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<?xml version="1.0" encoding="UTF-8"?> |
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<!--Xholon Workbook http://www.primordion.com/Xholon/gwt/ MIT License, Copyright (C) Ken Webb, Sun Jan 12 2020 19:09:31 GMT-0500 (Eastern Standard Time)--> |
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<XholonWorkbook> |
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<Notes><![CDATA[ |
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Xholon |
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------ |
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Title: Programming with Categories - Shape - essence of |
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Description: |
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Url: http://www.primordion.com/Xholon/gwt/ |
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InternalName: b8a7893bc4723388cd6e355069c9fd55 |
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Keywords: |
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My Notes |
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-------- |
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January 10, 2020 |
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TODO |
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- DONE add the SVG file to github |
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- extend the isomorphism (I) model |
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- add Fisher as Avatar or other JS behavior |
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- add Ocean and Fish |
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Azimuth Project [ref 4] |
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- uses Markdown for forum messages |
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- also latex with Mathjax |
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- to include SVG |
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To include images you can link to an external file using Markdown as follows: |
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  |
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You can also do it using HTML: for example, including this in your comment: |
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<img src = "http://math.ucr.edu/home/baez/mathematical/7_sketches/exercise_1.2.png"> |
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- |
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References |
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---------- |
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(1) http://brendanfong.com/programmingcats.html |
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Brendan Fong, Bartosz Milewski, and David Spivak, have just kicked off a new course at MIT called Programming with Categories. |
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Summary: |
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In this course we explain how category theory—a branch of mathematics known for its ability to organize the key |
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abstractions that structure much of the mathematical universe—has become useful for writing elegant and maintainable code. |
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In particular, we'll use examples from the Haskell programming language to motivate category-theoretic constructs, |
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and then explain these constructs from a more abstract and inclusive viewpoint. |
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Hands-on programming exercises will be used to demonstrate categorical ideas like "the universal property of products" |
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in working Haskell code. |
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(2) https://www.youtube.com/watch?v=jm3bJrULMqM |
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Programming with Categories - Lecture 2 |
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lecturer - Brendan Fong |
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(3) http://www.webgraphviz.com/ |
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I used this site to parse the Graphviz .dot notation, and display it as a Scalable Vector Graphics (SVG) image, which I then copied to this Xholon workbook page. |
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(4) https://forum.azimuthproject.org/ |
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) https://forum.azimuthproject.org/categories/-%20-%20Software |
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) https://forum.azimuthproject.org/discussion/966/graphviz#latest |
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) https://www.azimuthproject.org/azimuth/show/Forum+-+Guide |
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what is the azimuth project, and how to contribute |
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(5) https://forum.azimuthproject.org/categories/applied-category-theory-course |
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website for outside participants in the course |
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]]></Notes> |
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<markdown>< at MIT called *Programming with Categories*. |
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I don't live in Boston so I can't attend the course, but I am watching the videos as they come out. |
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In [lecture 2](https://www.youtube.com/watch?v=jm3bJrULMqM) (about time 5:30), Brendan introduces the informal concept of **Shape**. |
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I very much like this idea of shapes, where each Shape suggests "the essence of" a basic idea in Category Theory. |
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To me this provides a way to informally use these concepts. I can take this _essence_ and use it in any way that provides some practical value. |
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This is roughly the table that Brendan presented. |
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Shape Label | Essence Of | Objects | Arrows |
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----------- | ---------- | ------- | ------ |
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1 | object | 1 | 1 |
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P | pair | 2 | 2 |
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2 | arrow | 2 | 3 |
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I | isomorphism | 2 | 4 |
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You can use [Graphviz](https://graphviz.org/) .dot notation to roughly reproduce what Brendan drew on the blackboard. |
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```dot |
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// Graphviz specification |
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digraph G { |
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subgraph cluster_1 {x1->x1; label="1 object";} |
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subgraph cluster_P {xp->xp yp->yp; label="P pair";} |
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subgraph cluster_2 {x2->x2->y2->y2; label="2 arrow";} |
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subgraph cluster_I {xi->xi->yi->yi->xi; label="I isomorphism";} |
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} |
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``` |
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A number of online sites are able to parse Graphviz .dot notation, and display it as a Scalable Vector Graphics (SVG) image. |
