terms, to be able to write down and run as simulations qualitative differential equations that describe individual cases of systems with knowledge-directed behavior, systems which exhibit a progress toward a goal of knowledge. A knowledge-oriented system is one which maintains a knowledge base which figures into its behavior in a dual role, both as a guide to action and as the object of a system goal to increase the measure of its usefulness. An inquiry-driven system is one that develops its knowledge base in response to the differences existing between three aspects of state that may be projected or generated from its total state, components which might be called: expectations, observations, and intentions.

It is not clear at this point if there can be interesting classes of inquiry-driven systems which are purely deterministic, but a recognition of what such a system would be like might help to clarify the limits of the notion. In some sense a deterministic inquiry-driven system would fulfill a behaviorist dream. It would correspond to a scientific agent whose every action is predictable, even to the phenomena it will encounter, hypotheses it will entertain, and experiments it will perform as a consequence. If it is accepted that behaviorist proposals are tantamount to a restriction of methodology to the level of finite state description, then less elaborate characterizations of such systems are always available. Proper hypotheses, which are not just summaries of finite past experience but can refer to an infinite set of projected examples, are commonly associated with complexities in behavior that proceed from the essentially context-free level on up.

One important use of a system's current knowledge base is to project expectations of what is likely to be actualized in its experience, an image of what state it envisions probable. Another use of a system's personal knowledge base is to preserve intentions during the execution of series of actions, to keep a record of a current goal, a picture of what it would like to find actualized in its experience, an image of what state it envisions desirable. From these uses of images two kinds of differences crop up in the process of inquiry.

1.2.3.2. Surprises to Explain

One of the uses of a knowledge base is to support the generation of expectations. In return, one of the inputs to the operators which edit and update a knowledge base is the set of differences between expected and observed states. An inquiry-driven system requires a function to compare expected states, as represented in the images reflexively generated from present knowledge, and observed states, as represented in the images currently delivered as unquestioned records of actual happenings. In human terms this kind of discrepancy between expectation and observation is experienced as a surprise, and is usually felt as constituting an impulse toward an explanation that can reduce the sense of disparity. The specification of a particular inquiry-driven system would have to detail this relation between states of uncertainty and resultant actions.

1.2.3.3. Problems to Resolve

Since a system's determination of its own goals is a special case of knowledge in general, it is convenient to allocate a place for this kind of information in the knowledge component of an intelligent system. Thus, the intellectual component of a knowledge-oriented system may be allowed to preserve its intentions, the images of currently active goals. Often there is a difference between an actual state, as represented by the image developed in free space by a trusted process of observation, and an active goal, as represented by an image in the same space but cherished within the frame of intention or otherwise distinguished by an attentional affect. This situation represents a problem to be solved by the system through actions that effect changes on the level of its primary dynamics. The system chooses its trajectory in accord with reducing the difference between its active intentions and the observations that record actual conditions.

1.2.4. Simple Minded Systems

Of course, not every total manifold need have a nice factorization. It might be thought to dispense with such spaces immediately, to put them aside as not being reasonable. But it may not be possible to dismiss them quite so easily and summarily. Intelligent systems of this sort may end up being refractory to routine analysis and will have to be regarded as simple minded. That is, they may turn out to be simple in the way that algebraic objects are usually called simple, having no interesting proper factors of the same sort. Suppose there are such simple minded systems, otherwise deserving to be called intelligent but which have no proper factorization into the kind of gross dynamics and subtle dynamics that might correspond to the distinction ordinarily made between somatic and mental behavior. That is, they do not have their activity sorted into separate scenes of action: one for ordinary physical and thermal dynamics, another for information processing dynamics, symbolic operations, knowledge transformations, and so on up the scale. In the event that this idea of simplicity can be found to make sense, it is likely that simple minded systems would be deeply involved in or place extreme bounds on the structures of all intelligent systems.

