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<!DOCTYPE html>
<html lang="en-US">
<!-- Introduction to Pure See
author: Marius Buliga
last modified: 16.03.2021
//
-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="stylesheet" type="text/css" href="css/icequine-on-white.css">
<title>Pure See</title>
</head>
<body style="overflow: scroll;">
<script src="js/d3.v5.9.4.min.js"></script>
<script src="js/jquery.min.js"></script>
<script src="js/pagelook-vs.js"></script>
<script src="js/ioprep.js"></script>
<div class="header">
<span3>Cite as: <br></span3><div>Introduction to Pure See. <a href="http://imar.ro/~mbuliga/index.html">© Marius Buliga</a> (2020-2021), https://mbuliga.github.io/quinegraphs/puresee.html<span id="citeas"></span></div><br><br>
<span3>Draft! Version 06.06.2020</span3><br><br>
<span3>See also:</span3><br> <a href="https://chemlambda.github.io/index.html">All chemlambda projects</a><br><br><br><br>
<span3>
<h3>Pure See</h3>
is a geometrical, or cartographical lambda calculus. As primitives it uses variables, terms and a small number of special words: from, see, as, in, apply, with, note, over, free, origin. <br><br>
</span3>
<h3 id="Contents">Contents</h3>
<h4><a href="#Scalars">Scalars</a></h4>
<h4><a href="#VariablesAndTerms">Variables and terms</a></h4>
<h4><a href="#Commands">Commands</a></h4>
<h4><a href="#ConversionBetweenCommandsAndAlgebraicForms">Conversion between commands, functional and algebraic forms</a></h4>
<h4><a href="#ReductionOfTrigrams">Reduction of trigrams</a></h4>
<h4><a href="#MultiplicationByScalarsOrigin">Multiplication by scalars. Origin</a></h4>
<h4><a href="#ReductionByPassingToTheLimit">Reduction by passing to the limit</a></h4>
<h4><a href="#EmergentRewrites">Emergent rewrites</a></h4>
<br><br><br><br>
<h3 id="Scalars">Scalars</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
<span3> The following special words are scalars: from, see, as, in, apply, with, note, over. They are rational functions of a formal parameter denoted usually by "z":<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:500px;border:0">
<tr>
<th style="text-align: left;">special word
</th>
<th style="text-align: left;">as a rational function
</th>
</tr>
<tr>
<td>see</td><td> see[z] = z</td>
</tr>
<tr>
<td>from</td><td> from[z] = 1-z</td>
</tr>
<tr>
<td>apply</td><td> apply[z] = 1/z</td>
</tr>
<tr>
<td>with</td><td> with[z] = 1/(1-z)</td>
</tr>
<tr>
<td>note</td><td> note[z] = z/(z-1)</td>
</tr>
<tr>
<td>over</td><td> over[z] = (z-1)/z</td>
</tr>
<tr>
<td>as</td><td> as[z] = -1</td>
</tr>
<tr>
<td>in</td><td> in[z] = 1</td>
</tr>
</table>
<span3>
There are three special scalars. The scalar "in" is also denoted by "1". The two other special scalars are:
0 (which can be seen as the constant rational function equal to 0) and ∞. Any number of other constant functions may be considered as scalars.<br><br>
These scalars (with the exception of those special ones) generate all the scalars via two operations: composition of scalars and multiplication of scalars.<br><br>
Composition of scalars is defined as composition of functions<br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:350px;border:0"> k[l][z] = k[l[z]], for k,l scalars </p>
<span3>
Remark that with composition the set {see, from, apply, with, note, over} is the <a href="https://en.wikipedia.org/wiki/Cross-ratio#The_anharmonic_group">anharmonic group</a>. This group is isomorphic with the group of permutations of 3 elements.<br><br>
Multiplication of scalars is defined as multiplication of functions<br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:350px;border:0"> [k l][z] = k[z] l[z], for k,l scalars </p>
<span3>
Composition and multiplication of scalars 0 and ∞ with any other scalar is undefined.<br><br>
<h3 id="VariablesAndTerms">Variables and terms</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
Variables are denoted by a, b, c, ...
