Seven Sketches book - Example 1.56 (The tree of life)

xholonWorkbook.xml

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<!--Xholon Workbook http://www.primordion.com/Xholon/gwt/ MIT License, Copyright (C) Ken Webb, Sat Nov 23 2019 14:36:51 GMT-0500 (Eastern Standard Time)-->
<XholonWorkbook>

<Notes><![CDATA[
Xholon
------
Title: Seven Sketches book - Example 1.56 (The tree of life)
Description: 
Url: http://www.primordion.com/Xholon/gwt/
InternalName: dbe776b8d7a7c4a992880bc73c6454af
Keywords: 

My Notes
--------
November 20, 2019

see also my notes in my notebook for November 18 "Compositionality and Seven Sketches book", and other related subjects from the same time period

In this workbook, I explore example 1.56 in the Fong and Spivak book.
The example is about animal classifications, an area of biology, here informally called "the tree of life".
The example uses several math concepts: set, order, preorder, monotone map.
This is a simplified example, and almost certainly biologists do not actually think about animal classification as a "tree of life". [see ref 12]
"Taxonomy" seems to be the proper biology term for species classification. [see ref 13]
Note that this is a Mathematics example, and does not intend to be biologically accurate.

My questions, as a software analyst, designer, and developer, and as the developer of the Xholon platform, include:
 * How do I build a Xholon application that is true to example 1.56?
 * Can I model everything in Xholon?
 * If anything is missing in Xholon, what is it, and how can I add it in to my platform?
 * In general, how can I use math and applied category theory as concepts, patterns that can help guide software development?

An immediate challenge for Xholon:
- there seem to be two parallel class inheritance hierarchies

An important observation:
- There is a relationship between the concept of Set and the concept of Type/Class.

Something new for me (probably new) is the math concept of a monotone map.
As a software developer, I may find that this is a useful practical concept that might help me develop certain applications.
If I can use it, then it's an example of Applied Category Theory (ACT).
It's a math-inspired idea.

At the end of the day, what will be important in this Xholon application, is whether it can say something about individual animals,
for example my cat Licorice as he interacts with members of my family.

Some key and useful (for me, for Xholon, for my way of understanding things) ideas from example 1.56 and from that whole section in the book:
1. the explicit concept of a preorder, and of a preorder being applied to an existing set
2. the concept of a monotone map between two sets with preorders
3. the basic math approach of systematically building up more and more sophisticated structures, starting with one or more sets

TODO
- work through the math process of building up structure from sets
- formalize this as a Domain Specific Language (DSL)

TODO
This is incomplete, and only partially mathematically correct/rigorous
If this is rigorous, then I should be able to write a parser that will construct a valid Xholon application from the math notation (the DSL)
Constructing the tree of life (as specified in example 1.56)
-----------------------------
Assume two sets A and B.
A = {sapiens, habilis, homo, primate, lion, tiger, panthera, carnivore, mammal}
Set A has a binary operation ≤ that defines how the elements of the set are ordered.
The binary operation defines a set P of ordered pairs, which is a subset of AxA.  P = (A, ≤)
P = {(sapiens,homo), (habilis,homo), (lion,panthera), (tiger,panthera), (homo,primate), (panthera,carnivore), (primate,mammal), (carnivore,mammal)}
B = {individual, species, genus, family, order, class, phylum, kingdom}
Set B has a binary operation ≤ that defines how the elements of the set are ordered.
The binary operation defines a set Q of ordered pairs, which is a subset of BxB.
Q = {(individual,species), (species,genus), (genus,family), (family,order), (order,class), (class,phylum), (phylum,kingdom)}
The monotone map M consists of a set of ordered pairs that connect elements of set A with elements of set B. M is a subset of AxB.
M = {(sapiens,species), (habilis,species), (lion,species), (tiger,species), (homo,genus), (panthera,genus), (primate,order), (carnivore,order), (mammal,class)}

OR describe it this way
-----------------------
In mathematics, a Set is a collection of things.
	X = {thing1, thing2}
    Y = {}
    Z = {1,2,3,4,5}

In mathematics, a name is just a convenient label. In this document, "name" and "label" mean the same thing.

