| … | … | @@ -107,8 +107,9 @@ jigsaw much assmble so as to build be bridge between constraints. The |
| 107 | 107 | satisfiability algorithm must be fast, but also the bridge that is build |
| 108 | 108 | mst be minimal or optimal in some way. |
| 109 | 109 | |
| 110 | ### Minimization |
| 110 | 111 | The words "minimal" and "optimal" make me personally think of minimal |
| 111 | | surfaces: soap bubbles, for instance. Geodesics in Rimannian geometry. |
| 112 | surfaces: soap bubbles, for instance. Geodesics in Riemannian geometry. |
| 112 | 113 | Assemblies of jigsaws do not have smooth differentiable structure, but |
| 113 | 114 | still, some of the ideas carry over analogously, because there are |
| 114 | 115 | several natural ideas of "minimal" for jigaw assemblies. For example: |
| … | … | @@ -116,6 +117,7 @@ the assembly that uses the minimal number of pieces. Or, if we weight |
| 116 | 117 | each piece with e.g. log probability (i.e. "entropy" or "information") |
| 117 | 118 | then minimize the sum of that. |
| 118 | 119 | |
| 120 | ### Homtopic deformation |
| 119 | 121 | The general "feel" of continutity is taken up by the Scott topology, |
| 120 | 122 | where we can talk about the homotopic deformation of jigsaw aassemblies. |
| 121 | 123 | There is a very easy way of understanding this if you are a software |
| … | … | @@ -143,6 +145,7 @@ the programmer wrote the C/C++ code to do, the machine binary will do |
| 143 | 145 | the same thing. The transformation from one to the other was |
| 144 | 146 | accomplished in a very long sequence of small steps. |
| 145 | 147 | |
| 148 | ### Differentiability |
| 146 | 149 | What about differentiability? Even that has an analog: finite |
| 147 | 150 | differences. When collections of jigsaws can be assigned a number (e.g. |
| 148 | 151 | total number of pieces; total weight; length of shortest path; total |
| … | … | @@ -162,5 +165,60 @@ and when one has paths that can take you from here to there in some |
| 162 | 165 | quasi-continuous fashion, then one has some local difference-type |
| 163 | 166 | gradient that helps figure out how to move. |
| 164 | 167 | |
| 168 | ### What is a priori? |
| 169 | The above arguments is slippery and misleading. Starting from "nothing" |
| 170 | we arrive at some very complex and arcane notions derived by analogy |
| 171 | from a huge body of work in mathematical physics to conclude "ah ha! |
| 172 | Free energy! That's the Answer with a Capital A!" |
| 173 | |
| 174 | ### Self-assembly |
| 175 | But I don't want this. I want a system to self-assemble (necessairly |
| 176 | a system that has moved to a critical point.) The word "critical" in |
| 177 | the sense of self-assembly and avalanches for sand-piles at criticality, |
| 178 | but now, for jigsaw pieces, not sand-grains. |
| 179 | |
| 180 | The jigsaws are generalizations of structural parts: jigsaws are the |
| 181 | connectable parts of axioms, of inference rules, of lambda expressions, |
| 182 | of function terms, of relation terms, of logical connectives. |
| 183 | |
| 184 | I want, and I think it is possible, to have a system that self-reifies: |
| 185 | a self-assembly of pieces that can figure out how to reason, deduce, |
| 186 | prove, use logic and discover axioms. A system that can perceive |
| 187 | structure, in a "sensori-" way, and can manipulate structure, in a |
| 188 | "motor" kind of way. So a sensori-motor system that percives structure, |
| 189 | and acts on, manipulates that structure. |
| 190 | |
| 191 | ### Perceiving structure |
| 192 | So again, what is structure, and how is it perceived? Once again: it is |
| 193 | the pairwise relationships or correlations between pairs of "things". |
| 194 | The pair-wise connections form a network graph, of vertexes and edges. |
| 195 | This network graph is in rough correspondence with the "external world" |
| 196 | that is being perceived. A "world model" of sorts. |
| 197 | |
| 198 | Fix a vertex, and cut all the edges connecting to it in half. This gives |
| 199 | you a jigsaw piece. For each jigsaw piece; one for each vertex. |
| 200 | Categorize these: Many will be identical. The frequency (count) of |
| 201 | identical ones follows a Zipf power law distribution (as experimentally |
| 202 | observed, i.e. "in practice") |
| 203 | |
| 204 | Now, cluster them "by similarity". One of the easiest ways fo doing this |
| 205 | is by cosine-similarity. Cosine similarity requires vectors; where do |
| 206 | the vectors come from? They arise entirely naturally, in many ways. For |
| 207 | example, consider all jigsaws that have the label "A" for the center |
| 208 | vertex; these will have a large variety of labels "p,q,r..." on the |
| 209 | connectors. How many are there? Say there are `N(A | p,q,r...)` of |
| 210 | these. If we hold "A" fixed, while allowing "p,q,r..." to vary, this |
| 211 | forms a vector! The ket `|p,q,r,...)` is the basis elt of the vector. |
| 212 | Suppose the central vertex is labelled by "B": then `N(B | p,q,r...)` |
| 213 | is another vector, and the dot product is easy: |
| 214 | ``` |
| 215 | A . B = Sum_{p,q,r..} N(A | p,q,r...) N(B | p,q,r...) |
| 216 | ``` |
| 217 | In practice, (i.e. experimentally measured) the dot products are |
| 218 | gaussian distributed, i.e. the data is distributed uniformly on a very |
| 219 | high dimensional sphere. |
| 220 | |
| 221 | |
| 222 | |
| 165 | 223 | |
| 166 | 224 | pairs. |