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[WebGraphviz](http://www.webgraphviz.com/) is one such site. Copy the above text, paste it into the site, and click _Generate Graph!_. |
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The resulting SVG image looks like this: |
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 |
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I also like the idea of a **probe** (about time 48:00). |
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I like these ideas partly because they help to define a set of experiments I'm working on with Jen, a researcher in Cognitive Science at Carleton University in Ottawa. |
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We are exploring the concept of **exploitation**, starting from a set of very simple models. |
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We have worked out a basic definition of exploitation. It must involve at least two individual people, engaged in some sort of interaction. |
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In our model, developed using my software [Xholon](https://github.com/kenwebb/Xholon) as an [Agent-Based Modeling](https://en.m.wikipedia.org/wiki/Agent-based_model) tool, we start with a simplest possible scenario. |
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**Scenario 1** |
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In this scenario, there is one **World** that contains one **Island** on which may live one **Fisher** (Fisherman). |
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As yet, there is no **Ocean** and there are no **Fish**, so there's not yet much of a living for a Fisher. |
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Basically, there is just an Island, a single object. |
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This scenario implements the shape labeled **1**, and suggests the essence of **object**. |
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There is one object _Island_ and one arrow _identity_. |
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**Scenario P** |
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There is now a pair of islands, each isolated from the other. There are two objects, and two identity arrows. |
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This scenario implements the shape labeled **P**, and suggests the essence of **pair**. |
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**Scenario 2** |
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One island now provides a directed link to the other island, but no return link. |
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This scenario implements the shape labeled **2**, and suggests the essence of **arrow**. |
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**Scenario I** |
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Both islands provide a directed link to the other island. |
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This scenario implements the shape labeled **I**, and suggests the essence of **isomorphism**. |
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**Xholon** |
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I have implemented a Graphviz parser and model generator in my own software, based on the [ANTLR](https://www.antlr.org/) example [DOT grammar](https://github.com/antlr/grammars-v4/tree/master/dot). |
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It only runs locally at this point. |
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It generates the following XML from the Graphviz specification, which Xholon then parses into an internal structure of nodes and edges. |
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```xml |
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<TheSystem> |
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<Node roleName="x1"> |
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<port name="edge" index="0" connector="../Node[@roleName='x1']"/> |
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</Node> |
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<Node roleName="xp"> |
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<port name="edge" index="0" connector="../Node[@roleName='xp']"/> |
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</Node> |
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<Node roleName="yp"> |
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<port name="edge" index="0" connector="../Node[@roleName='yp']"/> |
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</Node> |
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<Node roleName="x2"> |
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<port name="edge" index="0" connector="../Node[@roleName='x2']"/> |
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<port name="edge" index="1" connector="../Node[@roleName='y2']"/> |
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</Node> |
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<Node roleName="y2"> |
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<port name="edge" index="0" connector="../Node[@roleName='y2']"/> |
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</Node> |
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<Node roleName="xi"> |
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<port name="edge" index="0" connector="../Node[@roleName='xi']"/> |
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<port name="edge" index="1" connector="../Node[@roleName='yi']"/> |
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</Node> |
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<Node roleName="yi"> |
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<port name="edge" index="0" connector="../Node[@roleName='yi']"/> |
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<port name="edge" index="1" connector="../Node[@roleName='xi']"/> |
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</Node> |
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</TheSystem> |
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``` |
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An Avatar is then able to examine and move through the resulting structure. The Avatar acts as a **probe**, somewhat as described in Brendan's lecture. |
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The Avatar is the single element of a Set, and is able to link to any of the nodes in the Shape set. It can learn about nodes and about the edges emanating from each node. |
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The following is a trace of actions performed ny the Avatar. |
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In this example, the Avatar is controlled directly by a human. |
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``` |
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You are the system Avatar avatar_42.You are in chameleon_0. |
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... |
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Moving to theSystem_392 |
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You are in theSystem_392. |
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You see x1:node_394 |
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You see xp:node_396 |
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You see yp:node_398 |
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You see x2:node_400 |
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You see y2:node_403 |
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You see xi:node_405 |
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You see yi:node_408 |
