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<!--Weave of 'Binary Predicate Term Details' generated by Inweb-->
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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#3">Chapter 3: Binary Predicates</a></li><li><b>Binary Predicate Term Details</b></li></ul></div>
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<p class="purpose">To keep track of requirements on the terms for a binary predicate.</p>
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<p class="commentary firstcommentary"><a id="SP1" class="paragraph-anchor"></a><b>§1. </b>Different BPs apply to different sorts of terms: for instance, the
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numerical less-than comparison applies to numbers, whereas containment
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applies to things. The two terms need not have the same domain: the
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"wearing" relation, as seen in
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</p>
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<blockquote>
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<p>Harry Smythe wears the tweed waistcoat.</p>
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</blockquote>
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<p class="commentary">is a binary predicate \(W(x_0, x_1)\) such that \(x_0\) ranges across people
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and \(x_1\) ranges across things.
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</p>
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<p class="commentary">It's therefore helpful to record requirements on any given term of a BP,
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and we do that with the following structure.<sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup>
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</p>
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<ul class="footnotetexts"><li class="footnote" id="fn:1"><p class="inwebfootnote"><sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> Unary predicates do not use this structure, because it is slightly more
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efficient to roughly duplicate this arrangement than to use it directly. But
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the ideas are exactly the same.
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<a href="#fnref:1" title="return to text"> ↩</a></p></li></ul>
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<pre class="definitions code-font"><span class="definition-keyword">default</span> <span class="constant-syntax">TERM_DOMAIN_CALCULUS_TYPE</span><span class="plain-syntax"> </span><span class="reserved-syntax">struct</span><span class="plain-syntax"> </span><span class="identifier-syntax">kind</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">typedef</span><span class="plain-syntax"> </span><span class="reserved-syntax">struct</span><span class="plain-syntax"> </span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">struct</span><span class="plain-syntax"> </span><span class="identifier-syntax">wording</span><span class="plain-syntax"> </span><span class="identifier-syntax">called_name</span><span class="plain-syntax">; </span><span class="comment-syntax"> "(called...)" name, if any exists</span>
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<span class="plain-syntax"> </span><span class="constant-syntax">TERM_DOMAIN_CALCULUS_TYPE</span><span class="plain-syntax"> *</span><span class="identifier-syntax">implies_infs</span><span class="plain-syntax">; </span><span class="comment-syntax"> the domain of values allowed</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">struct</span><span class="plain-syntax"> </span><span class="identifier-syntax">kind</span><span class="plain-syntax"> *</span><span class="identifier-syntax">implies_kind</span><span class="plain-syntax">; </span><span class="comment-syntax"> the kind of these values</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">struct</span><span class="plain-syntax"> </span><span class="reserved-syntax">i6_schema</span><span class="plain-syntax"> *</span><span class="identifier-syntax">function_of_other</span><span class="plain-syntax">; </span><span class="comment-syntax"> where one term can be deduced from the other</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">char</span><span class="plain-syntax"> *</span><span class="identifier-syntax">index_term_as</span><span class="plain-syntax">; </span><span class="comment-syntax"> usually null, but if not, used in Phrasebook index</span>
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<span class="plain-syntax">} </span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax">;</span>
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</pre>
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<ul class="endnotetexts"><li>The structure bp_term_details is accessed in 3/bp and here.</li></ul>
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<p class="commentary firstcommentary"><a id="SP2" class="paragraph-anchor"></a><b>§2. </b></p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> </span><span class="function-syntax">BPTerms::new</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">BPTerms::new</span></span>:<br/><a href="3-bptd.html#SP3">§3</a><br/>Binary Predicates - <a href="3-bp.html#SP9">§9</a><br/>The Equality Relation - <a href="3-ter.html#SP3">§3</a></span></button><span class="plain-syntax">(</span><span class="constant-syntax">TERM_DOMAIN_CALCULUS_TYPE</span><span class="plain-syntax"> *</span><span class="identifier-syntax">infs</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">called_name</span><span class="plain-syntax"> = </span><span class="identifier-syntax">EMPTY_WORDING</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">function_of_other</span><span class="plain-syntax"> = </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">implies_infs</span><span class="plain-syntax"> = </span><span class="identifier-syntax">infs</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">implies_kind</span><span class="plain-syntax"> = </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">index_term_as</span><span class="plain-syntax"> = </span><span class="identifier-syntax">NULL</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">bptd</span><span class="plain-syntax">;</span>
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<span class="plain-syntax">}</span>
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<span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> </span><span class="function-syntax">BPTerms::new_kind</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>
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<span class="plain-syntax"> </span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">called_name</span><span class="plain-syntax"> = </span><span class="identifier-syntax">EMPTY_WORDING</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">function_of_other</span><span class="plain-syntax"> = </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">implies_infs</span><span class="plain-syntax"> = </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">implies_kind</span><span class="plain-syntax"> = </span><span class="identifier-syntax">K</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">index_term_as</span><span class="plain-syntax"> = </span><span class="identifier-syntax">NULL</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">bptd</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>And there is also a fuller version, including the inessentials:
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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">bp_term_details</span><span class="plain-syntax"> </span><span class="function-syntax">BPTerms::new_full</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">BPTerms::new_full</span></span>:<br/>The Equality Relation - <a href="3-ter.html#SP3">§3</a></span></button><span class="plain-syntax">(</span><span class="constant-syntax">TERM_DOMAIN_CALCULUS_TYPE</span><span class="plain-syntax"> *</span><span class="identifier-syntax">infs</span><span class="plain-syntax">,</span>
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<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><span class="identifier-syntax">wording</span><span class="plain-syntax"> </span><span class="identifier-syntax">CW</span><span class="plain-syntax">, </span><span class="reserved-syntax">i6_schema</span><span class="plain-syntax"> *</span><span class="identifier-syntax">f</span><span class="plain-syntax">) {</span>
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<span class="plain-syntax"> </span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax"> = </span><a href="3-bptd.html#SP2" class="function-link"><span class="function-syntax">BPTerms::new</span></a><span class="plain-syntax">(</span><span class="identifier-syntax">infs</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">implies_kind</span><span class="plain-syntax"> = </span><span class="identifier-syntax">K</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">called_name</span><span class="plain-syntax"> = </span><span class="identifier-syntax">CW</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">.</span><span class="element-syntax">function_of_other</span><span class="plain-syntax"> = </span><span class="identifier-syntax">f</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">bptd</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="SP4" class="paragraph-anchor"></a><b>§4. </b>In a few cases BPs need to be created before the relevant domains are known,
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so that we must fill them in later, using the following:
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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">void</span><span class="plain-syntax"> </span><span class="function-syntax">BPTerms::set_domain</span><span class="plain-syntax">(</span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> *</span><span class="identifier-syntax">bptd</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>
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<span class="plain-syntax"> </span><span class="reserved-syntax">if</span><span class="plain-syntax"> (</span><span class="identifier-syntax">bptd</span><span class="plain-syntax"> == </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">) </span><span class="identifier-syntax">internal_error</span><span class="plain-syntax">(</span><span class="string-syntax">"no BPTD"</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">-></span><span class="identifier-syntax">implies_kind</span><span class="plain-syntax"> = </span><span class="identifier-syntax">K</span><span class="plain-syntax">;</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">-></span><span class="identifier-syntax">implies_infs</span><span class="plain-syntax"> = </span><span class="identifier-syntax">TERM_DOMAIN_FROM_KIND_FUNCTION</span><span class="plain-syntax">(</span><span class="identifier-syntax">K</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="SP5" class="paragraph-anchor"></a><b>§5. </b>Some BPs are such that \(B(x, y)\) can be true for more or less any
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combination of \(x\) and \(y\). Those can take a lot of storage and it is
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difficult to perform any reasoning about them, because knowing that
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\(B(x, y)\) is true doesn't give you any information about \(B(x, z)\).
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For instance, the BP created by
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</p>
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<blockquote>
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<p>Suspicion relates various people to various people.</p>
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</blockquote>
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<p class="commentary">is stored at run-time in a bitmap of \(P^2\) bits, where \(P\) is the number
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of people, and searching it ("if anyone suspects Harry") requires
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exhaustive loops, which incur some speed overhead as well.
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</p>
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<p class="commentary">But other BPs have special properties restricting the circumstances in
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which they are true, and in those cases we want to capitalise on that.
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"Contains" is an example of this. A single thing \(y\) can be (directly)
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inside only one other thing \(x\) at a time, so that if we know \(C(x, y)\)
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and \(C(w, y)\) then we can deduce that \(x=w\). We write this common value
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as \(f_0(y)\), the only possible value for term 0 given that term 1 is \(y\).
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Another way to say this is that the only possible pairs making \(C\) true
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have the form \(C(f_0(y), y)\).
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</p>
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<p class="commentary">And similarly for term 1. If we write \(T\) for the "on top of" relation
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then it turns out that there is a function \(f_1\) such that the only cases
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where \(T\) is true have the form \(T(x, f_1(x))\). Here \(f_1(x)\) is the thing
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which directly supports \(x\).
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</p>
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<p class="commentary">Containment has an \(f_0\) but not an \(f_1\) function; "on top of" has an
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\(f_1\) but not an \(f_0\). Many BPs (like "suspicion" above) have neither.
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</p>
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<p class="commentary">Note that if \(B\) does have an \(f_0\) function then its reversal \(R\) has an
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identical \(f_1\) function, and vice versa.