A term is either a variable or the result of an operation between terms. There are 6 operations: ◦ is "map" , • is "unmap", (lambda calculus style) abstraction, (lambda calculus style) application, > is "fi", < is "fox". All in all, a term is defined as:<br><br>
<table style="padding: 20px; margin-bottom: 18px; width:500px;border:0">
<tr>
<td>origin</td> <td></td><td> | <td> <td>(is asimilated with a variable)</td>
</tr>
<tr>
<td>a</td> <td>variable</td><td> | <td> <td></td>
</tr>
<tr>
<td>a ◦ b</td> <td>a, b terms</td> <td> | <td> <td> (map)</td>
</tr>
<tr>
<td>a • b </td> <td>a, b terms</td> <td> | <td> <td> (unmap)</td>
</tr>
<tr>
<td>a > b</td> <td>a, b terms</td> <td> | <td> <td> (fi)</td>
</tr>
<tr>
<td>a < b</td> <td>a, b terms</td> <td> | <td> <td> (fox)</td>
</tr>
<tr>
<td>λ a. b</td> <td>a, b terms</td> <td> | <td> <td> (abstraction)</td>
</tr>
<tr>
<td> a b </td> <td>b, b terms</td> <td> | <td> <td> (application)</td>
</tr>
</table>
<br><br>
<h3 id="Commands">Commands</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
<span3>Pure See uses mainly 6 commands, which correspond to the 6 operations. Each operation is an "algebraic form" of the associated command. There are several other commands, which don't have a functional form. <br><br>
As indicated in the images, each command can be associated to a directed interaction combinators (dirIC) node, as defined in:<br>
[<a href="https://mbuliga.github.io/quinegraphs/ic-vs-chem.html#icvschem">Alife properties of directed interaction combinators vs. chemlambda. Marius Buliga (2020), https://mbuliga.github.io/quinegraphs/ic-vs-chem.html#icvschem</a>]<br> or a <a href="https://chemlambda.github.io/index.html">chemlambda</a> [<a href="https://arxiv.org/abs/2003.14332">arXiv:2003.14332</a>] node. <br><br>
For the list of chemlambda rewrites and the history of versions see<br>
[<a href="https://mbuliga.github.io/quinegraphs/history-of-chemlambda.html">Graph rewrites, from emergent algebras to chemlambda. Marius Buliga (2020), https://mbuliga.github.io/quinegraphs/history-of-chemlambda.html</a>]<br><br>
The exceptions are the nodes named "D" and "FOX", first time used in <a href="https://mbuliga.github.io/kali24.html">kali24</a> (but see them described in the file <a href="https://github.com/mbuliga/quinegraphs/blob/master/js/chemistry.js">chemistry.js</a>) and the node "FIN", the fanin dual to the fanout FO.<br><br>
<table style="padding: 20px; margin-bottom: 18px; border:0">
<tr>
<th style="text-align: left;">command
</th>
<th style="text-align: left;">algebraic form
</th>
</th>
<th style="text-align: left;">functional form
</th>
<th style="text-align: left;">graphical form
</th>
</tr>
<tr>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">from e see a as b; </p>
<span3>
</td>
<td>
map
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">b = e ◦ a; </p>
<span3>
</td>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">from e see a as b; </p>
<span3>
</td>
<td>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/D.jpg" alt="from e see a as b" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
</td>
</tr>
<tr>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
see a from e as b;</p>
<span3>
</td>
<td>
abstraction
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
b = λ e. a;</p>
<span3>
</td>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
see a from e as b;</p>
<span3>
</td>
<td>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/L.jpg" alt="see a from e as b" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
</td>
</tr>
<tr>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
as b from e see a;</p>
<span3>
</td>
<td>
application
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
a = b e;</p>
<span3>
</td>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
apply b over e as a;</p>
<span3>
</td>
<td>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/A.jpg" alt="as b from e see a" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
</td>
</tr>
<tr>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
see a as b from e;</p>
<span3>
</td>
<td>
fi
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
e = a > b;</p>
<span3>
</td>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
note a with b as e;</p>
<span3>
</td>
<td>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/FI.jpg" alt="see a as b from e" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
</td>
</tr>
<tr>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
from e as b see a;</p>
<span3>
</td>
<td>
unmap
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
a = e • b;</p>
<span3>
</td>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
over e apply b as a;</p>
<span3>
</td>
<td>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/FOE.jpg" alt="from e as b see a" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
</td>
</tr>
<tr>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
as b see a from e;</p>
<span3>
</td>
<td>
fox
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
e = b < a;</p>
<span3>
</td>
<td>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
with b note a as e;</p>
<span3>
</td>
<td>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/FOX.jpg" alt="as b see a from e" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
</td>
</tr>
</table>
There are several supplementary commands: fanout and fanin:
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
in e as a free b;</p>
<span3>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/FO.jpg" alt="in e as a free b" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
in e free b as a;</p>
<span3>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/FIN.jpg" alt="in e free b as a" style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
an Arrow command:
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
in a as b;</p>
<span3>
and an input, output and free commands:
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
in a;</p>
<span3>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
as a;</p>
<span3>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:200px;border:0">
free a;</p>
<span3>
which correspond respectively to 1-valent nodes denoted respectively by FROUT, FRIN and T.<br><br>
</span3>
<h3 id="ConversionBetweenCommandsAndAlgebraicForms">Conversion between commands, functional and algebraic forms</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
<span3>
The functional form of a command is the command multiplied by a scalar.Example:<br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
apply b over e as a;</p>
<span3>
is the command </span3>as b from e see a;<span3> multiplied by the scalar </span3>apply as<span3>, i.e.
