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<!--Weave of 'Binding and Substitution' generated by Inweb-->
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<div class="breadcrumbs">
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<ul class="crumbs"><li><a href="../index.html">Home</a></li><li><a href="../services.html">Services</a></li><li><a href="index.html">calculus</a></li><li><a href="index.html#4">Chapter 4: Propositions</a></li><li><b>Binding and Substitution</b></li></ul></div>
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<p class="purpose">To substitute constants into propositions in place of variables, and to apply quantifiers to bind any unbound variables.</p>
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<ul class="toc"><li><a href="4-bas.html#SP1">§1. Status of variables</a></li><li><a href="4-bas.html#SP9">§9. Getting rid of free variables</a></li><li><a href="4-bas.html#SP14">§14. A footnote on variable 0</a></li><li><a href="4-bas.html#SP17">§17. Detect locals</a></li></ul><hr class="tocbar">
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<p class="commentary firstcommentary"><a id="SP1" class="paragraph-anchor"></a><b>§1. Status of variables. </b>In any proposition \(\phi\), we say that a variable \(v\) is "bound" if it
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appears as the variable governed by a quantifier: it is "free" if it
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does appear somewhere in \(\phi\) — either directly as a term or indirectly
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through a function application — and is not bound. For instance, in
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$$ \forall x : K(x) \land B(x, f_C(y)) $$
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the variable \(x\) is bound and the variable \(y\) is free.
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</p>
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<p class="commentary">In any given proposition, each of the 26 variables always satisfies exactly
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one of the following:
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</p>
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<ul class="items"><li>(a) it is unused if it is never mentioned as, or in, any term, and is not the
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variable of any quantifier;
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</li><li>(b) it is bound if it appears as the variable of any <span class="extract"><span class="extract-syntax">QUANTIFIER_ATOM</span></span>;
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</li><li>(c) it is free if it is used but not bound.
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</li></ul>
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<p class="commentary">The following shows some examples of operations on variables:
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</p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="plain-syntax">'</span><span class="element-syntax">new unary even</span><span class="plain-syntax">':</span><span class="string-syntax"> ok</span>
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<span class="plain-syntax">'</span><span class="element-syntax">variables in << >></span><span class="plain-syntax">':</span><span class="string-syntax"> valid:</span>
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<span class="plain-syntax">'</span><span class="element-syntax">variables in << even (x) >></span><span class="plain-syntax">':</span><span class="string-syntax"> valid: x free</span>
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<span class="plain-syntax">'</span><span class="element-syntax">set P to << Forall x IN< kind = number (x) IN>: (x == y) ^ Exists z: (y == z) >></span><span class="plain-syntax">':</span><span class="string-syntax"> p set to << ForAll x IN< kind=number(x) IN> : (x == y) ^ Exists z : (y == z) >></span>
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<span class="plain-syntax">'</span><span class="element-syntax">variables in P</span><span class="plain-syntax">':</span><span class="string-syntax"> valid: x bound y free z bound</span>
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<span class="plain-syntax">'</span><span class="element-syntax">variable unused in << >></span><span class="plain-syntax">':</span><span class="string-syntax"> x</span>
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<span class="plain-syntax">'</span><span class="element-syntax">variable unused in P</span><span class="plain-syntax">':</span><span class="string-syntax"> a</span>
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<span class="plain-syntax">'</span><span class="element-syntax">renumbering of << even (x) >></span><span class="plain-syntax">':</span><span class="string-syntax"> << even(x) >></span>
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<span class="plain-syntax">'</span><span class="element-syntax">renumbering of << even (z) >></span><span class="plain-syntax">':</span><span class="string-syntax"> << even(x) >></span>
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<span class="plain-syntax">'</span><span class="element-syntax">renumbering of << Exists z: even (z) ^ (z == y) >></span><span class="plain-syntax">':</span><span class="string-syntax"> << Exists x : even(x) ^ (x == y) >></span>
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</pre>
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<p class="commentary">It might seem logical to have a function which takes a proposition \(\phi\)
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and a variable \(v\) and returns its status — unused, free or bound. But this
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would be inefficient, since we want to work with all 26 at once, so instead
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we take a pointer to an array of <span class="extract"><span class="extract-syntax">int</span></span> which needs to have (at least, but
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probably exactly) 26 entries, and on exit each entry is set to one of the
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following.
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</p>
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<p class="commentary">In the course of doing that, it's easy to test whether variables are used
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properly — a bound variable should occur for the first time in its
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quantification, and should not reoccur once the subexpression holding the
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quantifier has finished. We return <span class="extract"><span class="extract-syntax">TRUE</span></span> if all is well, or <span class="extract"><span class="extract-syntax">FALSE</span></span> if not,
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writing the reason why not to <span class="extract"><span class="extract-syntax">err</span></span>.
