The purpose of this book is to develop a generative theory of shape that has two properties we regard as fundamental to intelligence –(1) maximization of transfer: whenever possible, new structure should be described as the transfer of existing structure; and (2) maximization of recoverability: the generative operations in the theory must allow maximal inferentiability from data sets. We shall show that, if generativity satis?es these two basic criteria of - telligence, then it has a powerful mathematical structure and considerable applicability to the computational disciplines. The requirement of intelligence is particularly important in the gene- tion of complex shape. There are plenty of theories of shape that make the generation of complex shape unintelligible. However, our theory takes the opposite direction: we are concerned with the conversion of complexity into understandability. In this, we will develop a mathematical theory of und- standability. The issue of understandability comes down to the two basic principles of intelligence - maximization of transfer and maximization of recoverability. We shall show how to formulate these conditions group-theoretically. (1) Ma- mization of transfer will be formulated in terms of wreath products. Wreath products are groups in which there is an upper subgroup (which we will call a control group) that transfers a lower subgroup (which we will call a ?ber group) onto copies of itself. (2) maximization of recoverability is insured when the control group is symmetry-breaking with respect to the ?ber group.

The purpose of this book is to develop a generative theory of shape that has two properties we regard as fundamental to intelligence –(1) maximization of transfer: whenever possible, new structure should be described as the transfer of existing structure; and (2) maximization of recoverability: the generative operations in the theory must allow maximal inferentiability from data sets. We shall show that, if generativity satis?es these two basic criteria of - telligence, then it has a powerful mathematical structure and considerable applicability to the computational disciplines. The requirement of intelligence is particularly important in the gene- tion of complex shape. There are plenty of theories of shape that make the generation of complex shape unintelligible. However, our theory takes the opposite direction: we are concerned with the conversion of complexity into understandability. In this, we will develop a mathematical theory of und- standability. The issue of understandability comes down to the two basic principles of intelligence - maximization of transfer and maximization of recoverability. We shall show how to formulate these conditions group-theoretically. (1) Ma- mization of transfer will be formulated in terms of wreath products. Wreath products are groups in which there is an upper subgroup (which we will call a control group) that transfers a lower subgroup (which we will call a ?ber group) onto copies of itself. (2) maximization of recoverability is insured when the control group is symmetry-breaking with respect to the ?ber group.

Recent years have seen dramatic progress in shape recognition algorithms applied to ever-growing image databases. They have been applied to image stitching, stereo vision, image mosaics, solid object recognition and video or web image retrieval. More fundamentally, the ability of humans and animals to detect and recognize shapes is one of the enigmas of perception. The book describes a complete method that starts from a query image and an image database and yields a list of the images in the database containing shapes present in the query image. A false alarm number is associated to each detection. Many experiments will show that familiar simple shapes or images can reliably be identified with false alarm numbers ranging from 10-5 to less than 10-300. Technically speaking, there are two main issues. The first is extracting invariant shape descriptors from digital images. Indeed, a shape can be seen from various angles and distances and in various lights.

Leyton's Process Grammar has been applied by scientists and engineers in many disciplines including medical diagnosis, geology, computer-aided design, meteorology, biological anatomy, neuroscience, chemical engineering, etc. This book demonstrates the following: The Process Grammar invents several entirely new concepts in biological morphology and manufacturing design, and shows that these concepts are fundamentally important. The Process Grammar has process-inference rules that give, to morphological transitions, powerful new causal explanations. Remarkably, the book gives a profound unification of biological morphology and vehicle design. The book invents over 30 new CAD operations that realize fundamentally important functions of a product. A crucial fact is that the Process Grammar is an example of the laws in Leyton's Generative Theory of Shape which give the ability to recover the design intents for which the shape features of a CAD model were created. The book demonstrates that the Process Grammar recovers important design intents in biological morphology and manufacturing design. In large-scale manufacturing systems, the recovery of design intents is important for solving the interoperability problem and product lifecycle management. This book is one of a series of books in Springer that elaborates Leyton's Generative Theory of Shape.

This book discusses the origins of ornamental art — illustrated by the oldest examples, dating mostly from the paleolithic and neolithic ages, and considered from the theory-of-symmetry point of view. Because of its multidisciplinary nature, it will interest a wide range of readers: mathematicians, artists, art historians, architects, psychologists, and anthropologists.The book represents the complete analysis of plane symmetry structures, so it can be used by artists as a guide to the creation of new symmetry patterns. Some parts of the contents (such as Chapter 4, about conformal symmetry, and Chapter 6, about modularity in art) give the reader an opportunity to develop computer programs for producing images illustrating the corresponding symmetry forms.

In putting 'morphonology' up for adoption as a chapitre particulier in 1929, Trubetzkoy started a debate regarding the boundary between phonology and morphology that has not ended yet. Essentially a record of a roundtable devoted to that boundary (Montréal, October 1994), Trubetzkoy's Orphan is a full and fascinating picture of some very important contemporary attempts to define it. In addition to papers that focus on it, the volume also contains important papers on the closely related topics of 'morphoprosody' and the 'lexicon', views from 'the floor' and 'the outside', and edited transcripts of the discussions that took place at the Montréal Roundtable. Intended both for practicising and future phonologists and morpho-logists, Trubetzkoy's Orphan is a valuable record of a very important debate regarding one of the most central questions in phonology and morphology.

Wolfgang Wildgen presents three perspectives on the evolution of language as a key element in the evolution of mankind in terms of the development of human symbol use. (1) He approaches this question by constructing possible scenarios in which mechanisms necessary for symbolic behavior could have developed, on the basis of the state of the art in evolutionary anthropology and genetics. (2) Non-linguistic symbolic behavior such as cave art is investigated as an important clue to the developmental background to the origin of language. Creativity and innovation and a population's ability to integrate individual experiments are considered with regard to historical examples of symbolic creativity in the visual arts and natural sciences. (3) Probable linguistic 'fossils' of such linguistic innovations are examined. The results of this study allow for new proposals for a 'protolanguage' and for a theory of language within a broader philosophical and semiotic framework, and raises interesting questions as to human consciousness, universal grammar, and linguistic methodology. (Series B)