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<html>
<head>
<title>Lambda Calculator - Prelude</title>
<link rel="stylesheet" type="text/css" href="css/lambda.css" />
<link rel="stylesheet" type="text/css" href="css/bnf.css" />
<!-- used these references in building this document/environment -->
<link rel="reference" type="text/html" href="http://www.w3.org/TR/REC-html40/sgml/entities.htm" /> <!-- for all those great entities -->
<link rel="reference" type="text/html" href="http://www.cs.bgu.ac.il/~elhadad/advpro/course4.html" />
<link rel="reference" type="text/html" href="http://en.wikipedia.org/wiki/Lambda_calculus" />
<link rel="reference" type="text/html" href="http://ling.ucsd.edu/~barker/Lambda/" />
<link rel="reference" type="application/x-pdf" href="http://www.cs.uiowa.edu/~slonnegr/plf/Book/Chapter5.pdf" />
<link rel="reference" type="application/x-pdf" href="http://www.inf.fu-berlin.de/lehre/WS03/alpi/lambda.pdf" />
<link rel="reference" type="application/x-pdf" href="http://www.cs.iastate.edu/~lumpe/CS342/Solutions/SolutionPS4.pdf" />
<link rel="reference" type="text/html" href="http://www.itu.dk/people/sestoft/lamreduce/index.html" />
<link rel="reference" type="text/html" href="http://ellemose.dina.kvl.dk/~sestoft/lamreduce/lamframes.html" />
<link rel="reference" type="text/html" href="http://pages.cpsc.ucalgary.ca/~robin/class/417/Assignment_2.html" />
</head>
<body>
<h3>HTML-DOM λ-calculator</h3>
<pre>
The idea is to use HTML elements to represent λ-calculus expressions,
and the Javascript runtime to evaluate them.
This is not a very effecient or even practical means of implementation.
However it does allow for a nice display of the expressions, and even
some style.
This page consists of itself, two CSS files to define some style and a single
Javascript file to implement all the λ-calculus machinery.
</pre>
<p>
<em>A STRING is any valid Javascript String</em>
</p>
<p>
<a name="quick-nav"><strong>Quick Navigation</strong></a><br />
<a href="#lambda-calculus-grammar">λ-calculus grammar</a> |
<a href="#html-dom-grammar">λ-calculus HTML-DOM grammar</a> |
<a href="#lambda-calculus-axioms">λ-calculus axioms</a> |
<a href="#lambda-calculus-substitution">λ-calculus substitution</a> |
<a href="#lambda-calculus-environment">λ-calculus environment</a> |
<a href="#lambda-calculus-expressions">λ-calculus expressions</a>
</p>
<a name="lambda-calculus-grammar"><strong>λ-calculus grammar</strong></a>
<span class="bnf-notation">
<dl>
<dt><expression></dt>
<dd><var></dd>
<dd><abstraction></dd>
<dd><application></dd>
</dl>
<dl>
<dt><var></dt>
<dd>STRING</dd>
</dl>
<dl>
<dt><abstraction></dt>
<dd>(λ<var>. <expression>)</dd>
</dl>
<dl>
<dt><application></dt>
<dd>(<expression> <expression>)</dd>
</dl>
</span>
<a name="html-dom-grammar"><strong>λ-calculus HTML-DOM grammar</strong></a><a href="#quick-nav">[Quick Nav]</a>
<span class="bnf-notation">
<dl>
<dt><expression></dt>
<dd><var></dd>
<dd><abstraction></dd>
<dd><application></dd>
</dl>
<dl>
<dt><var></dt>
<dd><VAR>STRING</VAR></dd>
</dl>
<dl>
<dt><abstraction></dt>
<dd><DL><DT><var></DT><DD><expression></DD></DL></dd>
</dl>
<dl>
<dt><application></dt>
<dd><OL><LI><expression></LI><LI><expression></LI><OL></dd>
</dl>
</span>
<a name="lambda-calculus-axioms"><strong>λ-calculus axioms</strong></a><a href="#quick-nav">[Quick Nav]</a>
<pre>
Evaluation in λ-calculus is defined by three simple reduction rules:
1. α-conversion
2. β-reduction
3. η-reduction
α-conversion can be defined semi-formally as:
E[x′/x] → E′ ≡ E ∧ x′ ∈ FV(E)
Which means in expression E, x can be rewritten as x′ yeilding
an equivalent expression E′ with the condition that x′ is not bound in E
(FV is a function which returns the set of free-variables in its argument)
β-reduction can be defined semi-formally as:
(E1 E2) → E1[E2/x] → E′
Which means that in the application (E1 E2), E2 can be substituted
for the formal parameter <em>x</em> in E1 to yeild a new expression E′
η-reduction can be defined semi-formally as:
λx. (E x) → E ∧ x ∉ FV(E)
Which means that if the body of a λ-abstraction is the application
of another expression to the formal argument, and the formal argument
does not a free variable in E than you can simply reduce the expression
to E.