Biologists classify animals using a standard set of group names (labels).
Set A provides examples of these.
	A = {sapiens, habilis, homo, primate, lion, tiger, panthera, carnivore, mammal}

Biologists have developed a strict system of group levels.
Set B provides all the group level names. I have added the non-standard level name "individual".
	B = {individual, species, genus, family, order, class, phylum, kingdom}

Each individual animal has a unique name.
	C = {Ken, Andrea, Leo, Rahh, animal123, animal124, animaldbe776b8d7a7c4a992880bc73c6454af}

Biologists combine group names to form a hierarchy, based on how specific the name is.
For example, "sapiens" is more specific than "homo" or "mammal".
In mathematics, one way to specify this is as a set of ordered pairs, generated using the operation ≤, which in this case actually means < (less than, more specific than, less general than).
The preorder relation < (or ≤) on the set A is a binary relation on A. (Definition 1.25)
We call a pair (X,≤) consisting of a set equipped with a preorder relation a preorder. (Definition 1.25)
P is a preorder.
	P = (A,<)
P' is a set of ordered pairs.
	P' = {(sapiens,homo), (habilis,homo), (lion,panthera), (tiger,panthera), (homo,primate), (panthera,carnivore), (primate,mammal), (carnivore,mammal)}

Note that the details of the binary relation (for the purposes of computer parsing) will only be known once the monotonic map is created.
In the software world, you need a Comparator that can be automatically called to determine if b is < > = to c.

The monotone map M consists of a set of ordered pairs that connect elements of set A with elements of set B. M is a subset of AxB.
	M = {(sapiens,species), (habilis,species), (lion,species), (tiger,species), (homo,genus), (panthera,genus), (primate,order), (carnivore,order), (mammal,class)}

Then I can compute:
	Q = (B, <)
	Q' = {(individual,species), (species,genus), (genus,family), (family,order), (order,class), (class,phylum), (phylum,kingdom)}

end
=====

complete product of AxA
≤          sapiens habilis homo primate lion tiger panthera carnivore mammal
---------- ------- ------- ---- ------- ---- ----- -------- --------- ------
sapiens    x
habilis    x
homo       x
primate    x
lion       x
tiger      x
panthera   x
carnivore  x
mammal     x

TODO add various exports
IH mammal _other,ChapTree
IH mammal MindMap

Preorder Comparator
-------------------
for each pair of elements (x,y) in the product of AxA
 if x.m < y.m
 then
  x < y
  add the pair (x,y) to the set P'

Note:
 - Java has the concept of a Comparator [ref 7]
 - JavaScript Array.prototype.sort provides something similar [ref 8]

once again - the steps in the correct order
----------
Step 1 - specify the group level names; these are not dependant on anything else
	B = {individual, species, genus, family, order, class, phylum, kingdom}
Step 2 - specify a preorder on B; the preorder is a subset of BxB
	Q = (B, <)
	Q' = {(individual,species), (species,genus), (genus,family), (family,order), (order,class), (class,phylum), (phylum,kingdom)}
Step 3 - specify the group names
	A = {sapiens, habilis, homo, primate, lion, tiger, panthera, carnivore, mammal}
Step 4 - specify a monotonic map; AxB
	M = {(sapiens,species), (habilis,species), (lion,species), (tiger,species), (homo,genus), (panthera,genus), (primate,order), (carnivore,order), (mammal,class)}
Step 5 - specify a preorder on A
	P = (A, <)
	P' = {(sapiens,homo), (habilis,homo), (lion,panthera), (tiger,panthera), (homo,primate), (panthera,carnivore), (primate,mammal), (carnivore,mammal)}
Plus - I need to write code (the Preorder Comparator) to automatically perform step 5

end
==========

To transform the Math/DSL notation into a JavaScript Object structure and JSON
------------
Math notation ->      JavaScript
simple set            Array
ordered pair          Object
set of ordered pairs  ?

