Models of Action

Mechanisms for Adaptive Behavior

Clive D.L. Wynne, John E.R. Staddon

• Publisher: Psychology Press
• Year: 2013
• eBook ISBN: 9781134787579
• Topic: Psychology
• Subtopic: Experimental Psychology
• Index: Psychology

This volume presents an international group of researchers who model animal and human behavior--both simple and complex. The models presented focus on such subjects as the pattern of eating in meals and bouts, the energizing and shaping impact of reinforcers on behavior, transitive inferential reasoning, responding to a compound stimulus, avoidance and escape learning, recognition memory, category formation, generalization, the timing of adaptive responses, and chromosomes exchanging information. The chapters are united by a common interest in adaptive behavior--whether of human, animal, or artificial system--and clearly demonstrate the rich variety of ways in which this fascinating area of research can be approached.

In so doing, the book demonstrates the range of thought that qualifies as theorizing in the contemporary study of the mechanisms of adaptive behavior. It has two purposes: to bring together a very wide range of approaches in one place and to give authors space to explain how their ideas developed. Journal literature often presents fully-formed theories with no explanation of how an idea came to have the shape in which it is presented. In this volume, however, leaders in different fields provide background on the development of their ideas. Where once psychologists and a few zoologists had this field to themselves, now various types of computer scientists have added great energy to the mix.

1

Learning the Language of Mind: Symbolic Dynamics for Modeling Adaptive Behavior

Ben Goertzel

College of Staten Island

Intelli Genesis Corp.

Symbolic dynamics is an analytical technique that reveals the linguistic structures hidden in the chaotic attractors of dynamical systems. I argue here that symbolic dynamics deserves a place at the center of cognitive and behavioral modeling.

The applicability of symbolic dynamics is explored on several different levels. After an exposition of the historical roots and basic concepts of symbolic dynamics, the relevance of these concepts to data analysis is explored by considering the problem of predicting mood fluctuations. Then, moving to the brain level, various techniques for constructing emergent grammar-producing neural networks are described. Next, the psynet model of psychological dynamics is outlined, and in this context it is argued that the recursively modular structure of the brain may perhaps reflect itself in emergent phrase-structure grammars. Finally, the role of symbolic dynamics as a unifier of various psychological modeling strategies is discussed, and in this light some new ideas on topological language spaces are introduced.

Language, which presumably evolved for the description of observable situations and personal feelings, has long since turned into something more powerful. It can be used to describe anything from the counterintuitive behavior of quarks to the imagined mating habits of space aliens. It is indispensable, not only to communication, but to many of our thought processes as well.

Even so, the languages that we use in everyday life are only a small subset of the realm of possible languages. In theoretical computer science, one finds the notion of a formal language—defined as a collection of entities (the alphabet) together with a collection of combination rules. The rules tell you which combinations of entities are correct and which are incorrect, or else which are more correct than others. The universe of formal languages contains all human languages, from Arabic to Yiddish, and an infinite realm of other languages as well, some extremely simple and some incomprehensibly complex.

What use are all these other languages, one might wonder, if no one can speak them? A principal use is in the theory of computer programming languages. What I argue here is that these formal languages are also essential for the modeling of mind and brain. The key to this relationship is the technique of symbolic dynamics.

Symbolic dynamics is a mathematical/computational tool for expressing patterns of change over time as formal languages. It appears in mathematics and physics as a tool for the study of nonlinear iterations and differential equations, and for the analysis of complex time-series data. It involves the transformation of a sequence of values (e.g., real vectors) representing the trajectory of some dynamical system, into a sequence of letters or symbols drawn from an abstract alphabet. The patterns in the system’s behavior make themselves apparent as linguistic patterns in the corpus of abstract symbol combinations. The emergent symbol system does not entirely capture the underlying nonlinear dynamics, but it captures the most significant abstract patterns in this dynamic.¹

I argue that symbolic dynamics deserves to play a prominent role in the study of complex adaptive behavior. It has the potential to clarify at least two important issues: the meaning of abstract models of mind/brain, and the relation between symbolic reasoning and nonlinear neurodynamics. Symbolic dynamics presents a new vision of behavioral science as the study of dynamical systems and the topology of formal language spaces.