</pre>
<a name="lambda-calculus-substitution"><strong>λ-calculus substitution</strong> (<em>This damn concept took me forever to get right.</em>)</a><a href="#quick-nav">[Quick Nav]</a>
<pre>
Substitution in λ-calculus is a rather involved process.
This is entirely because free variable can get captured during substitution.
Substitution is done recursively, and has the following rules:
Given application (λv.E M) we define E[M/v] <em>substitute M for v in E</em> as:
<strong>Variable Case</strong>
E ≡ v → E[M/v] → M
<em>E is equal to v, substitute M for E</em>
E ≡ x ∧ x ≠ v → E[M/v] → x
<em>E is a variable other than v, no-change to E</em>
<strong>Application Case</strong>
E ≡ (E<sub>1</sub> E<sub>2</sub>) → E[M/v] → (E<sub>1</sub>[M/v] E<sub>2</sub>[M/v])
<em>substitute recursively on E<sub>1</sub> and E<sub>2</sub></em>
<strong>Abstraction Case</strong>
E ≡ λx. E<sub>1</sub> ∧ v ∉ FV(E<sub>1</sub>) → E[M/v] → E
<em>v is bound in E<sub>1</sub>, no-change to E</em>
E ≡ λx. E<sub>1</sub> ∧ x ≠ v ∧ x ∉ FV(M) → E[M/v] → λx. E<sub>1</sub>[M/v]
<em>x is a variable other than v and x is not bound in M, substitute recursively on E<sub>1</sub></em>
E ≡ λx. E<sub>1</sub> ∧ x ≠ v ∧ x ∈ FV(M) ∧ z ≠ x ∧ z ∉ FV(M E<sub>1</sub>) → E ↔<sub>α</sub> λz. (E<sub>1</sub>[z/x] → E<sub>1</sub>[M/v])
<em>x is a variable other than v and x is not bound in M;
z is a variable other than x and z is not bound in M and E<sub>1</sub>, alpha convert x to z in E<sub>1</sub> then substitute recursively in E<sub>1</sub></em>
</pre>
<h3><a name="lambda-calculus-environment">λ-calculus environment</a><a href="#quick-nav">[Quick Nav]</a></h3>
<strong>TRUE</strong>
<div id="TRUE" class="lambda-exp">
<dl>
<dt><var>x</var></dt>
<dd>
<dl>
<dt><var>y</var></dt>
<dd>
<var>x</var>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>FALSE</strong>
<div id="FALSE" class="lambda-exp">
<dl>
<dt><var>x</var></dt>
<dd>
<dl>
<dt><var>y</var></dt>
<dd>
<var>y</var>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>IF</strong>
<div id="IF" class="lambda-exp">
<dl>
<dt><var>predicate</var></dt>
<dd>
<dl>
<dt><var>consequent</var></dt>
<dd>
<dl>
<dt><var>alternative</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>predicate</var></li>
<li><var>consequent</var></li>
</ol>
</li>
<li><var>alternative</var></li>
</ol>
</dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>AND</strong>
<div id="AND" class="lambda-exp">
<dl>
<dt><var>p</var></dt>
<dd>
<dl>
<dt><var>q</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>p</var></li>
<li><var>q</var></li>
</ol>
</li>
<li><var>FALSE</var>
</ol>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>OR</strong>
<div id="OR" class="lambda-exp">
<dl>
<dt><var>p</var></dt>
<dd>
<dl>
<dt><var>q</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>p</var></li>
<li><var>TRUE</var></li>
</ol>
</li>
<li><var>q</var></li>
</ol>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>NOT</strong>
<div id="NOT" class="lambda-exp">
<dl>
<dt><var>p</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>p</var></li>
<li><var>FALSE</var></li>
</ol>
</li>
<li><var>TRUE</var></li>
</ol>
</dd>
</dl>
</div>
<pre>
Remember, ZERO ≡ FALSE.
</pre>
<strong>ZERO?</strong>
<div id="ZERO?" class="lambda-exp">
<dl>
<dt><var>n</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>n</var></li>
<li>
<dl>
<dt><var>x</var></dt>
<dd><var>FALSE</var></dd>
</dl>
</li>
</ol>
</li>
<li><var>TRUE</var></li>
</ol>
</dd>
</dl>
</div>
<h4>Church Numerals</h4>
<pre>
SUCC allows us to define any number once we have 0.