A realm of understanding subject to a certain rule of analysis may have a boundary marked by simple but adamant exceptions to its further reign. Or it may not have a boundary, but that seems to verge on an order of understanding beyond the competence of computational systems. Whether the human form of finitude abides or infringes this limitation is something much discussed but not likely to be settled any time soon. In order to pursue the question of simplicity the form of analysis must be examined more carefully. The type of factorization that system-theoretic analogies suggested was gotten by locating a convenient stage at which to truncate or abridge the typical datum. This amounts to a projection of the data space onto a stage set by this process of critical evaluation. The fibers of this projection are the data sets that form the inverse images of points in its range.

In reflecting on the form of analysis that has naturally arisen at this point it appears to display the following character. An object is presented to contemplation in the light of a finite collection of features. If the object is found to possess every one of the listed features, this incurs the existence of another object, simpler in some sense, to which analytic attention is then shifted. It may be figuratively expressed that the analysis descends to a situation closer to the initial conditions or bounds to a site nearer the boundary conditions.

The cases of simple minded systems appear to contain at least the following two possibilities. First, a simple minded system may come into being already knowing itself perfectly, in which case all the irony of a Socrates would be lost on it, in terms of bringing it a wit closer to knowledge. The system already knows its whole manifold of possible states, that is, its knowledge component is in some sense complete, containing an answer to every possible dynamic puzzle that might be posed to it. Rather than an overwhelming richness of theory, this is more likely to arise from a structural poverty of the total space and a lack of capacity for the reception of questions that can be posed to it, as opposed to those posed about it. Second, a simple minded system might be born into an initial condition of ignorance, with the potential of reaching states of knowledge within its space, but these states may be discretely distributed in a continuous manifold. This means that states of knowledge could be achieved only by jumping directly to them, without the benefit of an error-controlled feedback process that allows a system to converge gradually upon the goals of knowledge.

1.3. Telos : Horizons and Further Applications

In its etymology, intelligence suggests a capacity that contains its goal (telos) within itself. [No, insert correction here.] Of course, it does not initially grasp that for which it reaches, does not always possess its goal, otherwise it would be finished from the start. So it must be that it contains only a knowledge of its goal. This need not be a perfect knowledge even of what defines the goal, leaving room for clarification in that dimension, also. Some thinkers on the question suspect that the capacity for setting goals may answer to another name: wisdom (sophia), prudence (phronesis), and even elegance (arete) are among the candidates often heard. If so, intelligence would have a relationship to this wisdom and sagacity that is analogous to the relationship of logic to ethics and esthetics. At least, this is how it appears from the standpoint of one philosophical tradition that recommends itself to me.

1.3.1. Logic, Ethics, Esthetics

The philosophy I find myself converging to more often lately is the pragmatism of C.S. Peirce and John Dewey. According to this account, logic, ethics, and esthetics form a concentric series of normative sciences, each a subdiscipline of the next. Logic tells how one ought to conduct one's reasoning in order to achieve the stated goals of reasoning in general. Thus logic is a special application of ethics. Ethics tells how one ought to conduct one's activities in general in order to achieve the good appropriate to each enterprise. What makes the difference between a normative science and a prescriptive dogma is whether this "telling" is based on actual inquiry into the relationship of conduct to result, or not.

In this view, logic and ethics do not set goals, they merely serve them. Of course, logic may examine the consistency of an arbitrary selection of goals in the light of what science tells about the likely repercussions in nature of trying to actualize them all. Logic and ethics may serve the criticism of certain goals by pointing out the deductive implications and probable effects of striving toward them, but it has to be some other science which finds and tells whether these effects are preferred and encouraged or detested and discouraged relative to a particular form of being.

The science which examines individual goods, species goods, and generic goods from an outside perspective must be an esthetic science. The capacity for inquiry into a subject must depend on the capacity for uncertainty about that subject. Esthetics is capable of inquiry into the nature of the good precisely because it is able to be in question about what is good. Whether conceived as empirical science or as experimental art, it is the job of esthetics to determine what might be good for us. Through the exploration of artistic media we find out what satisfies our own form of being. Through the expeditions of science we discover and further the goals of own species' evolution.