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
(apply as) (as b from e see a);</p>
<span3>
<span3>We can pass from algebraic forms to commands by the following rewrite (ALG):<br><br>
<table style="padding: 20px; margin-bottom: 18px; width:350px; border:0">
<tr>
<td>b = a;
</td>
<td> is replaced by
</td>
<td> in a as b;
</td>
</tr>
</table>
Examples:
<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<th style="text-align: left;">command
</th>
<th style="text-align: left;">algebraic form
</th>
<th style="text-align: left;">replace by
</th>
</tr>
<tr>
<td>from e see a as b;
</td>
<td> b = e ◦ a;
</td>
<td> in (e ◦ a) as b;
</td>
</tr>
<tr>
<td>see a from e as b;
</td>
<td> b = λ e. a;
</td>
<td> in (λ e. a) as b;
</td>
</tr>
<tr>
<td>as b from e see a;
</td>
<td> a = b e;
</td>
<td> in (b e) as a;
</td>
</tr>
<tr>
<td>see a as b from e;
</td>
<td> e = a > b;
</td>
<td> in (a > b) as e;
</td>
</tr>
<tr>
<td>from e as b see a;
</td>
<td> a = e • b;
</td>
<td> in (e • b) as a;
</td>
</tr>
<tr>
<td>as b see a from e;
</td>
<td> e = b < a;
</td>
<td> in (b < a) as e;
</td>
</tr>
</table>
Another rewrite is (COMB) which eliminates Arrow commands:<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<th style="text-align: left;">LHS pattern
</th>
<th style="text-align: left;">action
</th>
</tr>
<tr>
<td>in a as b;
</td>
<td> delete the command,<br> replace b by a
</td>
</tr>
</table>
<span3>
with the exception: delete in a as a;<br><br>
(ALG) and (COMB) may be combined into an evaluation rewrite (EVAL), for example<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<th style="text-align: left;">LHS pattern
</th>
<th style="text-align: left;">action
</th>
</tr>
<tr>
<td>see a from e as b;
</td>
<td> delete the command,<br> replace b by λ e. a
</td>
</tr>
</table>
<span3>
<h3 id="ReductionOfTrigrams">Reduction of trigrams</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
A trigram is any triple of commands of the form:<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<td> for[ε] a see[ε] b as[ε] e;
</td>
</tr>
<tr>
<td> for[μ] e see[μ] c as[μ] d;
</td>
</tr>
<tr>
<td> for[ε] u see[ε] w as[ε] c;
</td>
</tr>
</table>
<span3>
where μ, ε are scalars.<br><br>
The main rewrite (schema) of Pure See is (SHUFFLE), which replaces a trigram by another trigram:<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<th style="text-align: left;">LHS pattern
</th>
<th style="text-align: left;">RHS pattern
</th>
</tr>
<tr><td></td><td></td></tr>
<tr><td></td><td></td></tr>
<tr>
<td>
<table>
<tr>
<td> from[ε] a see[ε] b as[ε] e;
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[μ] e see[μ] c as[μ] d;
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[ε] u see[ε] w as[ε] c;
</td>
</tr>
</table>
</td>
<td>
<table>
<tr>
<td> from[μ] a see[μ] u as[μ] v;
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[ε] v see[ε] p as[ε] d;
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[μ] b see[μ] w as[μ] p;
</td>
</tr>
</table>
</td>
</tr>
</table>
<span3>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/trigram-dist1.jpg" alt="trigram for a DIST1" style="padding: 20px; margin-bottom: 18px; width:450px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
</div>
</div>
<h3 id="MultiplicationByScalarsOrigin">Multiplication by scalars. Origin</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
<span3>
The special word </span3>origin<span3> is a term. With respect to the </span3>origin<span3> we define the multiplication by the scalar μ of the term T:<br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
from[μ] origin see[μ] T as (μ T);</p>
<span3>