Total transformation process
----------------------------
example in the book ->
 my Math/DSL notation ->
 JavaScript Object structure ->
  JSON &
  Xholon app
Note: this is a gradual process of moving from Math to executable Code

JavaScript Object structure
---------------------------
// the game here is to specify all the parts as JavaScript arrays or objects, all nested within a single outermost object
const JSON_SPACE = 2; // < 1 means no pretty-printing
var mstruct = {};

// SET: A individual,species,genus,family,order,class,phylum,kingdom
mstruct["A"] = ["individual","species","genus","family","order","class","phylum","kingdom"];

// OPAIR: P A,<
mstruct["P"] = {"A":"<"};

// SETOPAIR: Q (individual,species),(species,genus),(genus,family),(family,order),(order,class),(class,phylum),(phylum,kingdom)
// Q is a subset of AxA
mstruct["Q"] = [{"individual":"species"},{"species":"genus"},{"genus":"family"},{"family":"order"},{"order":"class"},{"class":"phylum"},{"phylum":"kingdom"}];
// maybe mstruct["Q"] = [{"A":"A"},{"individual":"species"}, ...];
// or maybe mstruct["Q"] = [{"A.individual":"A.species"}, ...];

// SET: B sapiens,habilis,homo,primate,lion,tiger,panthera,carnivore,mammal
mstruct["B"] = ["sapiens","habilis","homo","primate","lion","tiger","panthera","carnivore","mammal"];

// SETOPAIR: M (sapiens,species),(habilis,species),(lion,species),(tiger,species),(homo,genus),(panthera,genus),(primate,order),(carnivore,order),(mammal,class)
// M is a subset of BxA
mstruct["M"] = [{"sapiens":"species"},{"habilis":"species"},{"lion":"species"},{"tiger":"species"},{"homo":"genus"},{"panthera":"genus"},{"primate":"order"},{"carnivore":"order"},{"mammal":"class"}];
// mstruct["M"] = [{"B":"A"},{"sapiens":"species"}, ...];
// mstruct["M"] = [{"B.sapiens":"A.species"}, ...];

// OPAIR: R B,<
mstruct["R"] = {"B":"<"};

// SETOPAIR: S (sapiens,homo),(habilis,homo),(lion,panthera),(tiger,panthera),(homo,primate),(panthera,carnivore),(primate,mammal),(carnivore,mammal)
// S is a subset of BxB
mstruct["S"] = [{"sapiens":"homo"},{"habilis":"homo"},{"lion":"panthera"},{"tiger":"panthera"},{"homo":"primate"},{"panthera":"carnivore"},{"primate":"mammal"},{"carnivore":"mammal"}];

console.log(mstruct);
var mstructJSON = JSON.stringify(mstruct, null, JSON_SPACE);
console.log(mstructJSON);


as JSON
-------
{"A":["individual","species","genus","family","order","class","phylum","kingdom"],"P":{"A":"<"},"Q":[{"individual":"species"},{"species":"genus"},{"genus":"family"},{"family":"order"},{"order":"class"},{"class":"phylum"},{"phylum":"kingdom"}],"B":["sapiens","habilis","homo","primate","lion","tiger","panthera","carnivore","mammal"],"M":[{"sapiens":"species"},{"habilis":"species"},{"lion":"species"},{"tiger":"species"},{"homo":"genus"},{"panthera":"genus"},{"primate":"order"},{"carnivore":"order"},{"mammal":"class"}],"R":{"B":"<"},"S":[{"sapiens":"homo"},{"habilis":"homo"},{"lion":"panthera"},{"tiger":"panthera"},{"homo":"primate"},{"panthera":"carnivore"},{"primate":"mammal"},{"carnivore":"mammal"}]}

