By Michael Leyton

The objective of this booklet is to improve a generative conception of form that has homes we regard as basic to intelligence –(1) maximization of move: every time attainable, new constitution could be defined because the move of present constitution; and (2) maximization of recoverability: the generative operations within the conception needs to permit maximal inferentiability from facts units. we will exhibit that, if generativity satis?es those simple standards of - telligence, then it has a strong mathematical constitution and significant applicability to the computational disciplines. The requirement of intelligence is very very important within the gene- tion of complicated form. there are many theories of form that make the iteration of advanced form unintelligible. besides the fact that, our idea takes the wrong way: we're excited about the conversion of complexity into understandability. during this, we are going to boost a mathematical conception of und- standability. the difficulty of understandability comes right down to the 2 simple rules of intelligence - maximization of move and maximization of recoverability. we will express easy methods to formulate those stipulations group-theoretically. (1) Ma- mization of move can be formulated when it comes to wreath items. Wreath items are teams within which there's an top subgroup (which we are going to name a regulate staff) that transfers a decrease subgroup (which we are going to name a ?ber crew) onto copies of itself. (2) maximization of recoverability is insured whilst the keep an eye on crew is symmetry-breaking with appreciate to the ?ber group.

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We shall say that the program is recoverable from the data set. Recoverability of the generative program places strong constraints on the inference rules by which recovery takes place, and on the programs that will be inferred. This, in turn, produces a theory of geometry that is very diﬀerent from the current theories of geometry. Essentially, the recoverability of generative operations from the data set means that the shape acts as a memory store for the operations. More strongly, we will argue that all memory storage takes place via geometry.

Unity comes from the maximization of transfer, and transfer is expressed as wreath products. 34 1. 19 Shape Generation by Group Extensions The concept of group extension is basic to our generative theory. A group extension takes a group G1 and adds to it a second group G2 to produce a third, more encompassing, group G, thus: G1 E G2 = G where E is the extension operation. See our book on group extensions, Leyton [98]. It is clear, looking back over the previous sections, that according to our theory: Shape generation proceeds by a sequence of group extensions.

13). What we have just said illustrates a basic point that will be made in Chapter 20: With respect to scientiﬁc structure, there is the following correspondence. Conservation Laws ←→ Wreath Products. Mathematically we will construct this by setting up a correspondence between any pair of commuting observables and the wreath product of their 1-parameter groups. 13 1. Transfer Maximization of Transfer In the preceding sections, we have seen that several major domains are structured by transfer. (1) Human perception: The human perceptual system is organized as an nfold wreath product of groups, G1 w G2 w .