Kaufmann (preface) Introduction to the Theory of Fuzzy Subsets, “Avertissement” / “Preface”

 

by Corry Shores

 

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[The following is summary. Unless otherwise noted, boldface is my own. Page citations refer to the French edition first / then the English. I apologize in advance for my distracting typos or other mistakes, because proofreading is incomplete.]

 

 

 

Summary of

 

Arnold Kaufmann

 

Introduction à la théorie des sous-ensembles flous

à l’usage des ingénieurs

(Fuzzy sets theory)

1. Eléments théoriques de base

/

Introduction to the Theory of Fuzzy Subsets.

Vol.1 Fundamental Theoretical Elements

 

Avertissement

Preface

(by Arnold Kaufmann)

 

 

Brief summary:

The elements of fuzzy subsets are members in an “uncertain fashion” rather than in the certain fashion of classical sets where elements either are or are not in the set. The theory worked out here is about fuzzy subsets and not fuzzy sets, because “the reference set will always be an ordinary set, that is, [...] a collection of well-specified and distinct objects” (xii/xiii-xiv). This theory of fuzzy subsets will prove especially useful for designing intelligent machines so that they can handle fuzzy information, like human minds can.

 

 

 

Summary

 

Kaufmann notes that our scientific knowledge of the world is limited to the models, representations, “more or less true” laws, and “acceptable approximations in the state of our knowledge” that we use to study the world (ix/xi). [I am not sure about his next point, so I will quote it. It might be that the only confirmation we have of one model is that made by means of another model, and although they continue to correct one another, there will not be a perfect representation of the world, at least until some great revolution in ideas allows for a better means of representation. Here is the quotation:]

And the model of something for one is not exactly the same model of this thing for another; the formula may remain the same, but the interpretation may be different. The universe is perceived with the aid of models that are indeed perfecting themselves through embodying one in another, at least until some revolution in ideas appears, no longer permitting a correct embodyment.

(ix/xi)

 

But human thinking, unlike computer cognition, is fuzzy. This is partly because we use global or parallel reasoning which is necessarily fuzzy (ix/xi). And there is a lot of room for alterations and adaptation in human learning (ix/xi).

 

Kaufmann then wonders how we might introduce this real fuzziness into our mathematical systems (ix/xi).

 

Kaufmann then distinguishes classical and fuzzy membership.

For a mathematician, what does the word fuzzy signify (or synonymous words)? This will mean that an element is a member of a subset only in an uncertain fashion; while, on the other hand, in mathematics we understand that there are only two acceptable situations for an element: being a member of or not being a member of a subset. Any normal logic, boolean logic, rests on this base: membership or nonmembership in a subset of a reference set.

(x/xi, italics his)

 

L.A. Zadeh’s innovation was to allow for “weighted membership. An element may then belong more or less to a subset, and, from there, introducing the fundamental concept, that of a fuzzy subset” (x/xii).

 

The multivalued or n-ary logics of Post (1921), Lukasiewicz (1937), and Moisil (1940) opened the way for fuzzy logic. The two schools of fuzzy logic that emerged are of Zadeh and Moisil (x/xii).

 

One objection to fuzzy logic is that what it accomplishes can be accomplished by other systems. But this objection can be raised for almost any important system. [That in itself does not diminish the value of any system, so it should not diminish the value of fuzzy logic.] (x/xii)

 

The theory of fuzzy subsets should be of great interest to scientists who study fuzzy systems like language and thought, but also to “the literati and artists, those who construct truth and beauty with fuzziness” (x/xii).

 

This book is designed to be as accessible as possible for those with a technical interest in the field (x/xii).

 

Kaufmann added many examples, although for some readers that might make the book too lengthy (xi/xii-xiii).

 

Fuzziness is here limited to variables and configurations, but one could extrapolate from this presentation other conceptual aspects of fuzziness. So, many disciplines could derive value from this material. (xi/xiii).

 

Linear computation machines will be able to handle the fuzzy problems we deal with in this book (xi-xii/xiii).

 

Kaufmann then addresses the question, why do we use the term fuzzy “subset” and not fuzzy “set”? He explains that this is because “the reference set will always be an ordinary set, that is, such as one defined intuitively in modern mathematics, that is again, a collection of well-specified and distinct objects. It is the subsets that will be fuzzy, as we shall see” (xii/xiii-xiv).