{
  "A": [
    "individual",
    "species",
    "genus",
    "family",
    "order",
    "class",
    "phylum",
    "kingdom"
  ],
  "P": {
    "A": "<"
  },
  "Q": [
    {
      "individual": "species"
    },
    {
      "species": "genus"
    },
    {
      "genus": "family"
    },
    {
      "family": "order"
    },
    {
      "order": "class"
    },
    {
      "class": "phylum"
    },
    {
      "phylum": "kingdom"
    }
  ],
  "B": [
    "sapiens",
    "habilis",
    "homo",
    "primate",
    "lion",
    "tiger",
    "panthera",
    "carnivore",
    "mammal"
  ],
  "M": [
    {
      "sapiens": "species"
    },
    {
      "habilis": "species"
    },
    {
      "lion": "species"
    },
    {
      "tiger": "species"
    },
    {
      "homo": "genus"
    },
    {
      "panthera": "genus"
    },
    {
      "primate": "order"
    },
    {
      "carnivore": "order"
    },
    {
      "mammal": "class"
    }
  ],
  "R": {
    "B": "<"
  },
  "S": [
    {
      "sapiens": "homo"
    },
    {
      "habilis": "homo"
    },
    {
      "lion": "panthera"
    },
    {
      "tiger": "panthera"
    },
    {
      "homo": "primate"
    },
    {
      "panthera": "carnivore"
    },
    {
      "primate": "mammal"
    },
    {
      "carnivore": "mammal"
    }
  ]
}

November 23, 2019

In definition 1.54 [ref 1], the preorders are labeled:
(A,≤A) and (B,≤B)
This makes it clear that there are two separate binary operations, ≤A that operates on set A, and ≤B that operates on set B.
Note that A and B should be subscripted in ≤A and ≤B .

An alternative
A is a set of bins. There are 8 bins.
Each element of set B is instead an element of one of the bins in set A.
ex: lion and tiger are subclasses of species
TODO think through the separate IH and CSH hierarchies: types and instances
In software development, these are analysis and design questions.
 As we scale-up the number of species (possibly millions) and individuals (possibly billions or trillions), what data structures are practical?
 But perhaps, in analysis and design, I can use the helpful organizing concepts of Math, and later write some code that will transform this to a more efficient implementation scheme?

possible CSH (with an implied order)
------------
species
 sapiens
 habilis
 lion
 tiger
genus
 homo
 panthers
family
order
 primate
 carnivore
class
 mammal
phylum
kingdom

Then we use ports to connect from left to right, for example from sapiens to homo.
This is a more classic Xholon formulation.
How do I decide which is better?
Set vs Types approach?
Can one approach be readily converted to the other?
Dealing with "order" is an important thing.

The basic pattern I'm exploring in this workbook, in terms of JavaScript:
 - sorting array A using a map from A to an already-sorted array B
 - each element in A points to an element in B
 - each element in B may point to another element in B (this defines the order)
 A -> B -> B
 - implicitly, this defines an order from elements in A to other elements in A   A -> A

Graphviz
--------
This Graphviz program uses the order of a b c d e, to display one two three four five in a corresponding order.
In this case, it's the Graphviz layout algorithm and program (dot) that lays out the graph in what it considers to be the "best" way.
see: http://www.webgraphviz.com/

digraph G {
  d b a e c
  a -> b -> c -> d -> e
  five four six two one three seven
  three -> c
  five -> e
  four -> d
  two -> b
  one -> a
}

This undirected cyclic graph does much the same.
graph G {
  d b a e c
  a -- b -- c -- d -- e -- a
  five four two one three
  three -- c
  five -- e
  four -- d
  two -- b
  one -- a
}

JavaScript arrays
-----------------
Here is the start of the same type of graph in JavaScript, based on two arrays a and b:
var a = [{},{},{},{},{}];
var b = [{},{},{},{},{}];

b[0].next = b[1];
b[1].next = b[2];
b[2].next = b[3];
b[3].next = b[4];
b[4].next = null;

a[0].b = b[1];


References
----------
(1) Fong, Brendan and Spivak, David I. (2019), "An Invitation to Applied Category Theory: Seven Sketches in Compositionality" book