</pre>
<strong>SUCC</strong>
<div id="SUCC" class="lambda-exp">
<dl>
<dt><var>n</var>
<dd>
<dl>
<dt><var>f</var></dt>
<dd>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li><var>f</var></li>
<li>
<ol>
<li>
<ol>
<li><var>n</var></li>
<li><var>f</var></li>
</ol>
</li>
<li><var>x</var></li>
</ol>
</li>
</ol>
</dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</div>
<pre>
Most examples only define ZERO - THREE in church numerals,
and since each numeral is just SUCC applied to the previous
one there is really no need to define more (really ZERO and
SUCC should be all we need)
But since its slow to have to compute ONE each time we need it,
a few are defined here just for example.
</pre>
<strong>ZERO</strong>
<em>≡ FALSE, Surprise!</em>
<div id="ZERO" class="lambda-exp">
<var>FALSE</var>
</div>
<strong>ONE</strong>
<div id="ONE" class="lambda-exp">
<dl>
<dt><var>f</var></dt>
<dd>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li><var>f</var></li>
<li><var>x</var></li>
</ol>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>TWO</strong>
<div id="TWO" class="lambda-exp">
<dl>
<dt><var>f</var></dt>
<dd>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li><var>f</var></li>
<li>
<ol>
<li><var>f</var></li>
<li><var>x</var></li>
</ol>
</li>
</ol>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>THREE</strong>
<div id="THREE" class="lambda-exp">
<dl>
<dt><var>f</var></dt>
<dd>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li><var>f</var></li>
<li>
<ol>
<li><var>f</var></li>
<li>
<ol>
<li><var>f</var></li>
<li><var>x</var></li>
</ol>
</li>
</ol>
</li>
</ol>
</dd>
</dl>
</dd>
</dl>
</div>
<pre>
What good are numbers if you can't add them?
</pre>
<strong>PLUS</strong>
<div id="PLUS" class="lambda-exp">
<dl>
<dt><var>m</var></dt>
<dd>
<dl>
<dt><var>n</var></dt>
<dd>
<dl>
<dt><var>f</var></dt>
<dd>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>m</var></li>
<li><var>f</var></li>
</ol>
</li>
<li>
<ol>
<li>
<ol>
<li><var>n</var></li>
<li><var>f</var></li>
</ol>
</li>
<li><var>x</var></li>
</ol>
</li>
</ol>
</dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</div>
<pre>
MULT could be defined independently of PLUS,
but I like how the dependency illustrates the concept
more clearly
</pre>
<strong>MULT</strong>
<div id="MULT" class="lambda-exp">
<dl>
<dt><var>m</var></dt>
<dd>
<dl>
<dt><var>n</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>m</var></li>
<li>
<ol>
<li><var>PLUS</var></li>
<li><var>n</var></li>
</ol>
</li>
</ol>
</li>
<li><var>ZERO</var></li>
</ol>
</dd>
</dl>
</dd>
</dl>
</div>
<pre>
PRED can also be defined in terms of PLUS and ZERO,
but I won't do that here
</pre>
<strong>PRED</strong>
<div id="PRED" class="lambda-exp">
<dl>
<dt><var>n</var></dt>
<dd>
<dl>
<dt><var>f</var></dt>
<dd>
<dl>
<dt><var>y</var></dt>
<dd>
<ol>
<li>
<ol>
<li>
<ol>
<li><var>n</var></li>
<li>
<dl>
<dt><var>g</var></dt>
<dd>
<dl>
<dt><var>h</var></dt>
<dd>
<ol>
<li><var>h</var></li>
<li>
<ol>
<li><var>g</var></li>
<li><var>f</var></li>
</ol>
</li>
</ol>
</dd>
</dl>
</dd>
</dl>
</li>
</ol>
</li>
<li>
<dl>
<dt><var>u</var></dt>
<dd><var>y</var></dd>
</dl>
</li>
</ol>
</li>
<li>
<dl>
<dt><var>u</var></dt>
<dd><var>u</var></dt>
</dl>
</li>
</ol>
</dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</div>
<h4>List Operations</h4>
<strong>CONS</strong>
<div id="CONS" class="lambda-exp">
<dl>
<dt><var>f</var></dt>
<dd>
<dl>
<dt><var>s</var></dt>
<dd>
<dl>
<dt><var>b</var></dt>
<dd>
<ol>
<li>
<ol>
<li><var>b</var></li>
<li><var>f</var></li>
</ol>
</li>
<li><var>s</var></li>
</ol>
</dd>
</dl>
</dd>
</dl>
</dd>
</dl>
</div>
<strong>HEAD</strong>
<div id="HEAD" class="lambda-exp">
<dl>
<dt><var>p</var></dt>
<dd>
<ol>
<li><var>p</var></li>
<li><var>TRUE</var></li>
</ol>
</dd>
</dl>
</div>
<strong>TAIL</strong>
<div id="TAIL" class="lambda-exp">
<dl>
<dt><var>p</var></dt>
<dd>
<ol>
<li><var>p</var></li>
<li><var>FALSE</var></li>
</ol>
</dd>
</dl>
</div>
<strong>NIL</strong>
<div id="NIL" class="lambda-exp">
<dl>
<dt><var>x</var></dt>
<dd><var>TRUE</var></dd>
</dl>
</div>
<pre>
Since the environment is the DOM, it makes sense that applications
can use variables which exist elsewhere.