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</p>
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<pre class="definitions code-font"><span class="definition-keyword">define</span> <span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax"> </span><span class="constant-syntax">1</span>
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<span class="definition-keyword">define</span> <span class="constant-syntax">FREE_VST</span><span class="plain-syntax"> </span><span class="constant-syntax">2</span>
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<span class="definition-keyword">define</span> <span class="constant-syntax">BOUND_VST</span><span class="plain-syntax"> </span><span class="constant-syntax">3</span>
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</pre>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::determine_status</span><button class="popup" onclick="togglePopup('usagePopup1')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup1">Usage of <span class="code-font"><span class="function-syntax">Binding::determine_status</span></span>:<br/><a href="4-bas.html#SP2">§2</a>, <a href="4-bas.html#SP3">§3</a>, <a href="4-bas.html#SP4">§4</a>, <a href="4-bas.html#SP5">§5</a>, <a href="4-bas.html#SP8">§8</a>, <a href="4-bas.html#SP10">§10</a><br/>Type Check Propositions - <a href="4-tcp.html#SP4_7">§4.7</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> *</span><span class="identifier-syntax">var_states</span><span class="plain-syntax">,</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">text_stream</span><span class="plain-syntax"> *</span><span class="identifier-syntax">err</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_VARIABLE</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">j</span><span class="plain-syntax">, </span><span class="identifier-syntax">unavailable</span><span class="plain-syntax">[26], </span><span class="identifier-syntax">blevel</span><span class="plain-syntax"> = </span><span class="constant-syntax">0</span><span class="plain-syntax">, </span><span class="identifier-syntax">valid</span><span class="plain-syntax"> = </span><span class="identifier-syntax">TRUE</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">j</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="plain-syntax"><26; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) { </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] = </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">; </span><span class="identifier-syntax">unavailable</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] = </span><span class="constant-syntax">0</span><span class="plain-syntax">; }</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><a href="4-ap.html#SP3" class="function-link"><span class="function-syntax">Atoms::is_opener</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">element</span><span class="plain-syntax">)) </span><span class="identifier-syntax">blevel</span><span class="plain-syntax">++;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><a href="4-ap.html#SP3" class="function-link"><span class="function-syntax">Atoms::is_closer</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">element</span><span class="plain-syntax">)) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">blevel</span><span class="plain-syntax">--;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">j</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="plain-syntax"><26; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">unavailable</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] > </span><span class="identifier-syntax">blevel</span><span class="plain-syntax">) </span><span class="identifier-syntax">unavailable</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] = -1;</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">j</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="function-syntax"><p-></span><span class="element-syntax">arity</span><span class="plain-syntax">; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">v</span><span class="plain-syntax"> = </span><a href="4-trm.html#SP8" class="function-link"><span class="function-syntax">Terms::variable_underlying</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">]));</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">v</span><span class="plain-syntax"> >= </span><span class="constant-syntax">26</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">WRITE_TO</span><span class="plain-syntax">(</span><span class="identifier-syntax">err</span><span class="plain-syntax">, </span><span class="string-syntax">"corrupted variable term"</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">valid</span><span class="plain-syntax"> = </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> } </span><span class="reserved-syntax">else</span><span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">v</span><span class="plain-syntax"> >= </span><span class="constant-syntax">0</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">unavailable</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] == -1) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">valid</span><span class="plain-syntax"> = </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">WRITE_TO</span><span class="plain-syntax">(</span><span class="identifier-syntax">err</span><span class="plain-syntax">, </span><span class="string-syntax">"%c unavailable"</span><span class="plain-syntax">, </span><span class="identifier-syntax">pcalc_vars</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">]);</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">element</span><span class="plain-syntax"> == </span><span class="constant-syntax">QUANTIFIER_ATOM</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] != </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">valid</span><span class="plain-syntax"> = </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">WRITE_TO</span><span class="plain-syntax">(</span><span class="identifier-syntax">err</span><span class="plain-syntax">, </span><span class="string-syntax">"%c used outside its binding"</span><span class="plain-syntax">, </span><span class="identifier-syntax">pcalc_vars</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">]);</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] = </span><span class="constant-syntax">BOUND_VST</span><span class="plain-syntax">; </span><span class="identifier-syntax">unavailable</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] = </span><span class="identifier-syntax">blevel</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> } </span><span class="reserved-syntax">else</span><span class="plain-syntax"> {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] == </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">) </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] = </span><span class="constant-syntax">FREE_VST</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">valid</span><span class="plain-syntax">;</span>
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<span class="plain-syntax">}</span>
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</pre>
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<p class="commentary firstcommentary"><a id="SP2" class="paragraph-anchor"></a><b>§2. </b>With just a little wrapping, this gives us the test of well-formedness.
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</p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::is_well_formed</span><button class="popup" onclick="togglePopup('usagePopup2')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup2">Usage of <span class="code-font"><span class="function-syntax">Binding::is_well_formed</span></span>:<br/><a href="4-bas.html#SP13">§13</a><br/>Type Check Propositions - <a href="4-tcp.html#SP4">§4</a><br/>Sentence Conversions - <a href="5-sc.html#SP2_14">§2.14</a>, <a href="5-sc.html#SP3_2">§3.2</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">text_stream</span><span class="plain-syntax"> *</span><span class="identifier-syntax">err</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[26];</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><a href="4-prp.html#SP10" class="function-link"><span class="function-syntax">Propositions::is_syntactically_valid</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">err</span><span class="plain-syntax">) == </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">;</span>
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|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><a href="4-bas.html#SP1" class="function-link"><span class="function-syntax">Binding::determine_status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">, </span><span class="identifier-syntax">err</span><span class="plain-syntax">);</span>
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<span class="plain-syntax">}</span>
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</pre>
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<p class="commentary firstcommentary"><a id="SP3" class="paragraph-anchor"></a><b>§3. </b>Occasionally we really do care only about one of the 26 variables:
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</p>
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<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::status</span><button class="popup" onclick="togglePopup('usagePopup3')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup3">Usage of <span class="code-font"><span class="function-syntax">Binding::status</span></span>:<br/><a href="4-bas.html#SP13_1">§13.1</a><br/>Sentence Conversions - <a href="5-sc.html#SP3_2">§3.2</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">v</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[26];</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">v</span><span class="plain-syntax"> == -1) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP1" class="function-link"><span class="function-syntax">Binding::determine_status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">];</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP4" class="paragraph-anchor"></a><b>§4. </b>To distinguish sentences from descriptions, the following can be informative:
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::number_free</span><button class="popup" onclick="togglePopup('usagePopup4')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup4">Usage of <span class="code-font"><span class="function-syntax">Binding::number_free</span></span>:<br/>Sentence Conversions - <a href="5-sc.html#SP2_14">§2.14</a>, <a href="5-sc.html#SP3_2">§3.2</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[26], </span><span class="identifier-syntax">j</span><span class="plain-syntax">, </span><span class="identifier-syntax">c</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP1" class="function-link"><span class="function-syntax">Binding::determine_status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">j</span><span class="plain-syntax">=0, </span><span class="identifier-syntax">c</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="plain-syntax"><26; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] == </span><span class="constant-syntax">FREE_VST</span><span class="plain-syntax">) </span><span class="identifier-syntax">c</span><span class="plain-syntax">++;</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">LOGIF</span><span class="plain-syntax">(</span><span class="identifier-syntax">PREDICATE_CALCULUS_WORKINGS</span><span class="plain-syntax">, </span><span class="string-syntax">"There %s %d free variable%s in $D\n"</span><span class="plain-syntax">,</span>
|
|
<span class="plain-syntax"> (</span><span class="identifier-syntax">c</span><span class="plain-syntax">==1)?</span><span class="string-syntax">"is"</span><span class="plain-syntax">:</span><span class="string-syntax">"are"</span><span class="plain-syntax">, </span><span class="identifier-syntax">c</span><span class="plain-syntax">, (</span><span class="identifier-syntax">c</span><span class="plain-syntax">==1)?</span><span class="string-syntax">""</span><span class="plain-syntax">:</span><span class="string-syntax">"s"</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">c</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP5" class="paragraph-anchor"></a><b>§5. </b>While this gives us a new variable which can safely be added to an existing
|
|
proposition:
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::find_unused</span><button class="popup" onclick="togglePopup('usagePopup5')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup5">Usage of <span class="code-font"><span class="function-syntax">Binding::find_unused</span></span>:<br/>Sentence Conversions - <a href="5-sc.html#SP3_7">§3.7</a><br/>Simplifications - <a href="5-smp.html#SP2_2">§2.2</a>, <a href="5-smp.html#SP3">§3</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[26], </span><span class="identifier-syntax">j</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP1" class="function-link"><span class="function-syntax">Binding::determine_status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">j</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="plain-syntax"><26; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] == </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">j</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="constant-syntax">25</span><span class="plain-syntax">; </span><span class="comment-syntax"> the best we can do: it avoids crashes, at least...</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP6" class="paragraph-anchor"></a><b>§6. </b>Another "vector operation" on variables: to renumber them throughout a
|
|
proposition according to a map array. If <span class="extract"><span class="extract-syntax">renumber_map[j]</span></span> is \(-1\), make
|
|
no change; otherwise each instance of variable \(j\) should be changed to
|
|
this new number.