(2) https://github.com/ATALLC/AppliedCategoryTheory

(3) https://forum.azimuthproject.org/discussion/1807/lecture-1-introduction

(4) https://forum.azimuthproject.org/discussion/comment/16314/#Comment_16314
    examples of notation

(5) http://cartographer.id/
    cartographer - a tool for string diagrammatic reasoning 
  ) http://cartographer.id/cartographer-calco-2019.pdf

(6) https://cranky-goldstine-6a507b.netlify.com/
    a tool

(7) https://docs.oracle.com/javase/8/docs/api/java/util/Comparator.html
    public interface Comparator<T>
    A comparison function, which imposes a total ordering on some collection of objects.
    Comparators can be passed to a sort method (such as Collections.sort or Arrays.sort) to allow precise control over the sort order.
    Comparators can also be used to control the order of certain data structures (such as sorted sets or sorted maps), or to provide an ordering for collections of objects that don't have a natural ordering.

(8) https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/sort
    arr.sort([compareFunction])
    compareFunction Optional
      Specifies a function that defines the sort order. If omitted, the array elements are converted to strings, then sorted according to each character's Unicode code point value.
      firstEl
        The first element for comparison.
      secondEl
        The second element for comparison.

(9) https://en.wikipedia.org/wiki/Monotonic_function
    In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.
    This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

(10) https://www.youtube.com/results?search_query=seven+sketches+in+compositionality+_
    a playlist of 21 videos with David and Brendan presenting topics from the book
   ) Seven Sketches in Compositionality, Lecture 1.1 (David)
    poset (partially ordered set)
    1:13 David gives a good example of a generative effect, and I find that he explains the concept better than it's explained in the book
         he draws on the blackboard and describes the example, providing a timeline that may make a good foundation for a Xholon model
         cascade effects (generative effects)
    2:50 partitions
    10:00 join, notion of order
    join, meet
    poset
    18:10 example "tree of life"
   ) Seven Sketches in Compositionality, Lecture 1.2 (David)
    0:00 continuation of "tree of life" example
    0:35 join of lion and tiger is panthera; not everything has a meet
    graphs -> posets  (functors in CT)

(11) https://github.com/DSLsofMath/DSLsofMath
    Domain Specific Languages of Mathematics 

(12) https://en.wikipedia.org/wiki/Tree_of_life
    The tree of life is a widespread myth (mytheme) or archetype in the world's mythologies, related to the concept of sacred tree more generally, and hence in religious and philosophical tradition. 

(13) https://en.wikipedia.org/wiki/Taxonomy_(biology)
    In biology, taxonomy (from Ancient Greek τάξις (taxis), meaning 'arrangement', and -νομία (-nomia), meaning 'method') is the science of naming,
    defining (circumscribing) and classifying groups of biological organisms on the basis of shared characteristics.
    Organisms are grouped together into taxa (singular: taxon) and these groups are given a taxonomic rank; groups of a given rank can be aggregated
    to form a super-group of higher rank, thus creating a taxonomic hierarchy.
    The principal ranks in modern use are domain, kingdom, phylum (division is sometimes used in botany in place of phylum), class, order, family, genus, and species.
    The Swedish botanist Carl Linnaeus is regarded as the founder of the current system of taxonomy,
    as he developed a system known as Linnaean taxonomy for categorizing organisms and binomial nomenclature for naming organisms.
    With the advent of such fields of study as phylogenetics, cladistics, and systematics,
    the Linnaean system has progressed to a system of modern biological classification based on the evolutionary relationships between organisms, both living and extinct. 