Outriggers to these excursions are given by the comparative study of biological species and the computational study of abstractly specified systems. These provide extra ways to find out what is the sensible goal of an individual system and what is the perceived good for a particular species of creature. It is especially interesting to learn about the relationships that can be represented internally to a system's development between the good of a system and the system's perception, knowledge, intuition, feeling, or whatever sense it may have of its goal. This amounts to asking the questions: What good can a system be able to sense for itself? How can a system discover its own best interests? How can a system achieve, from the evidence of experience, a cognizance, evidenced in behavior, of its own best interests?

1.3.2. Inquiry and Education

My joint work with Susan Awbrey speculates on the yield of AI technology for new seasons of inquiry-based approaches to education and research (Awbrey & Awbrey, 1990, '91, '92). A fruitful beginning can be made, we find, by returning to grounds that were carefully prepared by C.S. Peirce and John Dewey, and by asking how best to rework these plots with the implements that the intervening years have provided. There is currently being pursued a far-ranging diversity of work on the applications of AI to education, through research on problem solving performance (Smith, 1991), learner models and the novice-expert shift (Gentner & Stevens, 1983), the impact of cognitive strategies on instructional design (West, Farmer, & Wolff, 1991), the use of expert systems as teaching tools (Buchanan & Shortliffe, 1984), (Clancey & Shortliffe, 1984), and the development of intelligent tutoring systems (Sleeman & Brown, 1982), (Mandl & Lesgold, 1988). Other perspectives on AI's place in science, society, and the global scene may be sampled in (Wiener, 1950, 1964), (Ryan, 1974), (Simon, 1982), (Gill, 1986), (Winograd & Flores, 1986), and (Graubard, 1988).

1.3.3. Cognitive Science

Remarkably, seeds of a hybrid character, similar to what is sought in the intersection of AI and systems theory, were planted many years ago by one who explored the farther and nether regions of the human mind. This model blended recognizably cybernetic and cognitive ideas in a scheme that included associative networks for adaptive learning and recursion mechanisms for problem solving. But these ideas lay dormant and untended by their originator for over half a century. Sigmund Freud rightly estimated that this model would always be too simple-minded to help with the complex and subtle exigencies of his chosen practice. But the "Project" he wrote out in 1895 (Freud, 1954) is still more sophisticated, in its underlying computational structure, than many receiving serious study today in AI and cognitive modeling. Again, here is another stage, another window, where old ideas and directions may be worth a new look with the new 'scopes available, if only to provide a basis for informing the speculations that get a theory started.

The ideas of information processing, AI, cybernetics, and systems theory had more direct interactions, of course, with the development of cognitive science. A share of these mutual influences and crosscurrents may be traced through the texts and references in (Young, 1964, 1978), (de Bono, 1969), (Eccles, 1970), (Anderson & Bower, 1973), (Krantz, et al., 1974), (Johnson-Laird & Wason, 1977), (Lachman, Lachman, & Butterfield, 1979), (Wyer & Carlston, 1979), (Boden, 1980), (Anderson, 1981, '83, '90), (Schank, 1982), (Gentner & Stevens, 1983), (H. Gardner, 1983, 1985), (O'Shea & Eisenstadt, 1984), (Pylyshyn, 1984), (Bakeman & Gottman, 1986), (Collins & Smith, 1988), (Minsky & Papert, 1988), (Posner, 1989), (Vosniadou & Ortony, 1989), (Gottman & Roy, 1990), and (Newell, 1990).

1.3.4. Philosophy of Science

Continuing the angle of assault previously taken toward the abandoned mines of intellectual history, there are many other veins and lodes, subsided and shelved, that experts assay too low a grade for current standards of professional work. Yet many of these superseded courses and discredited vaults of theory are worth retooling and remining in the shape of computer models. Computational reenactments of these precept chapters in human thought, not just repetitions but analytic representations, could serve the purpose of school figures, training exercises and stock examples, to be used as instructional paradigm cases.