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Entered x1:node_394 |
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You are in x1:node_394. |
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You see passage edge0 to x1:node_394 |
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Moving along edge to x1:node_394 |
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Moving along edge to x1:node_394 |
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... |
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Moving to xp:node_396 |
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You are the system Avatar avatar_42.You are in xp:node_396. |
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You are in xp:node_396. |
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You see passage edge0 to xp:node_396 |
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Moving along edge to xp:node_396 |
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Moving along edge to xp:node_396 |
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... |
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Moving to yp:node_398 |
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You are in yp:node_398. |
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You see passage edge0 to yp:node_398 |
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Moving along edge to yp:node_398 |
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Moving along edge to yp:node_398 |
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... |
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Moving to x2:node_400 |
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You are in x2:node_400. |
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You see passage edge0 to x2:node_400 |
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You see passage edge1 to y2:node_403 |
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Moving along edge to x2:node_400 |
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Moving along edge to x2:node_400 |
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... |
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Moving along edge to y2:node_403 |
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You are in y2:node_403. |
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You see passage edge0 to y2:node_403 |
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Moving along edge to y2:node_403 |
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Moving along edge to y2:node_403 |
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... |
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You are in y2:node_403. |
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You see passage edge0 to y2:node_403 |
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Moving to xi:node_405 |
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You are in xi:node_405. |
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You see passage edge0 to xi:node_405 |
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You see passage edge1 to yi:node_408 |
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Moving along edge to xi:node_405 |
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Moving along edge to xi:node_405 |
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... |
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Moving along edge to yi:node_408 |
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Moving along edge to yi:node_408 |
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... |
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Moving along edge to xi:node_405 |
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Moving along edge to yi:node_408 |
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Moving along edge to yi:node_408 |
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... |
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Moving along edge to xi:node_405 |
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Moving along edge to xi:node_405 |
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... |
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``` |
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**Some background on myself** |
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I am a software developer (C, C++, Java, JavaScript, Assembly Language, etc.). |
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In the past few years I have become interested in how to apply ideas from Math and Category Theory for the development of software and for understanding the world around us. |
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A couple of years ago I met with David (and also met Brendan) to discuss some of these connections. |
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I've also benefited from David's book (Category Theory for the Sciences), |
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from Brendan and David's newer book (An Invitation to Applied Category Theory - Seven Sketches in Compositionality), |
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from Bartosz's website (piggies rule!), and from everybody's videos. |
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To understand anything from the Math world, it always works best for me to approach it from a constructive (building and running things/code) perspective. |
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I'm looking forward to learning some Haskell. |
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<hr/> |
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A briefer version of this for the Azimuth Project course website. |
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By labeling something as an _essence_, I feel I have permission to use that thing in any way that provides some practical value. |
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That thing can become a useful tool, rather than a "scary Math thing". |
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In [lecture 2](https://www.youtube.com/watch?v=jm3bJrULMqM) (about time 5:30), Brendan introduces a notion of **Shape**. |
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I like this idea. Each Shape (object, pair, arrow, isomorphism) suggests _the essence of_ a basic idea in Category Theory. |
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An essence is not formally defined. I can play with it in my own way, with tools I'm familiar with. |
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This is roughly the table that Brendan presented. |
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``` |
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| Shape Label | Essence Of | Objects | Arrows | |
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| ----------- | ----------- | ------- | ------ | |
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| 1 | object | 1 | 1 | |
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| P | pair | 2 | 2 | |
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| 2 | arrow | 2 | 3 | |
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| I | isomorphism | 2 | 4 | |
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``` |
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[Graphviz](https://graphviz.org/), can roughly reproduce what Brendan drew on the blackboard. |
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A number of online sites can parse Graphviz .dot notation, and display it as a Scalable Vector Graphics (SVG) image. |
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[WebGraphviz](http://www.webgraphviz.com/) is one such site. Copy any of the following four _digraph_ lines, paste it into the WebGraphviz site, click _Generate Graph!_, and view the SVG image (as shown below). |