(14) https://gist.github.com/kenwebb/9500085
    that pre-existing workbook provides an eexample of using childSuperClass
    based on that example, in the present workbook I use:
    <TreeLevelNames childSuperClass="TreeLevelName"/>

]]></Notes> 

<_-.XholonClass>
  <ActSystem/> <!-- Applied Category Theory (ACT) system -->
  
  <!--
Consider the set of all animal classifications, for example "tiger", "mammal", "sapiens", "carnivore", etc.
These are ordered by specificity: since "tiger" is a type of "mammal", we write tiger ≤ mammal.
The result is a preorder, which in fact forms a tree, often called the tree of life.
At the top of the following diagram we see a small part of it:
[ref 1]
In Xholon it's natural to represent this as an a class (type) inheritance hierarchy, for example:
 - Tiger is a subclass (subtype) of Mammal
-->
  <AnimalClassification>
    <mammal>
      <primate>
        <homo>
          <sapiens/>
          <habilis/>
        </homo>
      </primate>
      <carnivore>
        <panthera>
          <lion/>
          <tiger/>
        </panthera>
      </carnivore>
    </mammal>
  </AnimalClassification>
  
  <!-- the set of all animal classifications -->
  <AnimalClassifications/>
  
  <!--
At the bottom we see the hierarchical structure as a preorder.  [ref 1]
For lack of a better name, I will call this the set of tree level names.
Each subclass is a singleton.
I don't know yet whether this is a tree (phylum a subclass of kingdom), a directed acyclic graph (DAG) (subclasses have an order), or something else.
 - within the Xholon inheritance hierarchy, they are in fact just names, with an order yet to be specified,
   although weakly implied by the sequential listing of the names (speciesTRN is listed before genusTRN)
I'm appending TRN to the end of each name, to avoid conflict with other names in Xholon.

Note: Xholon already has a mechanism where I can specify that any item in the CSH that is contained with TreeLevelNames is in fact a subclass of TreeLevelName.
childSuperClass

TODO
- extend this to include individualTRN, so I can represent individual animals

- reverse the order of placement so that kingdomTRN is first and speciesTRN is last, to be consistent with the order under AnimalClassification
 - the math concepts of preorder and monotone map, provide strong support for this
 - alternatively, both lists could NOT be ordered from most specific to least specific
  - this would not work with the AnimalClassification hierarchy, which is a tree structure
 - this may not really matter; left and right, and up and down, do not necessarily imply the direction of an order, but only a relative ordering
  - ABCDEFG only implies that D is closer to C and E than it is to A or G

-->
  <!--
  <TreeLevelName>
    <speciesTRN/>
    <genusTRN/>
    <familyTRN/>
    <orderTRN/>
    <classTRN/>
    <phylumTRN/>
    <kingdomTRN/>
  </TreeLevelName>
  -->
  
  <!-- 
this appears to be the obvious way to order the items, using the math ideas as a guideline
also, I've added individualTRN
-->
  <TreeLevelName>
    <!-- these subclasses do not need to be explicity specified, because I use <TreeLevelNames childSuperClass="TreeLevelName"/> -->
    <!--<kingdomTRN/>
    <phylumTRN/>
    <classTRN/>
    <orderTRN/>
    <familyTRN/>
    <genusTRN/>
    <speciesTRN/>
    <individualTRN/>-->
  </TreeLevelName>
  
  <!-- the set of all tree level names -->
  <TreeLevelNames childSuperClass="TreeLevelName"/>
  
  <!-- a set of individual animals -->
  <Animals/>
  
  <MathParser superClass="script"/>
  
</_-.XholonClass>

<xholonClassDetails>
  
  <!-- this is unnecessary, and is only an example of the use of childSuperClass -->
  <!--<TreeLevelNames/>-->
  
  <MathParser><DefaultContent><![CDATA[
var me, spec, beh = {
  postConfigure: function() {
    me = this.cnode.parent();
    me.println("I am " + me.name());
    spec = me.first().text().trim();
    me.first().remove(); // remove the Attribute_String node
    this.parseSpec(spec);
    this.cnode.remove(); // remove this BigraphParser node
  },
  