 

Volume 1 presents the theory, while Volume 2 applies it to such areas as “fuzzy languages, fuzzy systems, fuzzy automata, fuzzy algorithms, machines and control, decision problems in a fuzzy universe, recognition of forms, problems of classification and selection, documentary research, etc.” (xii/xiv).

 

Kaufmann then thanks a number of people who helped with the production of this book (xii/xiv).

 

He especially thanks his son Alain for his corrections (xii/xiv).

 

Kaufmann notes that the human mind will “remain fuzzy and creative” (xii/xiv). [The French Avertissement ends here, and then there begins another one for the second edition. Part of that is found in English edition as a continuation of its Preface.]

 

Kaufmann then notes that he corrected a number of errors for this second edition (xiii/xiv). In the French edition he mentions some features of the text, like the extensive Bibliography, also his new volumes, and he calls upon his readers to work together on furthering our knowledge by means of these findings (xiii).

 

 

 

 

 

 

 

From:

 

Kaufmann, Arnold. 1975 [1973]. Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs (Fuzzy sets theory). 1: Eléments théoriques de base. Foreword by L.A. Zadeh. 2nd Edn. Paris: Masson.

 

Kaufmann, Arnold. 1975. Introduction to the Theory of Fuzzy Subsets. Vol.1: Fundamental Theoretical Elements. Foreword by L.A. Zadeh. English translation by D.L. Swanson. New York / San Francisco / London: Academic Press.

 

 

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Zadeh (Foreword) in Kaufmann Introduction to the Theory of Fuzzy Subsets, “Préface” / “Foreword”

 

by Corry Shores

 

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[Logic and Semantics, Entry Directory]

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[The following is summary. Unless otherwise noted, boldface is my own. Page citations refer to the French edition first / then the English. I apologize in advance for my distracting typos or other mistakes, because proofreading is incomplete.]

 

 

 

Summary of

 

Arnold Kaufmann

 

Introduction à la théorie des sous-ensembles flous

à l’usage des ingénieurs

(Fuzzy sets theory)

1. Eléments théoriques de base

/

Introduction to the Theory of Fuzzy Subsets.

Vol.1 Fundamental Theoretical Elements

 

Foreword

by L.A. Zadeh

 

 

Brief summary:

Fuzzy sets are “classes with unsharp boundaries in which the transition from membership to nonmembership is gradual rather than abrupt” (Zadeh ix). The reliance on classical sets in studies of human life and in human or artificial cognition has limited these efforts, because the real world and human thinking involve fuzziness.

 

 

 

Summary

 

The theory of fuzzy subsets tries to bring together precise mathematics with the “pervasive imprecision of the real world” (Zadeh v/ix). This is also an effort to better understand mental cognition.

 

At the time of this writing, artificial intelligence science has been unable to replicate the many types of human cognition (v/ix).

 

The reason for this is that human cognition has the ability to process imprecise data, while computers do not (v-vi/ix).

 

“The fundamental concept in mathematics is that of a set – a collection of objects” (vi/ix). However, Zadeh thinks that most human cognition uses fuzzy sets or subsets:

We have been slow in coming to the realization that much perhaps most, of human cognition and interaction with the outside world involves constructs which are not sets in the classical sense, but rather “fuzzy sets” (or subsets), that is, classes with unsharp boundaries in which the transition from membership to nonmembership is gradual rather than abrupt. Indeed, it may be argued that much of the logic of human reasoning is not the classical two-valued or even multivalued logic but a logic with fuzzy truths, fuzzy connectives, and fuzzy rules of inference.

(vi/ix)

 

Because we have sought precision in our scientific endeavors, we have tried to make the real world fit into mathematical models that leave no room for fuzziness. We have even tried to use such precision to understand human individual and social behavior. Zadeh thinks this is a doomed project (vi/ix).

In our quest for precision, we have attempted to fit the real world to mathematical models that make no provision for fuzziness. We have tried to describe the laws governing the behavior of humans, both singly and in groups, in mathematical terms similar to those employed in the analysis of inanimate systems. This, in my view, has been and will continue to be a misdirected effort, comparable to our long-forgotten searches for the perpetuum mobile and the philosopher’s stone.