|
|
</p>
|
|
|
|
<p class="commentary">In general, it is dangerous to renumber any variable to another which is
|
|
already used in the same proposition: that way we could accidentally change
|
|
"\(v\) is greater than \(w\)" into "\(w\) is greater than \(w\)", thus changing the
|
|
meaning.
|
|
</p>
|
|
|
|
<p class="commentary">Note that because <span class="extract"><span class="extract-syntax">QUANTIFIER_ATOM</span></span>s store the variable being quantified
|
|
as a term, the following changes quantification variables as well as
|
|
predicate terms, which is as it should be.
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="function-syntax">Binding::vars_map</span><button class="popup" onclick="togglePopup('usagePopup6')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup6">Usage of <span class="code-font"><span class="function-syntax">Binding::vars_map</span></span>:<br/><a href="4-bas.html#SP7">§7</a>, <a href="4-bas.html#SP8">§8</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> *</span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">, </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">preserving</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_VARIABLE</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">j</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="function-syntax"><p-></span><span class="element-syntax">arity</span><span class="plain-syntax">; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">pt</span><span class="plain-syntax"> = &(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">]);</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP6" class="function-link"><span class="function-syntax">Binding::term_map</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">, </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">preserving</span><span class="plain-syntax">) </span><a href="4-bas.html#SP6" class="function-link"><span class="function-syntax">Binding::term_map</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">preserving</span><span class="plain-syntax">, </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">prop</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
|
|
|
|
<span class="reserved-syntax">void</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::term_map</span><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">pt</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> *</span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">while</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">) </span><span class="identifier-syntax">pt</span><span class="plain-syntax">=&(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">-></span><span class="element-syntax">fn_of</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">nv</span><span class="plain-syntax"> = </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">variable</span><span class="plain-syntax">];</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">variable</span><span class="plain-syntax"> >= </span><span class="constant-syntax">0</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">nv</span><span class="plain-syntax"> >= </span><span class="constant-syntax">0</span><span class="plain-syntax">)) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">nv</span><span class="plain-syntax"> >= </span><span class="constant-syntax">26</span><span class="plain-syntax">) </span><span class="identifier-syntax">internal_error</span><span class="plain-syntax">(</span><span class="string-syntax">"malformed renumbering map"</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">variable</span><span class="plain-syntax"> = </span><span class="identifier-syntax">nv</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP7" class="paragraph-anchor"></a><b>§7. </b>The following takes any proposition and edits it so that the variables
|
|
used are the lowest-numbered ones; moreover, variables are introduced
|
|
in numerical order — that is, the first mentioned will be \(x\), then the
|
|
next introduced will be \(y\), and so on.
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="function-syntax">Binding::renumber</span><button class="popup" onclick="togglePopup('usagePopup7')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup7">Usage of <span class="code-font"><span class="function-syntax">Binding::renumber</span></span>:<br/><a href="4-bas.html#SP10">§10</a><br/>Sentence Conversions - <a href="5-sc.html#SP2_14">§2.14</a><br/>Simplifications - <a href="5-smp.html#SP13">§13</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">preserving</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_VARIABLE</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[26];</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">j</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="plain-syntax"><26; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] = -1;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">k</span><span class="plain-syntax"> = </span><span class="constant-syntax">0</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">j</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="function-syntax"><p-></span><span class="element-syntax">arity</span><span class="plain-syntax">; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">v</span><span class="plain-syntax"> = </span><a href="4-trm.html#SP8" class="function-link"><span class="function-syntax">Terms::variable_underlying</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">]));</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">v</span><span class="plain-syntax"> >= </span><span class="constant-syntax">0</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] == -1)) </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">] = </span><span class="identifier-syntax">k</span><span class="plain-syntax">++;</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><a href="4-bas.html#SP6" class="function-link"><span class="function-syntax">Binding::vars_map</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">, </span><span class="identifier-syntax">preserving</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP8" class="paragraph-anchor"></a><b>§8. </b>This more complicated routine renumbers bound variables in one proposition
|
|
in order to guarantee that none of them coincides with a variable used
|
|
in a second proposition. This is needed in order to take the conjunction of
|
|
two propositions, because "for all \(x\), \(x\) is a door" and "there exists \(x\)
|
|
such that \(x\) is a container" mean different things by \(x\); they can only
|
|
be combined in a single proposition if one of the \(x\) variables is changed
|
|
to, say, \(y\).
|
|
</p>
|
|
|
|
<p class="commentary">The surprising thing here is the asymmetry. Why do we only renumber to avoid
|
|
clashes with bound variables in <span class="extract"><span class="extract-syntax">prop</span></span> — why not free ones as well? The
|
|
answer is that we use a form of conjunction in Inform which assumes that a
|
|
free variable in \(\phi\) has the same meaning as it does in \(\psi\); thus in
|
|
conjoining "open" with "lockable" we assume that the same thing is meant
|
|
to be both open and lockable. If we renumbered to avoid clashes in free
|
|
variables, we would produce a proposition meaning that one unknown thing is
|
|
open, and another one lockable: that would have two free variables and be
|
|
much harder to interpret.
|
|
</p>
|
|
|
|
<p class="commentary">If we pass a <span class="extract"><span class="extract-syntax">query</span></span> parameter which is a valid variable number, the routine
|
|
returns its new identity when renumbered.