  act: function() {
    //myHeight.println(this.toString());
  },
  
  parseSpec: function(spec) {
    const arr = spec.split("\n"); // there must be 2 back-slashes here
    //console.log(arr);
    for (var i = 0; i < arr.length; i++) {
      const line = arr[i].trim();
      const larr = line.split("=");
      const larr0 = larr[0].trim();
      const larr1 = larr[1] ? larr[1].trim() : "";
      //me.println("XXX" + (i+1) + " " + larr0 + " = " + larr1);
      switch (larr1.charAt(0)) {
      case "{":
        this.parseSet(larr0, larr1.substring(1, larr1.length - 1).replace(/ /g, ""));
        break;
      case "(":
        this.parseOrderedPair(larr0, larr1.substring(1, larr1.length - 1).replace(/ /g, ""));
        break;
      default:
        //me.println("DEFAULT: " + line);
        if (line.length == 0) {
          // this is a blank line
        }
        else if (line.startsWith("#")) {
          me.println("COMMENT:" + line.substring(1));
        }
        break;
      }
    }
  },
  
  // A individual,species,genus,family,order,class,phylum,kingdom
  // Q (individual,species),(species,genus),(genus,family),(family,order),(order,class),(class,phylum),(phylum,kingdom)
  parseSet: function(setName, str) {
    if (str.startsWith("(")) {
      this.parseSetOfOrderedPairs(setName, str);
    }
    else {
      me.println("SET: " + setName + " " + str);
    }
  },
  
  // Q (individual,species),(species,genus),(genus,family),(family,order),(order,class),(class,phylum),(phylum,kingdom)
  parseSetOfOrderedPairs: function(setName, str) {
    me.println("SETOPAIR: " + setName + " " + str);
  },
  
  // P (B,<)
  parseOrderedPair: function(opairName, str) {
    me.println("OPAIR: " + opairName + " " + str);
  }
  
}
  ]]></DefaultContent></MathParser>
  
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</xholonClassDetails>

<ActSystem>
  
  <!-- TODO how do I represent the monotone map ? -->
  
  <TreeLevelNames>
    <individualTRN/>
    <speciesTRN/>
    <genusTRN/>
    <familyTRN/>
    <orderTRN/>
    <classTRN/>
    <phylumTRN/>
    <kingdomTRN/>
    <dummyTRN/>
  </TreeLevelNames>
  
  <!-- 
the animals are all individuals (individualTRN  ex: "Ken") that are members of specific species (speciesTRN  ex: sapiens)
individuals are not ordered in any specific way
individuals are connected to a genus, etc. by following the links up the AnimalClassification tree of life
 - the genus of sapiens "Ken" is homo, because the parent of sapiens is homo
 - the order of genus "homo" is primate, because the parent of homo is primate
 - etc.
-->
  <Animals>
    <sapiens roleName="Ken"/>
    <sapiens roleName="Andrea"/>
    <lion roleName="Leo"/>
    <tiger roleName="Rahh"/>
  </Animals>
  
  <MathStructure><Attribute_String><![CDATA[
# Step 1 - specify the group level names; these are not dependant on anything else
A = {individual, species, genus, family, order, class, phylum, kingdom}
# Step 2 - specify a preorder on A; the preorder is a subset of AxA
P = (A, <)
Q = {(individual,species), (species,genus), (genus,family), (family,order), (order,class), (class,phylum), (phylum,kingdom)}
# Step 3 - specify the group names
B = {sapiens, habilis, homo, primate, lion, tiger, panthera, carnivore, mammal}
# Step 4 - specify a monotonic map; AxB
M = {(sapiens,species), (habilis,species), (lion,species), (tiger,species), (homo,genus), (panthera,genus), (primate,order), (carnivore,order), (mammal,class)}
# Step 5 - specify a preorder on B
R = (B, <)
S = {(sapiens,homo), (habilis,homo), (lion,panthera), (tiger,panthera), (homo,primate), (panthera,carnivore), (primate,mammal), (carnivore,mammal)}
# Plus - I need to write code (the Preorder Comparator) to automatically perform step 5
]]></Attribute_String>
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  </MathStructure>
  
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  var a = 123;
  var b = 456;
  var c = a * b;
  if (console) {
    console.log(c);
  }
  //# sourceURL=ActSystembehavior.js
]]></ActSystembehavior>

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