The structure of the chapter is as follows. The following section traces the concept of symbolic dynamics back to Leibniz, and casts Leibniz’s philosophical speculations in mathematical form with a statement called the “Chaos Language Hypothesis.” The next section introduces some basic ideas from dynamical systems theory and symbolic dynamics, using the Baker map and its generalizations as an illustrative example. Then, I explore the possibility of tracking human moods using symbolic dynamics, using some recent empirical data as a launching point for speculative, illustrative examples. I next demonstrate that simple neural networks are capable of manifesting arbitrary formal grammars in their chaotic dynamics, and discuss two methods for training neural networks to display structured chaos: the genetic algorithm, which has proven successful in simple instances, and an untried technique called attractor pattern learning. Finally, I discuss the role of symbolic dynamics in validating qualitative models of adaptive behavior, and toward this end present a novel topology for spaces of formal languages.

PAST AND FUTURE OF SYMBOLIC DYNAMICS

Only in the past few decades has symbolic dynamics emerged as a useful mathematical tool. Its conceptual roots, however, go back at least to Leibniz (1969). Leibniz proposed a “universal character for ‘objects of imagination’”; he argued that “a kind of alphabet of human thoughts can be worked out and that everything can be discovered and judged by a comparison of the letters of this alphabet and an analysis of the words made from them” (pp. 221–222). This systematization of knowledge, he claimed, would lead to an appreciation of subtle underlying regularities in the mind and world: “[T]here is some kind of relation or order in the characters which is also in things … there is in them a kind of complex mutual relation or order which fits things; whether we apply one character or another, the products will be the same” (p. 225). In modern language, the universal characteristic was intended to provide for the mathematical description of complex systems like minds, thoughts, and bodies, and also to lead to the recognition of robust emergent properties in these systems, properties common to wide classes of complex systems. These emergent properties were to appear as linguistic regularities.

Leibniz did not get very far with this idea. He developed the language of formal logic (what is now called Boolean logic), but, like the logic-oriented cognitive psychologists and AI theorists of the 1960s–1980s, he was unable to build from the simple formulas of propositional logic to the complex, self-organizing systems that make up the everyday world. Today, with dynamical systems theory and other aspects of complex systems science, we have come much closer to realizing his ambitions. (One might argue, however, that in the intervening centuries, Leibniz’s work contributed to the partial fulfillment of his program. The formal logic that he developed eventually blossomed into modern logic, computer science, and formal language theory. It is computer power, above all, that has enabled us to understand what little we do about the structure and dynamics of complex systems.)

Leibniz and Abraham (1995) pointed out the remarkable similarity between Leibniz’s Universal Characteristic and the modern idea of a unified theory of complex system behavior; they singled out the concepts of attractors, bifurcations, Lyapunov exponents, and fractal dimensions as being important elements of the emerging universal characteristic. It is symbolic dynamics, however, that provides by far the most explicit and striking parallel between Leibniz’s ideas and modern complex systems science. Symbolic dynamics does precisely what Leibniz prognosticated: It constructs formal alphabets, leading to formal “words” and “sentences” that reveal the hidden regularities of dynamical systems. As yet it does not quite live up to Leibniz’s lofty aspirations—but there is reason to be optimistic.

Leibniz’s speculations resonate wonderfully with recent ideas about the possibility of finding unifying laws of complex systems science (Goerner, 1993; Goertzel, 1994). Many have hypothesized the existence of such laws, but no one has been particularly clear about what form such laws might take. Symbolic dynamics gives some concrete ideas in this direction. Leibniz’s idea of the Universal Characteristic, translated into modern terminology, suggests that there may be a small number of archetypal attractor structures common to a wide variety of complex systems. Mathematically speaking, the approximation of these archetypes should reveal itself as a clustering of inferred formal languages in formal language space (a notion that can be formalized by defining, as in the final section, an appropriate topology on formal language space).