(vi/ix)

 

Instead, Zadeh argues that we need to incorporate fuzziness into our concepts and techniques for studying reality and human life (ix).

What we need is a new point of view, a new body of concepts and techniques in which fuzziness is accepted as an all pervasive reality of human existence. Clearly, we need an understanding of how to deal with fuzzy sets within the framework of classical mathematics. More important, we have to develop novel methods of treating fuzziness in a systematic – but not necessarily quantitative – manner. Such methods could open many new frontiers in psychology, sociology, political science, philosophy, physiology, economics, | linguistics, operations research, management science, and other fields, and provide a basis for the design of systems far superior in artificial intelligence to those we can conceive today.

(vi-vii/ix-x)

 

Ladeh then praises Kaufmann’s text. It is thorough and lucid, and it is the “first systematic exposition” of fuzzy subset theory (vii/x).

 

This text will deal with the mathematical aspects of fuzzy subsets, and it should prove useful to engineers and artificial intelligence programmers, because among other things, it details the notion of fuzzy algorithms (vii/x).

 

Zadeh thinks this book will prove highly influential (x).

 

 

 

 

 

From:

L.A. Zadeh’s “Préface” / “Foreword”  in

 

Kaufmann, Arnold. 1975 [1973]. Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs (Fuzzy sets theory). 1: Eléments théoriques de base. Foreword by L.A. Zadeh. 2nd Edn. Paris: Masson.

 

Kaufmann, Arnold. 1975. Introduction to the Theory of Fuzzy Subsets. Vol.1: Fundamental Theoretical Elements. Foreword by L.A. Zadeh. English translation by D.L. Swanson. New York / San Francisco / London: Academic Press.

 

 

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Andy Clark. 8.2 of Being There, “What is this Thing Called Representation?,” summary

summary by Corry Shores
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Andy Clark

Being There:

Putting Brain, Body, and World Together Again

Ch.8

Being, Computing, Representing

Part 8.2
What is this Thing Called Representation?

Brief Summary:

There is a view in cognitive science that believes ‘thinking’ systems such as brains and thinking machines have internal representations that correlate with the external world but are also processed without the direct causal influence of external factors.

Summary

Clark will discuss “internal representations”. Cognitive scientists often refer to them as being housed by brains and computer models. The concept helped bridge connectionism and classical artificial intelligence. Both camps agreed there are such internal representational systems, however they disagreed on the precise nature of them. (142d) The classicists thought mental contents were “tokened as strings of symbols that could be read, copied, and moved by some kind of inner central processing unit” (143a); they believe then in a “chunky symbolic” inner economy of mental contents. Connectionists however “believed in a much more implicit style of internal representation: one that replaced strings of chunky, manipulable symbols with complex numerical vectors and basic operations of pattern recognition and pattern transformation” (144a)

Both views see mental contents as internal representations. Haugeland lists the criteria for an internal representational system. [Quoting Clark:]

(1) It must coordinate its behaviors with environmental features that are not always "reliably present to the system."

(2) It copes with such cases by having something else (in place of a signal directly received from the environment) "stand in" and guide bahavior in its stead.

(3) That "something else" is part of a more general representational scheme that allows the standing in to occur systematically and allows for a variety of related representational states (see Haugeland 1991, p. 62). [144b]

Now consider plants that track the sun with their leaves. The sun’s changing position itself guides the leaves’ motion. So because the plant system is controlled by an environmental feature that is reliably present to the system [perhaps, to the system’s behavior], it would not satisfy the first criterion. [Consider instead a solar panel programmed to turn with the sun’s position. It need not even ‘see’ where the sun is. It can be programmed according to astronomical predictions of its position. These coordinates ‘stand-in’ for the direct control of the sun’s actual position on the plant leaf movement.] Point two says that instead of the environmental feature, something else stands-in for it and guides the system’s behavior. [Now consider how before eating, we see the food, and our stomach produces gastric juices, then in their place comes food. In a way,] our gastric juices stand-in for the food, and in that way represent the future presence of food. However, the third criterion says that the something else that stands-in must be a part of a larger representational system, and the gastric juice does not. [Yet the sun’s coordinates fit within a larger system of geometrical representations.] Clark thinks that the role of the decouplability of the inner and outer states in determining behavior is overplayed. (144c.d)