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digraph 1 {x->x; label="1 object";} |
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digraph P {x->x y->y; label="P pair";} |
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digraph 2 {x->x->y->y; label="2 arrow";} |
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digraph I {x->x->y->y->x; label="I isomorphism";} |
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 |
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I wonder what other essences are lurking out there, or is this as far as we can go with this concept? |
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<hr/> |
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Some other Graphviz digraphs and graphs to try. |
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x and y, each with two arrows to the other node (identity arrows not shown) |
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graph G {x->y->x->y->x} |
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x and y, each with two undirected arcs |
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graph G {x--y--x--y--x} |
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x, y and z, where x has an arrow to y and an arrow to z |
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digraph G {x->y x->z} |
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digraph G {a b c} |
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digraph G {a->b->c} |
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cycle of three nodes, with identity arrows |
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digraph G {a->a->b->b->c->c->a} |
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cycle of three nodes, with double identity arrows |
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digraph G {a->a->a->b->b->b->c->c->c->a} |
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]]></markdown> |
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<_-.XholonClass> |
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<TheSystem/> |
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<Scenario/> <!-- Shape --> |
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<Island/> |
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</_-.XholonClass> |
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<xholonClassDetails> |
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<Scenario><Color>green</Color></Scenario> |
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<Island><Color>orange</Color></Island> |
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<Avatar><Color>red</Color></Avatar> |
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</xholonClassDetails> |
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<TheSystem> |
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<!-- manually created, based on the structure generated thru Graphviz --> |
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<Scenario roleName="1"> |
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<Island roleName="x1"> |
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<port name="edge" index="0" connector="../Island[@roleName='x1']"/> |
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</Island> |
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</Scenario> |
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<Scenario roleName="P"> |
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<Island roleName="xp"> |
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<port name="edge" index="0" connector="../Island[@roleName='xp']"/> |
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</Island> |
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<Island roleName="yp"> |
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<port name="edge" index="0" connector="../Island[@roleName='yp']"/> |
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</Island> |
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</Scenario> |
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<Scenario roleName="2"> |
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<Island roleName="x2"> |
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<port name="edge" index="0" connector="../Island[@roleName='x2']"/> |
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<port name="edge" index="1" connector="../Island[@roleName='y2']"/> |
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</Island> |
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<Island roleName="y2"> |
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<port name="edge" index="0" connector="../Island[@roleName='y2']"/> |
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</Island> |
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</Scenario> |
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<Scenario roleName="I"> |
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<Island roleName="xi"> |
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<port name="edge" index="0" connector="../Island[@roleName='xi']"/> |
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<port name="edge" index="1" connector="../Island[@roleName='yi']"/> |
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</Island> |
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<Island roleName="yi"> |
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<port name="edge" index="0" connector="../Island[@roleName='yi']"/> |
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<port name="edge" index="1" connector="../Island[@roleName='xi']"/> |
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</Island> |
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</Scenario> |
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</TheSystem> |
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<Blockbehavior implName="org.primordion.xholon.base.Behavior_gwtjs"><![CDATA[ |
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var a = 123; |
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var b = 456; |
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var c = a * b; |
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if (console) { |
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console.log(c); |
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} |
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//# sourceURL=Blockbehavior.js |
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]]></Blockbehavior> |
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<Heightbehavior implName="org.primordion.xholon.base.Behavior_gwtjs"><![CDATA[ |
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var myHeight, testing; |
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var beh = { |
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postConfigure: function() { |
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testing = Math.floor(Math.random() * 10); |
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myHeight = this.cnode.parent(); |
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}, |
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act: function() { |
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myHeight.println(this.toString()); |
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}, |
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toString: function() { |
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return "testing:" + testing; |
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} |
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} |
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//# sourceURL=Heightbehavior.js |
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]]></Heightbehavior> |
