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doc/build/html/references/gm-state-transitions.html
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<!DOCTYPE html>
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<html lang="en" data-content_root="../">
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<head>
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<meta charset="utf-8" />
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<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
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<title>G-Machine State Transition Rules — rl' documentation</title>
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<link rel="stylesheet" type="text/css" href="../_static/pygments.css?v=4f649999" />
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<link rel="stylesheet" type="text/css" href="../_static/alabaster.css?v=039e1c02" />
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<script src="../_static/documentation_options.js?v=5929fcd5"></script>
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<script src="../_static/doctools.js?v=888ff710"></script>
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<script src="../_static/sphinx_highlight.js?v=dc90522c"></script>
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<link rel="index" title="Index" href="../genindex.html" />
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<link rel="search" title="Search" href="../search.html" />
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<link rel="next" title="Template Instantiator State Transition Rules" href="ti-state-transitions.html" />
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<link rel="prev" title="The Template Instantiator" href="../commentary/ti.html" />
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<link rel="stylesheet" href="../_static/custom.css" type="text/css" />
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<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
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</head><body>
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<div class="document">
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<div class="documentwrapper">
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<div class="bodywrapper">
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<div class="body" role="main">
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<section id="g-machine-state-transition-rules">
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<h1>G-Machine State Transition Rules<a class="headerlink" href="#g-machine-state-transition-rules" title="Link to this heading">¶</a></h1>
|
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<section id="core-transition-rules">
|
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<h2>Core Transition Rules<a class="headerlink" href="#core-transition-rules" title="Link to this heading">¶</a></h2>
|
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<ol class="arabic">
|
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<li><p>Lookup a global by name and push its value onto the stack</p>
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<div class="math">
|
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<p><img src="../_images/math/18d81e1fc22e6feb876a993fe30fd9e8336f5bf6.svg" alt="\gmrule
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{ \mathtt{PushGlobal} \; f : i
|
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& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
f : a
|
||||
\end{bmatrix}
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
& m
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}"/></p>
|
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</div></li>
|
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<li><p>Allocate an int node on the heap, and push the address of the newly created
|
||||
node onto the stack</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/ed6af2733291dcf5001173a8dbaf8f59c817212a.svg" alt="\gmrule
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{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}"/></p>
|
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</div></li>
|
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<li><p>Allocate an application node on the heap, applying the top of the stack to
|
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the address directly below it. The address of the application node is pushed
|
||||
onto the stack.</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/607985174357bbbea2e2b47ab0694b2d21a71b35.svg" alt="\gmrule
|
||||
{ \mathtt{MkAp} : i
|
||||
& f : x : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NAp} \; f \; x
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Push a function’s argument onto the stack</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/643dd082a2b49d9ebdb87533aac62fdd342f4fd5.svg" alt="\gmrule
|
||||
{ \mathtt{Push} \; n : i
|
||||
& a_0 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a_n : a_0 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Tidy up the stack after instantiating a supercombinator</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/f02d55fb6c249a7da5a9df6b7e5c96ab6be7672b.svg" alt="\gmrule
|
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{ \mathtt{Slide} \; n : i
|
||||
& a_0 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a_0 : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>If the top of the stack is in WHNF (currently this just means a number) is on
|
||||
top of the stack, <code class="code docutils literal notranslate"><span class="pre">Unwind</span></code> considers evaluation complete. In the case
|
||||
where the dump is <strong>not</strong> empty, the instruction queue and stack is restored
|
||||
from the top.</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/2417422cdf63ed202b16462208f8a62630f99679.svg" alt="\gmrule
|
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{ \mathtt{Unwind} : \nillist
|
||||
& a : s
|
||||
& \langle i', s' \rangle : d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ i'
|
||||
& a : s'
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Bulding on the previous rule, in the case where the dump <strong>is</strong> empty, leave
|
||||
the machine in a halt state (i.e. with an empty instruction queue).</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/60f4e38bb9ff4a7bcb201e82cb956a3ccf064632.svg" alt="\gmrule
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a : s
|
||||
& \nillist
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ \nillist
|
||||
& a : s
|
||||
& \nillist
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>If an application is on top of the stack, <code class="code docutils literal notranslate"><span class="pre">Unwind</span></code> continues unwinding</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/b04167eaade600ab592e2a4441ffc27f5268cccf.svg" alt="\gmrule
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NAp} \; f \; x