But there is a further possibly. Many foregone projects were so complex that not everything was understood about their implications at the time they were rejected for some critical flaw or another. It is conceivable that new things might be learned about the global character of these precursory models from computer simulations of their axioms, leading principles, and general lines of reasoning. Even though their flaws were eventually detected by unaided analysis, their positive features and possible directions of amendment may not have been so easily appreciated. An extended reflection on the need for various kinds of reconstruction in and of philosophy, and the conditions for their meaningful application to unclear but present situations, may be found in (Dewey, 1948).

A prime example of a project awaiting this kind of salvage operation is the submerged edifice of Carnap's "world building" (1928, 1961), the remains of a mission dedicated to "the rational reconstruction of the concepts of all fields of knowledge on the basis of concepts that refer to the immediately given … the searching out of new definitions for old concepts" (1969, v). The illusory stability of the "immediately given" has never been more notorious than today. But the relevant character to be appreciated in this classical architecture is the degree of harmony and balance, the soundness in support of lofty design that subsists and makes itself evident in the relationship of one level to another. Much that is toxic in our intellectual environment today could be alleviated by a suitably analytic and perceptive movement to recycle, reclaim, and restore the artifacts and habitations of former times.

2. Conceptual Framework

2.1. Systems Theory and Artificial Intelligence

If the principles of systems theory are taken seriously in their application to AI, and if the tools that have been developed for dynamic systems are cast in with the array of techniques that are used in AI, a host of difficulties almost instantly arises. One obstacle to integrating systems theory and artificial intelligence is the bifurcation of approaches that are severally specialized for qualitative and quantitative realms, the unavoidable differences between boolean-discrete and real-continuous domains. My way of circumventing this obstruction will be to extend the compass of differential geometry and the rule of logic programming to what I see as a locus of natural contact. Continuing the inquiry to naturalize intelligent systems as serious subjects of dynamic systems theory, a whole series of further questions comes up:

1. What is the proper notion of state?
2. How is the knowledge component or the "intellectual property" of this state to be characterized?

In accord with customary definitions, the knowledge component would need to be represented as a projection onto a knowledge subspace. In those intelligences for whom not everything is knowledge, or at least for whom not everything is known at once, that is, the great majority of those we are likely to know, there must be an alternate projection onto another subspace. Some real difficulties begin here which threaten to entangle our own resources intelligence of irretrievably.

The project before me is simply to view intelligent systems as systems, to take the ostended substantive seriously. To succeed at this it will be necessary to answer several questions:

What is the proper notion of a state vector?

We need to analyze the state of the system into a knowledge component and a remaining or a sustaining component. This "everything else" component may be called the physical component so long as this does not prejudice the issue of a naturalistic aim, which seeks to understand all components as physis, that is, as coming under the original Greek idea of a natural process. Even the ordinary notion of a state vector, though continuing to be useful as a basis of analogy, may have to be challenged:

Are the state elements, the moments of the system's experience, really vectors?

Consider the common frame of a venn diagram, overlapping pools of elements arrayed on a nondescript plain, an arena of conventional measure but not routinely examined significance.

A certain figure of speech, a chiasmus, may be used to get this point across. The universe of discourse, as a system of objective realities, is something that is not yet perfectly described. And yet it can be currently described in the signs and the symbols of a discursive universe. By this is meant a formal language that is built up on terms that are taken to be simple. Yet the simplicity of the chosen terms is not an absolute property but a momentary expedient, a side-effect of their current interpretation.

2.2. Differential Geometry and Logic Programming

In this section I make a quick reconnaissance of the border areas between logic and geometry, charting a beeline for selected trouble spots. In the following sections I return to more carefully survey the grounds needed to address these problems and to begin settling this frontier.

2.2.1. Differences and Difficulties

Why have I chosen differential geometry and logic programming to try jamming together? A clue may be picked up in the quotation below. When the foundations of that ingenious duplex, AI and cybernetics, were being poured, one who was present placed these words in a cornerstone of the structure (Ashby, 1956, p. 9).