The multiplication of a command<br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:350px;border:0"> from[ε] u see[ε] w as[ε] c;</p>
<span3> by the scalar μ is defined as<br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; width:350px;border:0">
from[ε] (μ u) see[ε] (μ w) as[ε] (μ c);</p>
<span3>
via a SHUFFLE schema<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<th style="text-align: left;">LHS pattern
</th>
<th style="text-align: left;">RHS pattern
</th>
</tr>
<tr><td></td><td></td></tr>
<tr><td></td><td></td></tr>
<tr>
<td>
<table>
<tr>
<td> from[ε] origin see[ε] origin as[ε] origin;
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[μ] origin see[μ] c as[μ] (μ c);
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[ε] u see[ε] w as[ε] c;
</td>
</tr>
</table>
</td>
<td>
<table>
<tr>
<td> from[μ] origin see[μ] u as[μ] (μ u);
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[ε] (μ u) see[ε] (μ w) as[ε] (μ c);
</td>
</tr>
<tr><td></td></tr>
<tr>
<td> from[μ] origin see[μ] w as[μ] (μ w);
</td>
</tr>
</table>
</td>
</tr>
</table>
<span3>
</span3>
<h3 id="ReductionByPassingToTheLimit">Reduction by passing to the limit</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
We are allowed to "pass to the limit" with scalars, to one of the special scalars 0, 1, ∞. This passage to the limit is a rewrite.<br><br>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<th style="text-align: left;">LHS pattern
</th>
<th style="text-align: left; width:120px;">
</th>
<th style="text-align: left;">RHS pattern
</th>
</tr>
<tr><td></td><td></td><td></td></tr>
<tr><td></td><td></td><td></td></tr>
<tr>
<td>from[μ] e see[μ] a as[μ] b;
</td>
<td>μ → 1</td>
<td> in e; in a as b;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → 0</td>
<td>in a; in e as b;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → ∞</td>
<td>in e free a as b;
</td>
</tr>
<tr><td></td><td></td><td></td></tr>
<tr><td></td><td></td><td></td></tr>
<tr>
<td>see[μ] a from[μ] e as[μ] b;
</td>
<td>μ → 1</td>
<td> as e; in a as b;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → 0</td>
<td>in a as e free b;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → ∞</td>
<td>as b; in a as e;
</td>
</tr>
<tr><td></td><td></td><td></td></tr>
<tr><td></td><td></td><td></td></tr>
<tr>
<td>as[μ] b from[μ] e see[μ] a;
</td>
<td>μ → 1</td>
<td> in e; in b as a;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → 0</td>
<td>in b free e as a;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → ∞</td>
<td>as b; in e as a;
</td>
</tr>
<tr><td></td><td></td><td></td></tr>
<tr><td></td><td></td><td></td></tr>
<tr>
<td>see[μ] a as[μ] b from[μ] e;
</td>
<td>μ → 1</td>
<td> in a free b as e;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → 0</td>
<td>in a; in b as e;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → ∞</td>
<td>in b; in a as e;
</td>
</tr>
<tr><td></td><td></td><td></td></tr>
<tr><td></td><td></td><td></td></tr>
<tr>
<td>from[μ] e as[μ] b see[μ] a;
</td>
<td>μ → 1</td>
<td> in e as b free a;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → 0</td>
<td>as a; in e as b;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → ∞</td>
<td>as b; in e as a;
</td>
</tr>
<tr><td></td><td></td><td></td></tr>
<tr><td></td><td></td><td></td></tr>
<tr>
<td>as[μ] b see[μ] a from[μ] e;
</td>
<td>μ → 1</td>
<td> as e; in b as a;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → 0</td>
<td>as a; in b as e;
</td>
</tr>
<tr>
<td>
</td>
<td>μ → ∞</td>
<td>in b; as a free e;
</td>
</tr>
</table>
<span3>
<!--
We are also allowed to pass to the limit in the following quadrigram (this corresponds to axiom A4 of dilation structures <a href="https://arxiv.org/abs/0810.5042">[arXiv:0810.5042]</a>)<br><br>
