Regular Structures: Lectures in Pattern Theory Volume III
This is the third and final volume of the Lectures in Pattern Theory. Its two first chapters describe the 5cience- theoretic principles on which pattern theory rests. Chapter 3 is devoted to the algebraic study of regularity while Chapter 5 contains new results in metric pattern theory. Some brief remarks on topological image algebras can be found in Chapter 4. Two chapters deal with pattern synthesis: Chapter 6 on scientific hypothesis formation and Chapter 7 on social domination structures. In Chapter 8 we study taxonomic pat- terns, both their synthesis and analysis, while in the last chapter we investigate a pattern processor for doing semantic abduction. TABLE OF CONTENTS INTRODUCTION . . . . . CHAPTER I. PATTERNS: FROM CHAOS TO ORDER The search for regularity Some regular structures . . . The mathematical study of regularity. CHAPTE R 2. A PATTERN FORMALISM. 2.1. The principle of atomism. 2.2. The combinatory principle 2.3. The principle of observabi1ity. 2.4. The principle of realism. CHAPTER 3. ALGEBRA OF REGULAR STRUCTURES, Generator coordinates . . . Configuration coordinates . Connectors. . . . . . . . . Configuration homomorphisms Configuration categories. . Set operations in 5f(9i'). . Operations on images. . . . . . . . . . . Homomorphisms for given global regularity Representations by image isomorphisms CHAPTER 4, SOME TOPOLOGY OF IMAGE ALGEBRAS. A topology for configurations A topology for images . . Some examples . . . . . . CHAPTER 5. METRIC PATTERN THEORY. Regularity controlled probabilities Conditioning by regularity. . . . . Frozen patterns: finite G and n . . . Frozen patterns: infinite G and finite n. Quadratic energy function . . . . . . Frozen patterns: infinite G and n. . Asymptotically minimum energy . . . . . . Asymptotics for large configurations. . . Spectral density matrix for E = LINEAR(y) . . Factorization of the spectral density matrix. Representation of the random configurations . Spectral density matrix for E = LATTICE(y). . Factorization of the spectral density matrix in two dimensions . . . . . . . . . . . . . Representations of the random configurations in the two dimensional case . . . . . Laws of large numbers in pattern theory . . . Random dynamics for configurations. . . . . . CHAPTER 6. PATTERNS OF SCIENTIFIC HYPOTHESES. Hypotheses as regular structures. . . Patterns of statistical hypotheses. . Generators for statistical hypotheses Examples of configurations. . Hypotheses as images. . . . . . Image algebras of hypotheses. . Conclusions . . . . . . . . . . CHAPTER 7. SYNTHESIS OF SOCIAL PATTERNS OF DOMINATION 353 Patterns in mathematical sociology. Domination regularity . . , . . . Configuration dynamics. . . . . . System in equilibrium . . , . . . Large configurations - simulation results Large configurations - analytical results Further problems and extensions Appendix. . . . . . . CHAPTER 8. TAXONOMIC PATTERNS. . . . . A logic for taxonomic patterns. . . . Logic of taxonomic affinity patterns. . . Synthesis of taxonomic affinity patterns. Analysis of affinity patterns . . . . CHAPTER 9. PATTERNS IN MATHEMATICAL SEMANTICS Introduction. . . . . . . . . . . . . Introducing mathematical semantics. . . . Formalization through regular structures. Two special image algebras. The choice of language type for the study Semantic maps . . . . Special semantic maps Learning semantics. . Abduction of semantic maps. OUTLOOK. APPENDIX NOTES. . BIBLIOGRAPHY INDEX. . . .