But consider the way that the neurons in a rat’s brain process coded signals for which way the head is pointed. The system uses a general representational scheme, but Clark wonders if really this part of the system can function even if it were decoupled from “the continuous stream of proprioceptive signals from the rat’s body.” (145)

A strict application of Haugeland’s criteria will not help us understand the flows of information in such neuronal systems that come from the body. (145)

So Haugheland’s criteria is a bit too restrictive; nonetheless we still need a way to constrain the applicability of the concept of internal representation. For, we need to rule out simple cases of environmental control over the system’s behavior. Also, internal complexity in a system is also alone not enough to qualify as inner representation. In addition, a correlation between an inner state and some environmental parameter is insufficient for inner representation [recall the gastric juice example]. “It is thus important that the system uses the correlations in a way that suggests that the system of inner states has the function of carrying specific types of information.”  (146a)

So the fact that the tides correlate with the moon’s position does not mean that either represents the other; for, the correlation was neither designed nor evolved for the purpose of carrying information about the other’s variations. On the other hand, the neuronal activity in the rat’s brain does seem to have the purpose of carrying information about the head’s position. (146b.c)

So what will qualify an inner state as a representation will have to do with the role it plays in the system.

It may be a static structure or a temporally extended process. It may be local or highly distributed. It may be very accurate or woefully inaccurate. What counts is that it is supposed to carry a certain type of information and that its role relative to other inner systems and relative to the production of behavior is precisely to bear such information. (146cd)

So Clark proposes that we

call a processing story representationalist if it depicts whole systems of identifiable inner states (local or distributed) or processes (temporal sequences of such states) as having the function of bearing specific types of information about external or bodily states of affairs. (147a, boldface mine)

Consider such adaptive hook-ups as the sunflower tracking the sun’s position, or a robot seeking light. Clark thinks there is little to gain by calling such adaptive hook-ups representational.

Representation talk gets its foothold, I suggest, when we confront inner states that, in addition, exhibit a systematic kind of coordination with a whole space of environmental contingencies. In such cases it is illuminating to think of the inner states as a kind of code that can express the various possibilities and which is effectively "read" by other inner systems that need to be informed about the features being tracked. Adaptive hookup thus phases gradually into genuine internal representation as the hookup's complexity and systematicity increase. At the far end of this continuum we find Haugeland's creatures that can deploy the inner codes in the total absence of their target environmental features. Such creatures are the most obvious representers of their world, and are the ones able to engage in complex imaginings, off-line reflection, and counterfactual reasoning. Problems that require such capacities for their solution are representation hungry, in that they seem to cry out for the use of inner systemic features as stand-ins for external states of affairs. (147bc)

Those who like dynamic systems theory tend toward rejecting information-processing accounts “that identify specific inner states or processes as playing specific content-bearing roles.” (148b) They thus might endorse this radical thesis:

Thesis of Radical Embodied Cognition  Structured, symbolic, representational, and computational views of cognition are mistaken. Embodied cognition is best studied by means of noncomputational and nonrepresentational ideas and explanatory schemes involving, e.g., the tools of Dynamical Systems theory. (148c)

Many scientists already hold this view. (49a)

But Clark thinks such a strong reaction is unwarranted, and he will explain why in the following sections. (49b)

Andy Clark. Being There: Putting Brain, Body, and World Together Again. Cambridge, Massachusetts/London: MIT, 1997.

Andy Clark. 2.4 of Being There, “Soft Assembly and Decentralized Solutions,” summary

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Andy Clark

Being There:

Putting Brain, Body, and World Together Again

 

Ch.2

The Situated Infant

Part 2.4

Soft Assembly and Decentralized Solutions

Brief Summary:

Machines, including the ‘machinery’ of infant learning, can better adapt to complex and dynamic situations when information processing is decentralized.

Summary

Clark will discuss soft assembly in human development. (42)

Traditional robotic programming is hard assembled, because it does not make real-time adjustments, unlike the soft assembly of human motion.