</span3>
<table style="padding: 20px; margin-bottom: 18px; width:750px; border:0">
<tr>
<th style="text-align: left;">LHS pattern
</th>
<th style="text-align: left; width:120px;">
</th>
<th style="text-align: left;">RHS pattern
</th>
</tr>
<tr>
<td> α (for[μ] a see[μ] b as[μ] e);</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td> β (over[μ] e apply[μ] c as[μ] d);</td>
<td> μ → 0 </td>
<td> in a as b in w as d;</td>
</tr>
<tr>
<td> γ (for[μ] a see[μ] w as[μ] c);</td>
<td> </td>
<td> </td>
</tr>
</table>
<span3>
-->
<h3 id="EmergentRewrites">Emergent rewrites</h3>
<h4>(<a href="#Contents">back to contents</a>)</h4>
Here is exemplified the rewrite L-A (or the beta rewrite) as an emergent rewrite. You see the rewrite described in graphical form and in trigram form:<br><br>
</span3>
<div class="row">
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/emergent-LA.jpg" alt="emergent BETA" style="padding: 20px; margin-bottom: 18px; width:550px;border:0">
</div>
<div class="col-6 menu" style="text-align: center;">
<img src="img/puresee/trigram-LA.jpg" alt="trigram for LA" style="padding: 20px; margin-bottom: 18px; width:550px;border:0">
</div>
</div>
<span3>
COMB rewrites are also applied.<br><br>
Look in diagonal, from the top left to the bottom right of the figure which shows the graphical form of L-A as emergent. We may interpret the FOE node as the rewrite enzyme.<br><br>
DIST rewrites of chemlambda, dirIC or even those of <a href="https://mbuliga.github.io/kali24.html">kali24</a> are described in the file <a href="https://github.com/mbuliga/quinegraphs/blob/master/js/chemistry.js">chemistry.js</a>. They are particular forms of the possible DIST rewrites which are compatible with the SHUFFLE. <a href="https://mbuliga.github.io/rhs.html">Here is a tool</a> which generates all possible such rewrites. <br><br>
Let's concentrate on the dist1 rewrite. The following figure shows the graphical form of this rewrite:<br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; border:0">
<img src="img/puresee/dist1-commented.jpg" alt="dist1 rewrite commented" style="padding: 20px; margin-bottom: 18px; width:550px;border:0">
</p>
<span3>
In trigram form, the various nodes (i.e. commands) appear as lines: <br><br>
</span3>
<p style="padding: 20px; margin-bottom: 18px; border:0">
<img src="img/puresee/trigram-dist1.jpg" alt="trigram for a dist1" style="padding: 20px; margin-bottom: 18px; width:550px;border:0">
</p>
<span3>
But the LHS pattern of a dist1 has 2 nodes and the RHS pattern has 4 nodes. We can't use directly a SHUFFLE (which transforms 3 nodes into 3 nodes) in order to obtain this rewrite. We need first to create the term "c" (in the previous figure, the one which corresponds to the horizontal line from the bottom). Once created, this new line (i.e command) will allow the use of a SHUFFLE. <br><br>
The creation of this new line can be achieved by an emergent Reidemeister 2 rewrite:<br><br>
<p style="padding: 20px; margin-bottom: 18px; border:0">
<img src="img/puresee/emergent-dist1-2.jpg" alt="emergent R2-" style="padding: 20px; margin-bottom: 18px; width:750px;border:0">
</p>
And the SHUFFLE which finishes the dist1 rewrite is in the next figure. The right column of the figure shows the rewrite in terms of D nodes (dilations) and the column from the left shows the rewrite A-FOE from chemlambda, which is a particular dist1 rewrite.<br><br>
<p style="padding: 20px; margin-bottom: 18px; border:0">
<img src="img/puresee/emergent-dist1.jpg" alt="emergent DIST1" style="padding: 20px; margin-bottom: 18px; width:550px;border:0">
</p>
</span3>
</body>
</html>