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</p>
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<p class="commentary firstcommentary"><a id="SP6" class="paragraph-anchor"></a><b>§6. </b>We never in fact need to calculate the value of \(f_0(y)\) from \(y\) during
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compilation — only at run-time. So we store the function \(f_0(y)\) in an
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<a href="3-cs.html#SP2" class="internal">i6_schema</a> for the necessary run-time code. For example, this might be
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the schema <span class="extract"><span class="extract-syntax">ContainerOf(*1)</span></span>, which would code-generate to a function call.
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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">void</span><span class="plain-syntax"> </span><span class="function-syntax">BPTerms::set_function</span><span class="plain-syntax">(</span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> *</span><span class="identifier-syntax">bptd</span><span class="plain-syntax">, </span><span class="reserved-syntax">i6_schema</span><span class="plain-syntax"> *</span><span class="identifier-syntax">f</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">bptd</span><span class="plain-syntax"> == </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">) </span><span class="identifier-syntax">internal_error</span><span class="plain-syntax">(</span><span class="string-syntax">"no BPTD"</span><span class="plain-syntax">);</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">-></span><span class="identifier-syntax">function_of_other</span><span class="plain-syntax"> = </span><span class="identifier-syntax">f</span><span class="plain-syntax">;</span>
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<span class="plain-syntax">}</span>
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<span class="reserved-syntax">i6_schema</span><span class="plain-syntax"> *</span><span class="function-syntax">BPTerms::get_function</span><span class="plain-syntax">(</span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> *</span><span class="identifier-syntax">bptd</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">bptd</span><span class="plain-syntax"> == </span><span class="identifier-syntax">NULL</span><span class="plain-syntax">) </span><span class="identifier-syntax">internal_error</span><span class="plain-syntax">(</span><span class="string-syntax">"no BPTD"</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">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">function_of_other</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="SP7" class="paragraph-anchor"></a><b>§7. </b>The kind of a term is:
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</p>
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<pre class="displayed-code all-displayed-code code-font">
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<span class="identifier-syntax">kind</span><span class="plain-syntax"> *</span><span class="function-syntax">BPTerms::kind</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">BPTerms::kind</span></span>:<br/>Binary Predicates - <a href="3-bp.html#SP7">§7</a></span></button><span class="plain-syntax">(</span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> *</span><span class="identifier-syntax">bptd</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">bptd</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">NULL</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">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">implies_kind</span><span class="plain-syntax">) </span><span class="reserved-syntax">return</span><span class="plain-syntax"> </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">implies_kind</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">TERM_DOMAIN_TO_KIND_FUNCTION</span><span class="plain-syntax">(</span><span class="identifier-syntax">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">implies_infs</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="SP8" class="paragraph-anchor"></a><b>§8. </b>The table of relations in the index uses the textual name, so:
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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">void</span><span class="plain-syntax"> </span><span class="function-syntax">BPTerms::index</span><span class="plain-syntax">(</span><span class="identifier-syntax">OUTPUT_STREAM</span><span class="plain-syntax">, </span><span class="reserved-syntax">bp_term_details</span><span class="plain-syntax"> *</span><span class="identifier-syntax">bptd</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">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">index_term_as</span><span class="plain-syntax">) { </span><span class="identifier-syntax">WRITE</span><span class="plain-syntax">(</span><span class="string-syntax">"%s"</span><span class="plain-syntax">, </span><span class="identifier-syntax">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">index_term_as</span><span class="plain-syntax">); </span><span class="reserved-syntax">return</span><span class="plain-syntax">; }</span>
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<span class="plain-syntax"> </span><span class="identifier-syntax">wording</span><span class="plain-syntax"> </span><span class="identifier-syntax">W</span><span class="plain-syntax"> = </span><span class="identifier-syntax">EMPTY_WORDING</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">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">implies_infs</span><span class="plain-syntax">) </span><span class="identifier-syntax">W</span><span class="plain-syntax"> = </span><span class="identifier-syntax">TERM_DOMAIN_WORDING_FUNCTION</span><span class="plain-syntax">(</span><span class="identifier-syntax">bptd</span><span class="plain-syntax">-></span><span class="element-syntax">implies_infs</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">Wordings::nonempty</span><span class="plain-syntax">(</span><span class="identifier-syntax">W</span><span class="plain-syntax">)) </span><span class="identifier-syntax">WRITE</span><span class="plain-syntax">(</span><span class="string-syntax">"%W"</span><span class="plain-syntax">, </span><span class="identifier-syntax">W</span><span class="plain-syntax">); </span><span class="reserved-syntax">else</span><span class="plain-syntax"> </span><span class="identifier-syntax">WRITE</span><span class="plain-syntax">(</span><span class="string-syntax">"--"</span><span class="plain-syntax">);</span>
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<span class="plain-syntax">}</span>
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</pre>
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