A traditional robot arm, governed by a classical program, provides an example of "hard assembly." It commands a repertoire of moves, and its success depends on the precise placement, orientation, size, and other characteristics of the components it must manipulate. Human walking, in contrast, is soft-assembled in that it naturally compensates for quite major changes in the problem space. As Thelen and Smith point out, icy side- | walks, blisters, and high-heeled shoes all "recruit" different patterns of gait, muscle control, etc., while maintaining the gross goal of locomotion. Centralized control via detailed inner models or specifications seems, in general, to be inimical to such fluid, contextual adaptation. (42-43) […]

Multi-factor, decentralized approaches, in contrast, often yield such robust, contextual adaptation as a cost-free side effect. This is because such systems, as we saw, create actions from an "equal partners" approach in which the local environment plays a large role in selecting behaviors. In situations where a more classical, inner-model-driven solution would break down as a result of the model's incapacity to reflect some novel environment change, "equal partners" solutions often are able to cope because the environment itself helps to orchestrate the behavior. (43, boldface mine)

[Previously Clark describes childhood development and how many factors in both the child and his environment are equal partners in guiding its development.]

Pattie Maes invented a way for machines to determine among themselves how to distribute jobs, rather than having a centralized system handle all the data and make that decision. Each machine when creating a job asks the other machines to estimate how long they would take to perform it, and the machine most able given its current abilities and activities gets the job. Job scheduling then becomes an “emergent property” of the simple machine self-assessment and communication behaviors. (43d)

Thus

Soft assembly out of multiple, largely independent components yields a characteristic mix of robustness and variability. The solutions that emerge are tailored to the idiosyncrasies of context, yet they satisfy some general goal. This mix, pervasive throughout development, persists in mature problem solving and action. Individual variability should thus not be dismissed as "bad data" or "noise" that somehow obscures essential developmental patterns. Instead, it is, as Thelen and Smith insist, a powerful clue to the nature of underlying processes of soft assembly. (44a)

Thelen and Smith give the example of the development of child reaching behavior, where the factors and events leading up to the learned behavior vary widely between children even though the resulting behavior is similar for all. (44b)

One child began with fast flapping then dampened it. (44bc)

Another had to increase lift. (44c)

Other children exhibited other variations. The central nervous system is merely working with the physics and mechanics of the seemingly somewhat autonomous parts of the body that come to be adjusted. (44-45)

Clark writes:

the job is to learn to modulate parameters (such as stiffness) which will then interact with intrinsic bodily and environmental constraints so as to yield desired outcomes. In sum, the task is to learn how to soft-assemble adaptive behaviors in ways that respond to local context and exploit intrinsic dynamics. Mind, body, and world thus emerge as equal partners in the construction of robust, flexible behaviors. (45a.b boldface mine)

Andy Clark. Being There: Putting Brain, Body, and World Together Again. Cambridge, Massachusetts/London: MIT, 1997.

Cellular Automata and Dynamic Emergence in Humphreys "Synchronic and Diachronic Emergence"

by Corry Shores
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Paul Humphreys

"Synchronic and Diachronic Emergence"

Mind and Machines Dec. 2008

Abstract

Humphreys will contrast diachronic and synchronic emergence. He then discusses the historical aspect of Bedau's weak emergence. He argues that weak emergence is about token states and not types. Humphreys concludes by evaluating the weakness of weak emergence and discussing the lack of a unifying account for diachronic and synchronic emergence.

Introduction

Diachronic emergence differs from synchronic.

Diachronic emphasizes a novel phenomena's emergence across duration.

Synchronic stresses the lower level properties' and objects' coexistence with novel higher level ones.

Humphreys will argue:

1) Maybe someday we will be able to conceptualize a general sense of emergence that encompasses both diachronic and synchronic emergences. However, right now the two types of emergence are conceptually distinct. So for example, we know that the idea of synchronic emergence does not comply with diachronic; for, we cannot even say that synchronic emergences actually produce an emergence, because a historical development is always needed to produce the conditions for a synchronic emergence. So we might have two identical states, with only one being emergent, because its diachronic development provided for it.