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::renumber_bound</span><button class="popup" onclick="togglePopup('usagePopup8')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup8">Usage of <span class="code-font"><span class="function-syntax">Binding::renumber_bound</span></span>:<br/>Propositions - <a href="4-prp.html#SP15">§15</a><br/>Sentence Conversions - <a href="5-sc.html#SP2_13">§2.13</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">not_to_overlap</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">query</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">prop_vstates</span><span class="plain-syntax">[26], </span><span class="identifier-syntax">nto_vstates</span><span class="plain-syntax">[26], </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[26];</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">j</span><span class="plain-syntax">, </span><span class="identifier-syntax">next_unused</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP1" class="function-link"><span class="function-syntax">Binding::determine_status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop_vstates</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP1" class="function-link"><span class="function-syntax">Binding::determine_status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">not_to_overlap</span><span class="plain-syntax">, </span><span class="identifier-syntax">nto_vstates</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">);</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">j</span><span class="plain-syntax">=0, </span><span class="identifier-syntax">next_unused</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">j</span><span class="plain-syntax"><26; </span><span class="identifier-syntax">j</span><span class="plain-syntax">++)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">prop_vstates</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] == </span><span class="constant-syntax">BOUND_VST</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">nto_vstates</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] != </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">)) {</span>
|
|
<span class="plain-syntax"> </span><span class="named-paragraph-container code-font"><a href="4-bas.html#SP8_1" class="named-paragraph-link"><span class="named-paragraph">Advance to the next variable not used in either proposition</span><span class="named-paragraph-number">8.1</span></a></span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] = </span><span class="identifier-syntax">next_unused</span><span class="plain-syntax">++;</span>
|
|
<span class="plain-syntax"> } </span><span class="reserved-syntax">else</span><span class="plain-syntax"> </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] = -1;</span>
|
|
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP6" class="function-link"><span class="function-syntax">Binding::vars_map</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">query</span><span class="plain-syntax"> == -1) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> -1;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">query</span><span class="plain-syntax">] == -1) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">query</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">renumber_map</span><span class="plain-syntax">[</span><span class="identifier-syntax">query</span><span class="plain-syntax">];</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP8_1" class="paragraph-anchor"></a><b>§8.1. </b>Again, we fall back on variable 25 if we run out. (This can only happen if
|
|
the conjunction of the two propositions had 26 variables.)
|
|
</p>
|
|
|
|
<p class="commentary"><span class="named-paragraph-container code-font"><span class="named-paragraph-defn">Advance to the next variable not used in either proposition</span><span class="named-paragraph-number">8.1</span></span><span class="comment-syntax"> =</span>
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">k</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">k</span><span class="plain-syntax">=</span><span class="identifier-syntax">next_unused</span><span class="plain-syntax">; (</span><span class="identifier-syntax">k</span><span class="plain-syntax"><26) &&</span>
|
|
<span class="plain-syntax"> (!((</span><span class="identifier-syntax">prop_vstates</span><span class="plain-syntax">[</span><span class="identifier-syntax">k</span><span class="plain-syntax">] == </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">nto_vstates</span><span class="plain-syntax">[</span><span class="identifier-syntax">k</span><span class="plain-syntax">] == </span><span class="constant-syntax">UNUSED_VST</span><span class="plain-syntax">))); </span><span class="identifier-syntax">k</span><span class="plain-syntax">++) ;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">k</span><span class="plain-syntax"> == </span><span class="constant-syntax">26</span><span class="plain-syntax">) </span><span class="identifier-syntax">next_unused</span><span class="plain-syntax"> = </span><span class="constant-syntax">25</span><span class="plain-syntax">; </span><span class="reserved-syntax">else</span><span class="plain-syntax"> </span><span class="identifier-syntax">next_unused</span><span class="plain-syntax"> = </span><span class="identifier-syntax">k</span><span class="plain-syntax">;</span>
|
|
</pre>
|
|
<ul class="endnotetexts"><li>This code is used in <a href="4-bas.html#SP8">§8</a>.</li></ul>
|
|
<p class="commentary firstcommentary"><a id="SP9" class="paragraph-anchor"></a><b>§9. Getting rid of free variables. </b>Propositions with free variables are vague, and Inforn tries to minimise its
|
|
use of them. Whole verb phrases such as "the tree is in the Courtyard" can in
|
|
general become propositions with no free variables, while descriptions such as
|
|
"open containers which are in lighted rooms" will become propositions in which
|
|
only variable 0, <span class="extract"><span class="extract-syntax">x</span></span>, is free.
|
|
</p>
|
|
|
|
<p class="commentary">Here we see two ways to remove a free variable from a proposition:
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="plain-syntax">'</span><span class="element-syntax">new unary even</span><span class="plain-syntax">':</span><span class="string-syntax"> ok</span>
|
|
<span class="plain-syntax">'</span><span class="element-syntax">set P to << Forall x IN< kind = number (x) IN>: (x == y) ^ Exists z: (y == z) >></span><span class="plain-syntax">':</span><span class="string-syntax"> p set to << ForAll x IN< kind=number(x) IN> : (x == y) ^ Exists z : (y == z) >></span>
|
|
<span class="plain-syntax">'</span><span class="element-syntax">variables in P</span><span class="plain-syntax">':</span><span class="string-syntax"> valid: x bound y free z bound</span>
|
|
<span class="plain-syntax">'</span><span class="element-syntax">binding of P</span><span class="plain-syntax">':</span><span class="string-syntax"> << Exists y : ForAll x IN< kind=number(x) IN> : (x == y) ^ Exists z : (y == z) >></span>
|
|
<span class="plain-syntax">'</span><span class="element-syntax">variables in binding of P</span><span class="plain-syntax">':</span><span class="string-syntax"> valid: x bound y bound z bound</span>
|
|
<span class="plain-syntax">'</span><span class="element-syntax">substitution of y = 777 in P</span><span class="plain-syntax">':</span><span class="string-syntax"> << ForAll x IN< kind=number(x) IN> : (x == '777') ^ Exists z : ('777' == z) >></span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP10" class="paragraph-anchor"></a><b>§10. </b>The first way is "binding". Suppose <span class="extract"><span class="extract-syntax">x</span></span> is free and we do not know its
|
|
value. We can at least put <span class="extract"><span class="extract-syntax">Exists x :</span></span> at the front of the proposition, thus
|
|
saying only "well, it's something". This does the equivalent of turning "open
|
|
containers which are in lighted rooms" into "an open container is in a lighted
|
|
room", by applying the existential quantifier to anything free.