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& f : a : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>When a supercombinator is on top of the stack (and the correct number of
|
||||
arguments have been provided), <code class="code docutils literal notranslate"><span class="pre">Unwind</span></code> sets up the stack and jumps to
|
||||
the supercombinator’s code (<img class="math" src="../_images/math/90592e982322fa08f1b8ae0e6bcdaca11fc8de59.svg" alt="\beta" style="vertical-align: -4px"/>-reduction)</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/2dc5e48f0f26f08073e6e1cdd70ff876c6ec2434.svg" alt="\gmrule
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a_0 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_0 : \mathtt{NGlobal} \; n \; c \\
|
||||
a_1 : \mathtt{NAp} \; a_0 \; e_1 \\
|
||||
\vdots \\
|
||||
a_n : \mathtt{NAp} \; a_{n-1} \; e_n \\
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ c
|
||||
& e_1 : \ldots : e_n : a_n : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Pop the stack, and update the nth node to point to the popped address</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/e299a147e6e958303111e813e82a3507ded25437.svg" alt="\gmrule
|
||||
{ \mathtt{Update} \; n : i
|
||||
& e : f : a_1 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_1 : \mathtt{NAp} \; f \; e \\
|
||||
\vdots \\
|
||||
a_n : \mathtt{NAp} \; a_{n-1} \; e_n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& f : a_1 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_n : \mathtt{NInd} \; e
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Pop the stack.</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/aec9099fc8c459eb433947bd6c9cc0f96fe38f98.svg" alt="\gmrule
|
||||
{ \mathtt{Pop} \; n : i
|
||||
& a_1 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Follow indirections while unwinding</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/7648bfc0796ff33966fbc002c446b2242d296fd9.svg" alt="\gmrule
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NInd} \; a'
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a' : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Allocate uninitialised heap space</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/5380d8227d0a06608b108a26b4a618972b4b7e28.svg" alt="\gmrule
|
||||
{ \mathtt{Alloc} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a_1 : \ldots : a_n : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_1 : \mathtt{NUninitialised} \\
|
||||
\vdots \\
|
||||
a_n : \mathtt{NUninitialised} \\
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Evaluate the top of the stack to WHNF</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/ce745b5797483cfcb6ff4816364d91d3d5b6e6c4.svg" alt="\gmrule
|
||||
{ \mathtt{Eval} : i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a : \nillist
|
||||
& \langle i, s \rangle : d
|
||||
& h
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Reduce a primitive binary operator <img class="math" src="../_images/math/43434266ad5b9f36d523d70ff88db5c4e9366b03.svg" alt="*" style="vertical-align: 0px"/>.</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/176a0e2360b1e342041d8129c3a264321ae806ce.svg" alt="\gmrule
|
||||
{ * : i
|
||||
& a_1 : a_2 : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_1 : x \\
|
||||
a_2 : y
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a' : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a' : (x * y)
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Reduce a primitive unary operator <img class="math" src="../_images/math/604a42b94739b04fa9559a97c3703816bf57aa5c.svg" alt="\neg" style="vertical-align: 0px"/>.</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/6de99ea1d6d13dbeedcdfcf7e3807cdd4f2e8594.svg" alt="\gmrule
|
||||
{ \neg : i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : x
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a' : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a' : (\neg x)
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>Consider unapplied functions to be in WHNF</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/702f28367db2bbf757f819d55f238962369384ad.svg" alt="\gmrule
|
||||
{ \mathtt{Unwind} : \nillist
|
||||
& a_0 : \ldots : a_n : \nillist
|
||||
& \langle i, s \rangle : d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a_0 : \mathtt{NGlobal} \; k \; c
|
||||
\end{bmatrix}
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a_n : s
|
||||
& d
|
||||
& h
|
||||
& m \\
|
||||
\SetCell[c=2]{c}
|
||||
\text{when $n < k$}
|
||||
}"/></p>
|
||||
</div></li>
|
||||
</ol>
|
||||
</section>
|
||||
<section id="extension-rules">
|
||||
<h2>Extension Rules<a class="headerlink" href="#extension-rules" title="Link to this heading">¶</a></h2>
|
||||
<ol class="arabic">
|
||||
<li><p>A sneaky trick to enable sharing of <code class="code docutils literal notranslate"><span class="pre">NNum</span></code> nodes. We note that the
|
||||
global environment is a mapping of plain old strings to heap addresses.
|
||||
Strings of digits are not considered valid identifiers, so putting them on
|
||||
the global environment will never conflict with a supercombinator! We abuse
|
||||
this by modifying Core Rule 2 to update the global environment with the new
|
||||
node’s address. Consider how this rule might impact garbage collection
|
||||
(remember that the environment is intended for <em>globals</em>).</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/0fc7c9df5cc4f923f6953e615c1f28a6531c0b68.svg" alt="\gmrule
|
||||
{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
\begin{bmatrix}
|
||||
a : \mathtt{NNum} \; n
|
||||
\end{bmatrix}
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
n' : a
|
||||
\end{bmatrix}
|
||||
\\
|
||||
\SetCell[c=5]{c}
|
||||
\text{where $n'$ is the base-10 string rep. of $n$}
|
||||
}"/></p>
|
||||
</div></li>
|
||||
<li><p>In order for the previous rule to be effective, we are also required to take
|
||||
action when a number already exists in the environment:</p>
|
||||
<div class="math">
|
||||
<p><img src="../_images/math/d56941a661a880391bd898f16e5dcf9c0b8a7e59.svg" alt="\gmrule
|
||||
{ \mathtt{PushInt} \; n : i
|
||||
& s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\begin{bmatrix}
|
||||
n' : a
|
||||
\end{bmatrix}
|
||||
}
|
||||
{ i
|
||||
& a : s
|
||||
& d
|
||||
& h
|
||||
& m
|
||||
\\
|
||||
\SetCell[c=5]{c}
|
||||
\text{where $n'$ is the base-10 string rep. of $n$}
|
||||
}"/></p>
|
||||
</div></li>
|
||||
</ol>
|
||||
</section>
|
||||
</section>
|
||||
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
|
||||
<div class="sphinxsidebarwrapper">
|
||||
<h1 class="logo"><a href="../index.html">rl'</a></h1>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
<h3>Navigation</h3>
|
||||
<p class="caption" role="heading"><span class="caption-text">Index</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../glossary.html">Glossary</a></li>
|
||||
</ul>
|
||||
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|
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