2) There is a sort of emergence that we may characterize as "weak." Bedau offers this definition,

Weak emergence is the view that a system’s macro properties can be explained by its micro properties but only in an especially complicated way. (2008)

Humphreys says that the weak emergence account can explain a lot about diachronically emergent computational forms. But it can explain more if we were to add some of his suggestions.

3) To explain 'pattern emergence,' we need a satisfactory account of conceptual emergence. Humphrey's will only here discuss the philosophical implications.

Pattern Emergence

Pattern emergence

involves the appearance in a system of novel structure that results from the temporal evolution of the system. Pattern emergence is a common phenomenon in computational models such as agent based simulations and cellular automata, and it is widely agreed within the complexity theory literature that these patterns count as examples of emergent phenomena.

Humphrey's will use examples of pattern emergence in his argument. They are cellular automata with a spatial structure. He uses models rather than real world phenomena, because models are simpler to handle. These examples will also help us deal with the sociological notion of methodological individualism. Emergences from patters begin from the bottom up. So we are not concerned with a centralized 'top down' set of laws or rules that govern individuals' behavior.

Cellular Automata

Humphrey's illustrates weak emergence with a cellular automaton (CA). His example is two dimensional, so I will begin with an simpler one dimensional version. Cellular automata are little living abstract machines. They appear as patterns of squares that change according to a finite set of rules. Their space is a grid. In our one-dimensional version, it is a row of squares.

We apply transformation rules to the squares in that row, and place the changes in a new row below it.

So let's begin with this row, with just one darkened square in it.

We see that the black square has two neighbors: a white square on either side.

The other white squares have white neighbors.

We will go through each square, and depending on the contents of the neighborhood, we will transform the center block in the following row according to these rules. [Click to enlarge]

So we begin with the first neighborhood, and we see that we transform the white center block into another white block.

Now we have this for the next row so far:

And we do the same for the next neighborhood.

Now our second row is:

The next neighborhood has a black box on the one end, so we transform the center block to a dark one.

Now we have

Same for the original darkened box for the first row:

Which gives us

We continue this process for the rest of the row, and we obtain the full second row.

Now we repeat the procedure, this time only attending to the contents of the second row. We then obtain for the third row:

And then for the fourth row:

And next

Now notice the white "T" shape in the lower right side in the image below. This will appear as a triangle, and many more such triangles of various sizes will show as we continue the process.

Also, we presume from the beginning that the grid extends infinitely, so our picture above may continue expanding outward. The patterns that emerge from this automata are pseudo-random. Nonetheless, the character of its randomness emerges as the iterations proliferate. [Credits for the following images are given below. Click to enlarge. Thank you sources. Image 1]

[2]

[3]

[4], [This needs to be clicked to see what is going on, it's large.]

Humphrey has us consider a more complex two dimensional example. We are to imagine that

Over a wide range of initial states of the CA, appropriate updating rules can produce randomly distributed arrays of colored cells, stable patterns that persist across time, and dynamic patterns that evolve over time.

We are to suppose that the given transformation instructions will eventually produce the following form [5]:

To arrive at this form, we presumably had to carry-out the transformation operations many many times. What is important is that we cannot deduce that this form will come about merely by knowing the initial conditions. We always need to know the formation's preceding step, and its preceding step, and so on until we arrive back at the starting situation. Astronomers, however, can predict the next solar eclipses without needing to know the sun's positions in between eclipses. Such predictions are computationally "compressible," where cellular automata are "incompressible."

I shall examine weak emergence in detail in a moment, but the essence of the idea is that a state of a system is weakly emergent just in case that state can be produced only through a step-by-step simulation of the system. In other words, the process that leads up to the state is computationally incompressible. In yet other words, unlike the prediction of future solar eclipses for which the computational difficulty of prediction is almost independent of how far into the future the eclipse will take place, predictions of future states of computationally incompressible systems must run through each of the intermediate time steps between the initial state and the predicted state. Letting the computational model work out its own development is thus the only effective way to discover how the system’s states evolve. The philosophical motivation for accepting this criterion as capturing a certain kind of emergence draws on the philosophical tradition that emphasizes the essential unpredictability of emergent phenomena. (emphasis mine)

Properties of Pattern Emergence

For there to be emergence, something must emerge from something else. The bow-tie pattern above emerged from the pattern of its rule-based re-iterations. If instead we made a stamp with that image, and stamped it on paper, we would not say that the formation emerged on the paper.