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="function-syntax">Binding::bind_existential</span><button class="popup" onclick="togglePopup('usagePopup9')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup9">Usage of <span class="code-font"><span class="function-syntax">Binding::bind_existential</span></span>:<br/>Sentence Conversions - <a href="5-sc.html#SP2_3">§2.3</a>, <a href="5-sc.html#SP2_9">§2.9</a>, <a href="5-sc.html#SP3_7">§3.7</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">,</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">preserving</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[26], </span><span class="identifier-syntax">j</span><span class="plain-syntax">;</span>
|
|
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP7" class="function-link"><span class="function-syntax">Binding::renumber</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">preserving</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP1" class="function-link"><span class="function-syntax">Binding::determine_status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">var_states</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">);</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">j</span><span class="plain-syntax">=25; </span><span class="identifier-syntax">j</span><span class="plain-syntax">>=0; </span><span class="identifier-syntax">j</span><span class="plain-syntax">--)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">var_states</span><span class="plain-syntax">[</span><span class="identifier-syntax">j</span><span class="plain-syntax">] == </span><span class="constant-syntax">FREE_VST</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">prop</span><span class="plain-syntax"> = </span><a href="4-prp.html#SP18" class="function-link"><span class="function-syntax">Propositions::insert_atom</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">,</span>
|
|
<span class="plain-syntax"> </span><a href="4-ap.html#SP6" class="function-link"><span class="function-syntax">Atoms::QUANTIFIER_new</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">exists_quantifier</span><span class="plain-syntax">, </span><span class="identifier-syntax">j</span><span class="plain-syntax">, </span><span class="constant-syntax">0</span><span class="plain-syntax">));</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">prop</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP11" class="paragraph-anchor"></a><b>§11. </b>The second way is "substitution", for use when we do know the value of the
|
|
free variable we want to remove. We replace every mention of it with sone
|
|
other term: but as we shall see, this is trickier than it seems.
|
|
</p>
|
|
|
|
<p class="commentary">We begin with two utility routines to substitute into the variable "underneath"
|
|
a given term.
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::substitute_v_in_term</span><button class="popup" onclick="togglePopup('usagePopup10')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup10">Usage of <span class="code-font"><span class="function-syntax">Binding::substitute_v_in_term</span></span>:<br/><a href="4-bas.html#SP13">§13</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">pt</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">v</span><span class="plain-syntax">, </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">t</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">variable</span><span class="plain-syntax"> == </span><span class="identifier-syntax">v</span><span class="plain-syntax">) { *</span><span class="identifier-syntax">pt</span><span class="plain-syntax"> = *</span><span class="identifier-syntax">t</span><span class="plain-syntax">; </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">TRUE</span><span class="plain-syntax">; }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><a href="4-bas.html#SP11" class="function-link"><span class="function-syntax">Binding::substitute_v_in_term</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">-></span><span class="element-syntax">fn_of</span><span class="plain-syntax">), </span><span class="identifier-syntax">v</span><span class="plain-syntax">, </span><span class="identifier-syntax">t</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
|
|
|
|
<span class="reserved-syntax">void</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::substitute_nothing_in_term</span><button class="popup" onclick="togglePopup('usagePopup11')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup11">Usage of <span class="code-font"><span class="function-syntax">Binding::substitute_nothing_in_term</span></span>:<br/>Simplifications - <a href="5-smp.html#SP2_2">§2.2</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">pt</span><span class="plain-syntax">, </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">t</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax">#</span><span class="identifier-syntax">ifdef</span><span class="plain-syntax"> </span><span class="identifier-syntax">DETECT_NOTHING_CALCULUS_CALLBACK</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">constant</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">DETECT_NOTHING_CALCULUS_CALLBACK</span><span class="plain-syntax">(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">constant</span><span class="plain-syntax">))) { *</span><span class="identifier-syntax">pt</span><span class="plain-syntax"> = *</span><span class="identifier-syntax">t</span><span class="plain-syntax">; </span><span class="reserved-syntax">return</span><span class="plain-syntax">; }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">) </span><a href="4-bas.html#SP11" class="function-link"><span class="function-syntax">Binding::substitute_nothing_in_term</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">-></span><span class="element-syntax">fn_of</span><span class="plain-syntax">), </span><span class="identifier-syntax">t</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax">#</span><span class="identifier-syntax">endif</span>
|
|
<span class="plain-syntax">}</span>
|
|
|
|
<span class="reserved-syntax">void</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::substitute_term_in_term</span><button class="popup" onclick="togglePopup('usagePopup12')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup12">Usage of <span class="code-font"><span class="function-syntax">Binding::substitute_term_in_term</span></span>:<br/>Simplifications - <a href="5-smp.html#SP3">§3</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">pt</span><span class="plain-syntax">, </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">t</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">constant</span><span class="plain-syntax">) { *</span><span class="identifier-syntax">pt</span><span class="plain-syntax"> = *</span><span class="identifier-syntax">t</span><span class="plain-syntax">; </span><span class="reserved-syntax">return</span><span class="plain-syntax">; }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">) </span><a href="4-bas.html#SP11" class="function-link"><span class="function-syntax">Binding::substitute_term_in_term</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">-></span><span class="element-syntax">fn_of</span><span class="plain-syntax">), </span><span class="identifier-syntax">t</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP12" class="paragraph-anchor"></a><b>§12. </b>Now the main procedure. This is one of those deceptive problems where the
|
|
actual algorithm is obvious, but the circumstances when it can validly be
|
|
applied are less so.
