DIMACS Technical Report 2007 - 09 September 2007 Process Grammar

This report gives an exposition of the Process-Grammar, published originally in the journal Artificial Intelligence in 1988, together with a description of some of the subsequent applications of the grammar in meteorology, biology, computer-aided design, chemical engineering, and geology. The Process-Grammar is a means of recovering the process-history of a smooth shape from its curvature extrema, and expressing that evolution in terms of transitions at those extrema. The inference of history follows from the Symmetry-Curvature Duality Theorem of Leyton (1987), which states that, to each curvature extremum, there is a differential symmetry axis leading to and terminating at that extremum; and from an inference rule that states that the symmetry axis is the record of a process. The Process-Grammar expresses the relationship between any two stages in the shape’s history as an extrapolation of the processes inferred by the theorem. 1 Extraction of History from Shape The purpose of this paper is to describe a grammar that I published in the journal Artificial Intelligence in 1988. The grammar is essentially a theorem I proved that any smooth shape evolution of a smooth 2D curve can be expressed in terms of six types of transitions at curvature extrema. These transitions constitute what I call the Process-Grammar. After I published the grammar, the grammar, and the mathematics on which it is based, was applied by scientists in many disciplines: Radiology, meteorology, computer vision, chemical engineering, geology, computer-aided design, robotics, anatomy, botany, forensic science, architecture, abductive reasoning, linguistics, mechanical engineering, computer graphics, archaeology, etc. Let us begin by understanding the purpose for which the grammar was developed: inferring history from shape; e.g., from the shapes of tumors, embryos, clouds, etc. For example, the shape shown in Fig 1 can be understood as the result of various processes such as protrusion, indentation, squashing, resistance. My book Symmetry, Causality, Mind (MIT Press) was essentially a 630-page rule-system for deducing the past history that formed any shape. The Process-Grammar is part of that rule-system – that related to the use of curvature extrema. Figure 1: Shape as history. 2 The PISA Symmetry Analysis It is first necessary to understand how symmetry can be defined in complex shape. Clearly, in a simple shape, such as an equilateral triangle, a symmetry axis is easy to define. One simply places a straight mirror across the shape such that one half is reflected onto the other. The straight line of the mirror is then defined to be a symmetry axis of the shape. However, in a complex shape, it is often impossible to place a mirror that will reflect one half of the figure onto the other. Fig 1, is an