This train of thought leads to the following formal hypothesis:

Chaos Language Hypothesis: The formal languages implicit in the trajectories of natural (in particular, psychological and social) dynamical systems show a strong tendency to “cluster” in the space of formal languages.

And this hypothesis suggests the following three-stage research program:

1. By computational analysis of data obtained from empirical studies and mathematical models, try to isolate the archetypal formal languages underlying complex psychological and social systems.

2. Analyze these languages to gain an intuitive and mathematical understanding of their structure.

3. Correlate these languages, as far as possible, with other quantitative and qualitative characterizations of the systems involved.

I suggest that this research program may be particularly fruitful in the context of cognitive and behavioral modeling. The brain, complex as it is, has proved particularly resistant to detailed quantitative analysis. It may be that linguistic analysis is the only way to go.

DYNAMICS, CHAOS, AND LINGUISTIC PATTERN

Symbolic dynamics has emerged out of the branch of mathematics known as dynamical systems theory. In this section, I review some concepts from dynamical systems theory and explain how symbolic dynamics connects the mathematics of differential equations and discrete iterations with the mathematics of formal languages.

Dynamics, in the most general sense, is the study of how things change over time. Mathematical dynamical systems theory is concerned mainly with the behavior of systems whose change over time is governed by briefly stated equations. It also, however, gives us deep qualitative insight into the behavior of the more complex dynamical systems that we see in the real world.

The key concept of dynamical systems theory is the attractor. An attractor is, quite simply, a characteristic behavior of a system. The striking insight of dynamical systems theory is that, for many mathematical and real-world dynamical systems, the initial state of the system is almost irrelevant. No matter where the system starts from, it will eventually drift into one of a small set of characteristic behaviors, a small number of attractors.

Some systems have fixed-point attractors, meaning that they drift into certain equilibrium conditions and stay there. Some systems have periodic attractors, meaning that after an initial transient period, they lock into a cyclic pattern of oscillation between a certain number of fixed states. Finally, some systems have attractors that are neither fixed points nor limit cycles, and are hence called strange attractors. The most complex systems possess all three kinds of attractors, so that different initial conditions lead not only to different behaviors, but to different types of behavior. To complicate things further, strange attractors come supplied with invariant measures, indicating the frequency with which different regions of the attractor are visited. Some dynamical systems have very simple attractors hosting subtly structured invariant measures; an example of this is given shortly.²

The formal definition of strange attractor is a matter of some contention. Rather than giving a mathematical definition, I prefer to give a dictionary definition that captures the common usage of the word. A strange attractor of a dynamic, as I use the term, is a collection of states that is: (a) invariant under the dynamic, in the sense that if one’s initial state is in the attractor, so will be all subsequent states; (b) attracting in the sense that states that are near to the attractor but not in it will tend to get nearer to the attractor as time progresses; and (c) not a fixed point or limit cycle.

A great deal of attention has been paid to the fact that some dynamical systems are chaotic, meaning that, despite being at bottom deterministic, they are capable of passing many statistical tests for randomness. They look random. Under some d...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Contributors
• Preface
• 1 Learning the Language of Mind: Symbolic Dynamics for Modeling Adaptive Behavior
• 2 Neural Substrates of Adaptively Timed Reinforcement, Recognition, and Motor Learning
• 3 Can the Whole Be Something Other Than the Sum of Its Parts?
• 4 The First Principle of Reinforcement
• 5 Using Biology to Solve a Problem in Automated Machine Learning
• 6 The Frightening Complexity of Avoidance: A Neural Network Approach
• 7 In Praise of Parsimony
• 8 A Minimal Model of Transitive Inference
• Author Index
• Subject Index