</pre>
<h3><a name="lambda-calculus-expressions">λ-calculus expression</a><a href="#quick-nav">[Quick Nav]</a></h3>
<strong>Sanity Check our Number System</strong><br />
<button onclick="prelude.compute_toplevel(document.getElementById('SUCC-ZERO-test'))">Reduce</button>
<div id="SUCC-ZERO-test" class="lambda-exp">
<ol>
<li><var>SUCC</var></li>
<li><var>ZERO</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('SUCC-TWO-test'))">Reduce</button>
<div id="SUCC-TWO-test" class="lambda-exp">
<ol>
<li><var>SUCC</var></li>
<li><var>TWO</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('PLUS-test'))">Reduce</button>
<div id="PLUS-test" class="lambda-exp">
<ol>
<li>
<ol>
<li><var>PLUS</var></li>
<li><var>TWO</var></li>
</ol>
</li>
<li><var>THREE</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('MULT-test'))">Reduce</button>
<div id="MULT-test" class="lambda-exp">
<ol>
<li>
<ol>
<li><var>MULT</var></li>
<li><var>TWO</var></li>
</ol>
</li>
<li><var>TWO</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('PRED-test'))">Reduce</button>
<div id="PRED-test" class="lambda-exp">
<ol>
<li>
<ol>
<li><var>PRED</var></li>
<li>
<ol>
<li><var>SUCC</var></li>
<li><var>ZERO</var></li>
</ol>
</li>
</ol>
</li>
<li><var>ONE</var></li>
</ol>
</div>
<strong>Testing Logic</strong><br />
<button onclick="prelude.compute_toplevel(document.getElementById('TRUE-test'))">Reduce</button>
<div id="TRUE-test" class="lambda-exp">
<ol>
<li>
<ol>
<li><var>TRUE</var></li>
<li><var>1</var></li>
</ol>
</li>
<li><var>-1</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('FALSE-test'))">Reduce</button>
<div id="FALSE-test" class="lambda-exp">
<ol>
<li>
<ol>
<li><var>FALSE</var></li>
<li><var>1</var></li>
</ol>
</li>
<li><var>-1</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('IF-test'))">Reduce</button>
<div id="IF-test" class="lambda-exp">
<ol>
<li>
<ol>
<li>
<ol>
<li><var>IF</var></li>
<li><var>TRUE</var></li>
</ol>
</li>
<li><var>Truth</var></li>
</ol>
</li>
<li><var>Fiction</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('AND-test'))">Reduce</button>
<div id="AND-test" class="lambda-exp">
<ol>
<li>
<ol>
<li><var>AND</var></li>
<li><var>TRUE</var></li>
</ol>
</li>
<li><var>FALSE</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('OR-test'))">Reduce</button>
<div id="OR-test" class="lambda-exp">
<ol>
<li>
<ol>
<li><var>OR</var></li>
<li><var>FALSE</var></li>
</ol>
</li>
<li><var>TRUE</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('NOT-test'))">Reduce</button>
<div id="NOT-test" class="lambda-exp">
<ol>
<li><var>NOT</var></li>
<li><var>FALSE</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('FALSE-FALSE-test'))">Reduce</button>
<div id="FALSE-FALSE-test" class="lambda-exp">
<ol>
<li><var>FALSE</var></li>
<li><var>FALSE</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('ZERO?-test-1'))">Reduce</button>
<div id="ZERO?-test-1" class="lambda-exp">
<ol>
<li><var>ZERO?</var></li>
<li><var>ZERO</var></li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('ZERO?-test-2'))">Reduce</button>
<div id="ZERO?-test-2" class="lambda-exp">
<ol>
<li><var>ZERO?</var></li>
<li>
<ol>
<li><var>SUCC</var></li>
<li><var>ZERO</var></li>