The most fundamental concept in cybernetics is that of "difference", either that two things are recognisably different or that one thing has changed with time.

A deliberate continuity of method extends from this use of difference in goal-seeking behavior to the baby steps of AI per se, namely, the use of difference-reduction methods in the form of what is variously described as means-ends analysis, goal regression, or general problem solving.

2.2.1.1. Distance and Direction

Legend tells us that the primal twins of AI, the strife-born siblings of Goal-Seeking and Hill-Climbing, began to stumble and soon came to grief on certain notorious obstacles. The typical scenario runs as follows.

At any moment in time the following question is posed:
In this problem space how ought one choose to operate
in order to forge of one's current state a new update
that has hopes of being nearer to one's engoaled fate?

But before Jack and Jill can start up the hill they will need a whole bucket of prior notions to prime the pump. There must be an idea of distance, in short, a metric function defined on pairs of states in the problem space. There must be an idea of direction, a longing toward a goal that informs the moment, that fixes a relation of oriented distances to transition operators on states. Stated in linguistic terms the directive is a factor that commands and instructs. It arranges a form of interpretation that endows disparities with a particular sense of operational meaning.

Intelligent systems do not get to prescribe the problem spaces that will be thrown their way by nature, society, and the outside world in general. These nominal problems would hardly constitute problems if this were the case. Thus it pays to consider how intelligent systems might evolve to cast ever wider nets of competence in the spaces of problems that they can handle. Striving to adapt the differential strategies of classical cybernetics and of early AI to "soaring" new heights (Newell, 1990), to widening gyres of ever more general problem spaces, there comes a moment when the predicament thickens but the atmosphere of theory and the wings of artifice do not.

2.2.1.2. Topology and Metric

Topology is the most unconstrained study of spaces, beginning as it does with spaces that have barely enough hope of geometric structure to deserve the name of spaces (Kelley, 1961). An attention to this discipline inspires caution against taking too lightly the issue of a metric. There is no longer any reason to consider the question of a metric to be a trivial one, something whose presence and character can be taken for granted. For each space that can be contemplated there arises a typical suite of questions about the existence and the uniqueness of a possible metric. Some spaces are not metrizable at all (Munkres, sec. 2-9). Those that are may have a multitude of different metrics defined on them. My own sampling of differential methods in AI, both smooth and chunky style, suggests to me that this multiplicity of possible metrics is the ingredient that conditions one of their chief sticking points, a computational viscosity that consistently sticks in the craw of computers. Unpalatable if not intractable, it will continue to gum up the works, at least until some way is found to dissolve the treacle of complexity that downs our best theories.

2.2.1.3. Relevant Measures

Differences between problem states are not always defined. And even when they are, relevant differences are not always defined in the manner that would form the most obvious choice. Relevant differences are differences that make a difference, in the well-known pragmatist phrase, bearing on the problem and the purpose at hand. The qualification of relevance adds information to the abstractly considered problem space. This extra information has import for the selection of a relevant metric, but nothing says it will ever determine a unique metric suited to a given situation. Relevant metrics are generally defined on semantic features of the problem domain, involving pragmatic equivalence classes of objects. Measures of distinction defined on syntactic features, in effect, on the language that is used to discuss the problem domain, are subject to all of the immaterial differences and the accidental collision of expression that acts to compound the computational confusion and distraction.

When the problem of finding a fitting metric develops the intensity to cross a critical threshold, a strange situation is constellated. The new level of problemhood is noticed as an afterthought but may have a primeval reality about it in its own right, its true nature. The new circle of problem states may circumscribe and underlie the initial focus of attention. Can the problem of finding a suitable metric for the original problem space be tackled by the same means of problem solving that worked on the assumption of a given metric? A reduction of that sort is possible but is hardly ever guaranteed. The problem of picking the best metric for the initial problem space may be as difficult as the problem first encountered. And ultimately there is always the risk of reaching a level of circumspection where the problem space of last resort has no metric definable.

2.2.2. Logic with a Difference

In view of the importance of differential ideas in systems theory and against the