It is another token of the same pattern, but that token is not emergent because it is generated instantaneously.

This example tells us three things:

1) Pattern emergence is largely a "historical" phenomenon: "whether an instance of a pattern is emergent or not depends essentially upon the process that generated it." Thus we cannot look at the synchronic elements of some formation and determine whether or not it is emergent.

2) It cannot be synchronic, because that means we can look at a formation and determine the emergence. It also means that two of the same formations would bring about the same emergence. But we see that both the stamp and the automata have the same formation, but the stamp did not result from a pattern emergence.

3) If we looked at the two identical bow-tie patterns, one from automata, the other from an ink-stamp, and if we say there is the same formation, we are dealing with a type of formation. But we saw that in the one instance there was pattern emergence, but in the other instance there was not. Hence, "pattern emergence is about tokens or instances of patterns, not about types."

Weak Emergence

So we see with the automata that there are lower level circumstances and higher level properties. We call the lower level circumstances micro-facts and the higher level we call the macro-level. On the micro-level of cellular automata, there are only squares. So the property of being "bow-tie shaped" can only occur on the macro-level. We call such a property that can only appear on the higher level a nominally emergent property. We obtained it only by running the simulation that re-iterated the transformation rules. And we may characterize the individual square on the micro-level only in terms of its location and such intrinsic traits as it being black or white. The macro-level entity then is the aggregate of all the micro-constitutent states taken together along with their spatial relations. Because the whole of the macro-level structure can be reduced to the conglomeration of the micro-level states and locations, we call such a system a "locally reducible system."

Humphreys then defines weak emergence as (citing Bedau):

Assume that P is a nominally emergent property possessed by some locally reducible system S. Then P is weakly emergent if and only if P is derivable from all of S’s micro facts but only by simulation.’

We need to run the simulation, because the pattern must be computationally incompressible. Hence the bow-tie structure exemplifies weak emergence.

For the most part, this definition captures the sorts of emergences found in dynamical systems theory and in complexity theory. However, to fully apply, we need to supplement it in two ways.

1) We need to distinguish end states that are non-random from those which have "predicates picking out genuine macro-level properties."

1a) If the pattern begins random and ends random, that does not qualify as a novel emergence. For, even though each random pattern on its own is unique, all random patterns are of the type "random." So nothing is new when random transitions to random.

1b) If the pattern begins structured and ends ordered, then here also there would not be an emergence. For, we consider emergences to result from self-organization. But a turn to chaos is a self-disorganization. One thing that makes emergences so interesting is that they defy the Second Law of Thermodynamics (entropy). So we will not consider a random outcome from an ordered beginning to be emergent. It must have structure to be emergent. But there is a continuum between order and chaos, structure and random, so there is no clear way to make this distinction.

2) Maintaining the automata in a perpetually random state is not difficult to achieve. So we should not devalue the concept of weak emergence by including such cases.

Hence Humphreys offers this revised definition for weak emergence:

P is a non-random property of the system S that is distinct from any property possessed by the initial state of S.

Micro-stable and Micro-dynamic Patterns

Humphreys says that synchronic features are neither sufficient nor necessary conditions for pattern emergence.

Broadly speaking, there are two types of pattern emergences.

1) Micro-stable patterns.

Micro-stable patterns emerge when the computational process no longer develops because it has reached a stable non-random pattern which the transformation principles no longer alter. Our bow-tie pattern is such a case where there was a terminal form that maintained its exact shape.

2) Micro-dynamic patterns.

Micro-dynamic patterns emerge when a non-random pattern emerges, and that pattern itself cycles invariantly, on account of the transformation substitutions falling into a consistent pattern. To view a micro-dynamic pattern, scroll to the bottom of this page, and run the simulator.

We may distinguish three sub-types of micro-dynamic patterns.