|
|
</p>
|
|
|
|
<p class="commentary">The difficulty depends on the term \(T\) being substituted in for the variable
|
|
\(v\). In general every term is a chain of functions with, right at the end,
|
|
either a constant or a variable. If a constant is underneath, there is no
|
|
problem at all. But if there is a variable underneath \(T\) — a VUT, as we
|
|
say below — then it's possible that the substitution introduces circularities
|
|
which would make it invalid. If that happens, we run into this:
|
|
</p>
|
|
|
|
<pre class="definitions code-font"><span class="definition-keyword">define</span> <span class="identifier-syntax">DISALLOW</span><span class="plain-syntax">(</span><span class="identifier-syntax">msg</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">verify_only</span><span class="plain-syntax">) { *</span><span class="identifier-syntax">allowed</span><span class="plain-syntax"> = </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">; </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">prop</span><span class="plain-syntax">; }</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">internal_error</span><span class="plain-syntax">(</span><span class="identifier-syntax">msg</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP13" class="paragraph-anchor"></a><b>§13. </b>So the routine is intended to be called twice: once to ask if the situation
|
|
looks viable, and once to perform the substitution itself.
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="function-syntax">Binding::substitute_term</span><button class="popup" onclick="togglePopup('usagePopup13')"><span class="comment-syntax">?</span><span class="popuptext" id="usagePopup13">Usage of <span class="code-font"><span class="function-syntax">Binding::substitute_term</span></span>:<br/><a href="4-bas.html#SP15">§15</a><br/>Simplifications - <a href="5-smp.html#SP13">§13</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">v</span><span class="plain-syntax">, </span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> </span><span class="identifier-syntax">t</span><span class="plain-syntax">,</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">verify_only</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> *</span><span class="identifier-syntax">allowed</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> *</span><span class="identifier-syntax">changed</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_VARIABLE</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">);</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">verify_only</span><span class="plain-syntax">) *</span><span class="identifier-syntax">allowed</span><span class="plain-syntax"> = </span><span class="identifier-syntax">TRUE</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">v</span><span class="plain-syntax"><0) || (</span><span class="identifier-syntax">v</span><span class="plain-syntax">>=26)) </span><span class="identifier-syntax">DISALLOW</span><span class="plain-syntax">(</span><span class="string-syntax">"variable substitution out of range"</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><a href="4-bas.html#SP2" class="function-link"><span class="function-syntax">Binding::is_well_formed</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">) == </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">DISALLOW</span><span class="plain-syntax">(</span><span class="string-syntax">"substituting into malformed prop"</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="named-paragraph-container code-font"><a href="4-bas.html#SP13_1" class="named-paragraph-link"><span class="named-paragraph">Make sure the substitution would not fail because of a circularity</span><span class="named-paragraph-number">13.1</span></a></span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">verify_only</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">prop</span><span class="plain-syntax">;</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">LOGIF</span><span class="plain-syntax">(</span><span class="identifier-syntax">PREDICATE_CALCULUS_WORKINGS</span><span class="plain-syntax">,</span>
|
|
<span class="plain-syntax"> </span><span class="string-syntax">"Substituting %c = $0 in: $D\n"</span><span class="plain-syntax">, </span><span class="identifier-syntax">pcalc_vars</span><span class="plain-syntax">[</span><span class="identifier-syntax">v</span><span class="plain-syntax">], &</span><span class="identifier-syntax">t</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">);</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">i</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">i</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">i</span><span class="function-syntax"><p-></span><span class="element-syntax">arity</span><span class="plain-syntax">; </span><span class="identifier-syntax">i</span><span class="plain-syntax">++)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><a href="4-bas.html#SP11" class="function-link"><span class="function-syntax">Binding::substitute_v_in_term</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[</span><span class="identifier-syntax">i</span><span class="plain-syntax">]), </span><span class="identifier-syntax">v</span><span class="plain-syntax">, &</span><span class="identifier-syntax">t</span><span class="plain-syntax">))</span>
|
|
<span class="plain-syntax"> *</span><span class="identifier-syntax">changed</span><span class="plain-syntax"> = </span><span class="identifier-syntax">TRUE</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> }</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><a href="4-bas.html#SP2" class="function-link"><span class="function-syntax">Binding::is_well_formed</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">) == </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">internal_error</span><span class="plain-syntax">(</span><span class="string-syntax">"substitution made malformed prop"</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">prop</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP13_1" class="paragraph-anchor"></a><b>§13.1. </b>The problem we might find, then, is that setting \(v=T\) will be circular
|
|
because \(T\) itself depends on \(v\). There are two ways this can happen: first,
|
|
\(T\) might be directly a function of \(v\) itself, i.e., the VUT might be \(v\);
|
|
second, \(T\) might be a function of some variable \(w\) which, by being quantified
|
|
after \(v\), is allowed to depend on it, in some way that we can't determine.