</ol>
</li>
</ol>
</div>
<strong>A full LISP will have to wait until the entry system is better</strong><br />
<button onclick="prelude.compute_toplevel(document.getElementById('HEAD-CONS-test'))">Reduce</button>
<div id="HEAD-CONS-test" class="lambda-exp">
<ol>
<li><var>HEAD</var></li>
<li>
<ol>
<li>
<ol>
<li><var>CONS</var></li>
<li><var>TWO</var></li>
</ol>
</li>
<li><var>NIL</var></li>
</ol>
</li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('TAIL-CONS-test'))">Reduce</button>
<div id="TAIL-CONS-test" class="lambda-exp">
<ol>
<li><var>HEAD</var></li>
<li>
<ol>
<li><var>TAIL</var></li>
<li>
<ol>
<li>
<ol>
<li><var>CONS</var></li>
<li><var>ONE</var></li>
</ol>
</li>
<li>
<ol>
<li>
<ol>
<li><var>CONS</var></li>
<li><var>TWO</var></li>
</ol>
</li>
<li><var>NIL</var></li>
</ol>
</li>
</ol>
</li>
</ol>
</li>
</ol>
</div>
<strong>Here is a test of the reduction algorithm</strong><br />
<button onclick="prelude.compute_toplevel(document.getElementById('lambda-calc')); return false">Reduce</button>
<div id="lambda-calc" class="lambda-exp">
<ol>
<li>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li>
<var>x</var>
</li>
<li>
<var>World</var>
</li>
</ol>
</dd>
</dl>
</li>
<li>
<var>Hello</var>
</li>
</ol>
</div>
<strong>Testing the free variable checkers</strong><br />
<button onclick="alert(prelude.occurs_free(document.getElementById('yy-test'),document.getElementById('yy-test-y')) == false)">Assert y is not free</button>
<div class="lambda-exp">
<ol>
<li>
<dl id="yy-test">
<dt><var>x</var>
<dd>
<dl>
<dt><var>y</var>
<dd>
<ol>
<li><var>x</var></li>
<li><var>y</var></li>
</ol>
</dd>
</dl>
</dd>
</dl>
</li>
<li><var id="yy-test-y">y</var></li>
</ol>
</div>
<button onclick="alert(prelude.occurs_bound(document.getElementById('yz-test'),document.getElementById('yz-test-y')) == false)">Assert z is free</button>
<div class="lambda-exp">
<ol>
<li>
<dl id="yz-test">
<dt><var>x</var>
<dd>
<dl>
<dt><var>y</var>
<dd>
<ol>
<li><var>x</var></li>
<li><var>y</var></li>
</ol>
</dd>
</dl>
</dd>
</dl>
</li>
<li><var id="yz-test-y">z</var></li>
</ol>
</div>
<strong>Test FV</strong>
<button onclick="console.log(prelude.FV(document.getElementById('fv-test-abx')))">Reveal FV set (<em>requires FireBug enabled</em>)</button>
<div class="lambda-exp">
<dl id="fv-test-abx">
<dt><var>x</var></dt>
<dd>
<ol>
<li><var>x</var></li>
<li>
<ol>
<li><var>y</var></li>
<li><var>z</var></li>
</ol>
</li>
</ol>
</dd>
</dl>
</div>
<strong>Testing reduction</strong><br />
<button onclick="prelude.compute_toplevel(document.getElementById('reduce-test-exp'))">Reduce</button>
<div id="reduce-test-exp" class="lambda-exp">
<ol>
<li>
<dl>
<dt><var>P</var>
<dd>
<ol>
<li><var>P</li>
<li><var>j</li>
</ol>
</dd>
</dl>
</li>
<li>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li><var>Q</var></li>
<li><var>x</var></li>
</ol>
</dd>
</dl>
</li>
</ol>
</div>
<button onclick="prelude.compute_toplevel(document.getElementById('reduce-test-exp-2'))">Reduce</button>
<div id="reduce-test-exp-2" class="lambda-exp">
<ol>
<li>
<dl>
<dt><var>x</var></dt>
<dd>
<ol>
<li><var>P</var></li>
<li><var>x</var></li>
</ol>
</dd>
</dl>
</li>
<li><var>j</var></li>
</ol>
</div>