2a) Recirculating autonomy

The micro-dynamic pattern exhibits recirculating autonomy if its same consistent parts cycle around in a constant pattern. We see this for example when we heat a fluid between two hot plates. Inter-locking columns of circulating fluid cycle in their cellular regions. These are calledBénard convection cells. [6]

[7]

[8]

[9]

Animation can be found here.

Another example of recirculating autonomy is Couette flow. This comes-about when we place a fluid between two concentric rotating cylinders that are moving at different speeds. "Vortex rolls form when the velocity gradient exceeds a critical value."

[10]

[11], [12]

[13]

In recirculating stability, a structure emerges that consists of a fixed collection of entities, and this structure remains constant while the micro-level entities undergo dynamic cycling across duration.

2) Transient autonomy

In transient autonomy, the macro-structure emerges and persists. But unlike recirculating autonomy where the micro-entities remained the same while changing location, in transient autonomy the micro-constituents are continually substituted by the same type of entity. We may see this in river flows, for example, where the ripples repeat, but always with new water molecules. [14]

3) Equivalent class autonomy

Equivalent class autonomy occurs when the macro-level pattern is sustained by the cycling of lower level entities that are of the same general type. for example, if our river ripples came to be replaced by wine, beer, broth, etc.

Hence there are two roles for micro-process dynamics in pattern formation and perpetuation.

A) Bringing about the initial formation of the structured pattern. This role is important when considering pure diachronic emergence.

B) The persistence of the pattern while the micro-dynamics continue. This role draws upon both diachronic and synchronic emergence, because the stability of the cycling dynamics suggests a trans-temporal 'synchronic' form that manifests dynamically.

For these cycling patterns, again, see the animation at the bottom of this page. Humphreys offers an example where we begin with a random distribution of values [15]

and after 500 iterations, a spiral pattern emerges [16]

Humphreys then wonders if we can characterize this formation as "spiral" as though there were something about it to make it different from any other spatial pattern of cells.

Pattern Emergence and Supervenience

Consider if the bow-tie were a different color. Would it be the same pattern? It's not clear what our criteria will be for distinguishing patterns.

If I am correct about the role that this kind of pattern persistence plays in diachronic emergence, then whether the criteria for pattern identity turn out to be objective, subjective, pragmatic, or based on some other ground, that feature will automatically carry over to computational diachronic emergence itself.

Humphreys, Paul. "Synchronic and Diachronic Emergence." Minds and Machines. Vol.18, Number 4, December, 2008, pp.431-442.

More information and online text available at:

http://www.springerlink.com/content/d442431150343t17/?p=f1cef51a00d346b582d2c3ad1386c814π=1

Image credits:

[1], [3]

http://mathworld.wolfram.com/Rule30.html

[2]

http://www.jeffcook.com/skimboard/index.html

(The Author writes: Wolfram's "A New Kind of Science." ... The first 50 iterations of Rule 30 are shown on page 25 of Wolfram's book.)

[4]

http://commons.wikimedia.org/wiki/File:Rule_30_2000Generations.gif

[5], [15], [16]

Humphreys, Paul. "Synchronic and Diachronic Emergence." Minds and Machines. Vol.18, Number 4, December, 2008, pp.431-442.

http://www.springerlink.com/content/d442431150343t17/fulltext.html

[6]

http://hnsttl.blogspot.com/2008/10/downward-causation-in-fluids.html

[7]

http://web.phys.tue.nl/nl/de_faculteit/capaciteitsgroepen/transportfysica/fluid_dynamics_lab/turbulence_vortex_dynamics/rotating_rayleigh_benard_convection/

Schematic drawing of a Rayleigh-B\'enard cell. The red-shaded areas of the cell show regions of hot fluid, while the blue areas indicate cold fluid. Adapted from L. Kadanoff, Physics Today 54, 34 (2001).

[8]

http://www.etl.noaa.gov/about/eo/science/convection/RBCells.html

[9]

http://www.catea.gatech.edu/grade/mecheng/mod8/mod8.html

[10]

http://serve.me.nus.edu.sg/limtt/TCF_0.8.JPG

[11], [12]

http://www.mech.northwestern.edu/fac/lueptow/TC_Rich_new.html

[13]

http://www.vacet.org/gallery/images_video/flux_400.png

[14]

http://riverweaver.net/