|
|
</p>
|
|
|
|
<p class="commentary">The general rule, then, is that \(T\) can contain only constants or variables
|
|
which are free within and after the scope of \(v\). (If \(w\) is bound
|
|
outside the scope of \(v\) but after it, this means \(w\) didn't exist at the
|
|
time that \(v\) did, and the attempted substitution would produce a proposition
|
|
which isn't well-formed — \(w\) would occur before its quantifier.) We can
|
|
check this condition pretty easily, it turns out:
|
|
</p>
|
|
|
|
<p class="commentary"><span class="named-paragraph-container code-font"><span class="named-paragraph-defn">Make sure the substitution would not fail because of a circularity</span><span class="named-paragraph-number">13.1</span></span><span class="comment-syntax"> =</span>
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">verify_only</span><span class="plain-syntax"> == </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">) && (</span><a href="4-bas.html#SP3" class="function-link"><span class="function-syntax">Binding::status</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">v</span><span class="plain-syntax">) == </span><span class="constant-syntax">BOUND_VST</span><span class="plain-syntax">))</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">DISALLOW</span><span class="plain-syntax">(</span><span class="string-syntax">"substituting bound variable"</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">vut</span><span class="plain-syntax"> = </span><a href="4-trm.html#SP8" class="function-link"><span class="function-syntax">Terms::variable_underlying</span></a><span class="plain-syntax">(&</span><span class="identifier-syntax">t</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">vut</span><span class="plain-syntax"> >= </span><span class="constant-syntax">0</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">v_has_been_seen</span><span class="plain-syntax"> = </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">v</span><span class="plain-syntax"> == </span><span class="identifier-syntax">vut</span><span class="plain-syntax">) </span><span class="identifier-syntax">DISALLOW</span><span class="plain-syntax">(</span><span class="string-syntax">"resubstituting same variable"</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">v_has_been_seen</span><span class="plain-syntax"> == </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">i</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">i</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">i</span><span class="function-syntax"><p-></span><span class="element-syntax">arity</span><span class="plain-syntax">; </span><span class="identifier-syntax">i</span><span class="plain-syntax">++)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><a href="4-trm.html#SP8" class="function-link"><span class="function-syntax">Terms::variable_underlying</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[</span><span class="identifier-syntax">i</span><span class="plain-syntax">])) == </span><span class="identifier-syntax">v</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">v_has_been_seen</span><span class="plain-syntax"> = </span><span class="identifier-syntax">TRUE</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">element</span><span class="plain-syntax"> == </span><span class="constant-syntax">QUANTIFIER_ATOM</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[0].</span><span class="element-syntax">variable</span><span class="plain-syntax"> == </span><span class="identifier-syntax">vut</span><span class="plain-syntax">) &&</span>
|
|
<span class="plain-syntax"> (</span><span class="identifier-syntax">v_has_been_seen</span><span class="plain-syntax">))</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">DISALLOW</span><span class="plain-syntax">(</span><span class="string-syntax">"substituted value may be circular"</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> }</span>
|
|
</pre>
|
|
<ul class="endnotetexts"><li>This code is used in <a href="4-bas.html#SP13">§13</a>.</li></ul>
|
|
<p class="commentary firstcommentary"><a id="SP14" class="paragraph-anchor"></a><b>§14. A footnote on variable 0. </b>Because of the special status of \(x\) (variable 0) — the one allowed to be
|
|
free in SN-propositions — we sometimes need to know about it. The range
|
|
of a bound variable can be found by looking at its quantifier, but a free
|
|
variable can remain ambiguous. The presence of a "kind" atom will explicitly
|
|
solve the problem for us; if we don't find one, though, we will simply have
|
|
to assume that the set of objects is the domain of \(x\). (We return <span class="extract"><span class="extract-syntax">NULL</span></span>
|
|
here, but that's the assumption which the caller will have to make.)
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="identifier-syntax">kind</span><span class="plain-syntax"> *</span><span class="function-syntax">Binding::kind_of_variable_0</span><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_VARIABLE</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><a href="2-kp.html#SP3" class="function-link"><span class="function-syntax">KindPredicates::is_kind_atom</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">)) && (</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[0].</span><span class="element-syntax">variable</span><span class="plain-syntax"> == </span><span class="constant-syntax">0</span><span class="plain-syntax">))</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><a href="2-kp.html#SP3" class="function-link"><span class="function-syntax">KindPredicates::get_kind</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP15" class="paragraph-anchor"></a><b>§15. </b>And a quick way to substitute it:
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="function-syntax">Binding::substitute_var_0_in</span><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> *</span><span class="identifier-syntax">spec</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">bogus</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><a href="4-bas.html#SP13" class="function-link"><span class="function-syntax">Binding::substitute_term</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="constant-syntax">0</span><span class="plain-syntax">, </span><a href="4-trm.html#SP4" class="function-link"><span class="function-syntax">Terms::new_constant</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">spec</span><span class="plain-syntax">), </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">, </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">, &</span><span class="identifier-syntax">bogus</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP16" class="paragraph-anchor"></a><b>§16. </b>If we are willing to work a little harder:
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="identifier-syntax">kind</span><span class="plain-syntax"> *</span><span class="function-syntax">Binding::infer_kind_of_variable_0</span><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_VARIABLE</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">element</span><span class="plain-syntax"> == </span><span class="constant-syntax">PREDICATE_ATOM</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">arity</span><span class="plain-syntax"> == </span><span class="constant-syntax">1</span><span class="plain-syntax">) && (</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[0].</span><span class="element-syntax">variable</span><span class="plain-syntax"> == </span><span class="constant-syntax">0</span><span class="plain-syntax">)) {</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">unary_predicate</span><span class="plain-syntax"> *</span><span class="identifier-syntax">up</span><span class="plain-syntax"> = </span><span class="identifier-syntax">RETRIEVE_POINTER_unary_predicate</span><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">predicate</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">kind</span><span class="plain-syntax"> *</span><span class="identifier-syntax">K</span><span class="plain-syntax"> = </span><a href="2-upf.html#SP8" class="function-link"><span class="function-syntax">UnaryPredicateFamilies::infer_kind</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">up</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">K</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">K</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
|
|
</pre>
|
|
<p class="commentary firstcommentary"><a id="SP17" class="paragraph-anchor"></a><b>§17. Detect locals. </b>Properly speaking, this has nothing to do with variables,
|
|
but it solves a similar problem.
|
|
</p>
|
|
|
|
<p class="commentary">Here we search a proposition to look for any term involving a local variable.
|
|
This is used to verify past tense propositions, which cannot rely on local
|
|
values because their contents may have been wiped and reused many times
|
|
since the time with which the proposition is concerned.