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<Brickbehavior implName="org.primordion.xholon.base.Behavior_gwtjs"><![CDATA[ |
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$wnd.xh.Brickbehavior = function Brickbehavior() {} |
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$wnd.xh.Brickbehavior.prototype.postConfigure = function() { |
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this.brick = this.cnode.parent(); |
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this.iam = " red brick"; |
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}; |
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$wnd.xh.Brickbehavior.prototype.act = function() { |
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this.brick.println("I am a" + this.iam); |
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}; |
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//# sourceURL=Brickbehavior.js |
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]]></Brickbehavior> |
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<Brickbehavior implName="org.primordion.xholon.base.Behavior_gwtjs"><![CDATA[ |
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console.log("I'm another brick behavior"); |
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]]></Brickbehavior> |
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<SvgClient><Attribute_String roleName="svgUri"><![CDATA[data:image/svg+xml, |
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<svg width="100" height="50" xmlns="http://www.w3.org/2000/svg"> |
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<g> |
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<title>Block</title> |
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<rect id="PhysicalSystem/Block" fill="#98FB98" height="50" width="50" x="25" y="0"/> |
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<g> |
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<title>Height</title> |
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<rect id="PhysicalSystem/Block/Height" fill="#6AB06A" height="50" width="10" x="80" y="0"/> |
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</g> |
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</g> |
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</svg> |
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<svg width="499pt" height="172pt" viewBox="0.00 0.00 499.00 172.00" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> |
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<g id="graph1" class="graph" transform="scale(1 1) rotate(0) translate(4 168)"> |
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<title>G</title> |
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<polygon fill="white" stroke="white" points="-4,5 -4,-168 496,-168 496,5 -4,5"></polygon> |
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<g id="graph2" class="cluster"><title>cluster_1</title> |
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<polygon fill="none" stroke="black" points="8,-80 8,-156 96,-156 96,-80 8,-80"></polygon> |
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<text text-anchor="middle" x="52" y="-139.4" font-family="Times,serif" font-size="14.00">1 object</text> |
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</g> |
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<g id="graph3" class="cluster"><title>cluster_P</title> |
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<polygon fill="none" stroke="black" points="104,-80 104,-156 282,-156 282,-80 104,-80"></polygon> |
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<text text-anchor="middle" x="193" y="-139.4" font-family="Times,serif" font-size="14.00">P pair</text> |
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</g> |
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<g id="graph4" class="cluster"><title>cluster_2</title> |
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<polygon fill="none" stroke="black" points="290,-8 290,-156 378,-156 378,-8 290,-8"></polygon> |
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<text text-anchor="middle" x="334" y="-139.4" font-family="Times,serif" font-size="14.00">2 arrow</text> |
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</g> |
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<g id="graph5" class="cluster"><title>cluster_I</title> |
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<polygon fill="none" stroke="black" points="386,-8 386,-156 483,-156 483,-8 386,-8"></polygon> |
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<text text-anchor="middle" x="434.5" y="-139.4" font-family="Times,serif" font-size="14.00">I isomorphism</text> |
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</g> |
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<!-- x1 --> |
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<g id="node2" class="node"><title>x1</title> |
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<ellipse fill="none" stroke="black" cx="43" cy="-106" rx="27" ry="18"></ellipse> |
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<text text-anchor="middle" x="43" y="-101.8" font-family="Times,serif" font-size="14.00">x1</text> |
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</g> |
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<!-- x1->x1 --> |
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<g id="edge3" class="edge"><title>x1->x1</title> |
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<path fill="none" stroke="black" d="M62.895,-118.432C75.688,-121.675 88,-117.531 88,-106 88,-97.6218 81.5006,-93.1433 73.0395,-92.5644"></path> |
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<polygon fill="black" stroke="black" points="72.5019,-89.1004 62.895,-93.5679 73.191,-96.0665 72.5019,-89.1004"></polygon> |
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</g> |
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<!-- xp --> |
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<g id="node5" class="node"><title>xp</title> |
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<ellipse fill="none" stroke="black" cx="229" cy="-106" rx="27" ry="18"></ellipse> |
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<text text-anchor="middle" x="229" y="-101.8" font-family="Times,serif" font-size="14.00">xp</text> |
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</g> |
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<!-- xp->xp --> |
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<g id="edge6" class="edge"><title>xp->xp</title> |
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<path fill="none" stroke="black" d="M248.895,-118.432C261.688,-121.675 274,-117.531 274,-106 274,-97.6218 267.501,-93.1433 259.039,-92.5644"></path> |
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<polygon fill="black" stroke="black" points="258.502,-89.1004 248.895,-93.5679 259.191,-96.0665 258.502,-89.1004"></polygon> |
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</g> |
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<!-- yp --> |
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<g id="node7" class="node"><title>yp</title> |
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<ellipse fill="none" stroke="black" cx="139" cy="-106" rx="27" ry="18"></ellipse> |
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<text text-anchor="middle" x="139" y="-101.8" font-family="Times,serif" font-size="14.00">yp</text> |
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</g> |
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<!-- yp->yp --> |
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<g id="edge8" class="edge"><title>yp->yp</title> |
|
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