|
|
</p>
|
|
|
|
<pre class="displayed-code all-displayed-code code-font">
|
|
<span class="plain-syntax">#</span><span class="identifier-syntax">ifdef</span><span class="plain-syntax"> </span><span class="identifier-syntax">CORE_MODULE</span>
|
|
<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::detect_locals</span><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_prop</span><span class="plain-syntax"> *</span><span class="identifier-syntax">prop</span><span class="plain-syntax">, </span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> **</span><span class="identifier-syntax">example</span><span class="plain-syntax">) {</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_VARIABLE</span><span class="plain-syntax">(</span><span class="identifier-syntax">pl</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">i</span><span class="plain-syntax">, </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax"> = </span><span class="constant-syntax">0</span><span class="plain-syntax">;</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">TRAVERSE_PROPOSITION</span><span class="plain-syntax">(</span><span class="identifier-syntax">pl</span><span class="plain-syntax">, </span><span class="identifier-syntax">prop</span><span class="plain-syntax">)</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">i</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">i</span><span class="function-syntax"><pl-></span><span class="element-syntax">arity</span><span class="plain-syntax">; </span><span class="identifier-syntax">i</span><span class="plain-syntax">++)</span>
|
|
<span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax"> =</span>
|
|
<span class="plain-syntax"> </span><a href="4-bas.html#SP17" class="function-link"><span class="function-syntax">Binding::detect_local_in_term</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">pl</span><span class="plain-syntax">-></span><span class="element-syntax">terms</span><span class="plain-syntax">[</span><span class="identifier-syntax">i</span><span class="plain-syntax">]), </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">, </span><span class="identifier-syntax">example</span><span class="plain-syntax">);</span>
|
|
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::detect_local_in_term</span><span class="plain-syntax">(</span><span class="reserved-syntax">pcalc_term</span><span class="plain-syntax"> *</span><span class="identifier-syntax">pt</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">, </span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> **</span><span class="identifier-syntax">example</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">)</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax"> += </span><a href="4-bas.html#SP17" class="function-link"><span class="function-syntax">Binding::detect_local_in_term</span></a><span class="plain-syntax">(&(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">function</span><span class="plain-syntax">-></span><span class="element-syntax">fn_of</span><span class="plain-syntax">), </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">, </span><span class="identifier-syntax">example</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">constant</span><span class="plain-syntax">)</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax"> += </span><a href="4-bas.html#SP17" class="function-link"><span class="function-syntax">Binding::detect_local_in_spec</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">pt</span><span class="plain-syntax">-></span><span class="element-syntax">constant</span><span class="plain-syntax">, </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">, </span><span class="identifier-syntax">example</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">;</span>
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<span class="plain-syntax">}</span>
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<span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="function-syntax">Binding::detect_local_in_spec</span><span class="plain-syntax">(</span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> *</span><span class="identifier-syntax">spec</span><span class="plain-syntax">, </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">, </span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> **</span><span class="identifier-syntax">example</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">spec</span><span class="plain-syntax"> == </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">Lvalues::get_storage_form</span><span class="plain-syntax">(</span><span class="identifier-syntax">spec</span><span class="plain-syntax">) == </span><span class="identifier-syntax">LOCAL_VARIABLE_NT</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">example</span><span class="plain-syntax">) && (*</span><span class="identifier-syntax">example</span><span class="plain-syntax"> == </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">)) *</span><span class="identifier-syntax">example</span><span class="plain-syntax"> = </span><span class="identifier-syntax">spec</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> ++</span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">Lvalues::get_storage_form</span><span class="plain-syntax">(</span><span class="identifier-syntax">spec</span><span class="plain-syntax">) == </span><span class="identifier-syntax">NONLOCAL_VARIABLE_NT</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">nonlocal_variable</span><span class="plain-syntax"> *</span><span class="identifier-syntax">nlv</span><span class="plain-syntax"> = </span><span class="identifier-syntax">Node::get_constant_nonlocal_variable</span><span class="plain-syntax">(</span><span class="identifier-syntax">spec</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">NonlocalVariables::is_global</span><span class="plain-syntax">(</span><span class="identifier-syntax">nlv</span><span class="plain-syntax">) == </span><span class="identifier-syntax">FALSE</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> ((</span><span class="identifier-syntax">example</span><span class="plain-syntax">) && (*</span><span class="identifier-syntax">example</span><span class="plain-syntax"> == </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">)) *</span><span class="identifier-syntax">example</span><span class="plain-syntax"> = </span><span class="identifier-syntax">spec</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> ++</span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> }</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">Specifications::is_phrasal</span><span class="plain-syntax">(</span><span class="identifier-syntax">spec</span><span class="plain-syntax">)) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> *</span><span class="identifier-syntax">inv</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">LOOP_THROUGH_INVOCATION_LIST</span><span class="plain-syntax">(</span><span class="identifier-syntax">inv</span><span class="plain-syntax">, </span><span class="identifier-syntax">spec</span><span class="plain-syntax">-></span><span class="identifier-syntax">down</span><span class="plain-syntax">-></span><span class="identifier-syntax">down</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">tc</span><span class="plain-syntax"> = </span><span class="identifier-syntax">Invocations::get_no_tokens</span><span class="plain-syntax">(</span><span class="identifier-syntax">inv</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="reserved-syntax">int</span><span class="plain-syntax"> </span><span class="identifier-syntax">i</span><span class="plain-syntax">=0; </span><span class="identifier-syntax">i</span><span class="plain-syntax"><</span><span class="identifier-syntax">tc</span><span class="plain-syntax">; </span><span class="identifier-syntax">i</span><span class="plain-syntax">++) {</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> *</span><span class="identifier-syntax">param</span><span class="plain-syntax"> = </span><span class="identifier-syntax">Invocations::get_token_as_parsed</span><span class="plain-syntax">(</span><span class="identifier-syntax">inv</span><span class="plain-syntax">, </span><span class="identifier-syntax">i</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax"> +=</span>
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<span class="plain-syntax"> </span><a href="4-bas.html#SP17" class="function-link"><span class="function-syntax">Binding::detect_local_in_spec</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">param</span><span class="plain-syntax">, </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">, </span><span class="identifier-syntax">example</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> }</span>
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|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> }</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">for</span><span class="plain-syntax"> (</span><span class="identifier-syntax">parse_node</span><span class="plain-syntax"> *</span><span class="identifier-syntax">p</span><span class="plain-syntax"> = </span><span class="identifier-syntax">spec</span><span class="plain-syntax">-></span><span class="identifier-syntax">down</span><span class="plain-syntax">; </span><span class="identifier-syntax">p</span><span class="plain-syntax">; </span><span class="identifier-syntax">p</span><span class="plain-syntax"> = </span><span class="identifier-syntax">p</span><span class="plain-syntax">-></span><span class="element-syntax">next</span><span class="plain-syntax">)</span>
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|
<span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax"> +=</span>
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|
<span class="plain-syntax"> </span><a href="4-bas.html#SP17" class="function-link"><span class="function-syntax">Binding::detect_local_in_spec</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">p</span><span class="plain-syntax">, </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">, </span><span class="identifier-syntax">example</span><span class="plain-syntax">);</span>
|
|
<span class="plain-syntax"> </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">locals_count</span><span class="plain-syntax">;</span>
|
|
<span class="plain-syntax">}</span>
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<span class="plain-syntax">#</span><span class="identifier-syntax">endif</span>
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</pre>
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