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<head>
<link href="../favicon.ico" rel="shortcut icon" type="image/x-icon"/>
<link href="../style.css" rel="stylesheet"/>
<title>Specification: BQN inferred properties</title>
</head>
<div class="nav"><a href="https://github.com/mlochbaum/BQN">BQN</a> / <a href="../index.html">main</a> / <a href="index.html">spec</a></div>
<h1 id="specification-bqn-inferred-properties">Specification: BQN inferred properties</h1>
<p>BQN includes some simple deductive capabilities: detecting the type of empty array elements, and the Undo (<code><span class='Modifier'>⁼</span></code>) and Under (<code><span class='Modifier2'>⌾</span></code>) modifiers. These tasks are a kind of proof-based or constraint programming, and can never be solved completely (some instances will be undecidable) but can be solved in more instances by ever-more sophisticated algorithms. To allow implementers to develop more advanced implementations while offering some stability and portability to programmers, two kinds of specification are given here. First, constraints are given on the behavior of inferred properties. These are not exact and require some judgment on the part of the implementer. Second, behavior for common or useful cases is specified more precisely. Non-normative suggestions are also given as a reference for implementers.</p>
<p>For the specified cases, the given functions and modifiers refer to those particular representations. It is not necessary to detect equivalent representations, for example to reduce <code><span class='Paren'>(</span><span class='Function'>+-×</span><span class='Paren'>)</span><span class='Modifier'>⁼</span></code> to <code><span class='Function'>∨</span><span class='Modifier'>⁼</span></code>. However, it is necessary to identify computed functions and modifiers: for example <code><span class='Function'>F</span><span class='Modifier'>⁼</span></code> when the value of <code><span class='Function'>F</span></code> in the expression is <code><span class='Function'>∨</span></code>, or <code><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>⊑∧</span><span class='Ligature'>‿</span><span class='Function'>∨</span><span class='Paren'>)</span><span class='Modifier'>⁼</span></code>.</p>
<h2 id="undo">Undo</h2>
<p>The Undo modifier <code><span class='Modifier'>⁼</span></code>, given an operand <code><span class='Function'>𝔽</span></code> and argument <code><span class='Value'>𝕩</span></code>, and possibly a left argument <code><span class='Value'>𝕨</span></code>, finds a value <code><span class='Value'>y</span></code> such that <code><span class='Value'>𝕩</span><span class='Function'>≡</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Value'>y</span></code>, that is, an element of the pre-image of <code><span class='Value'>𝕩</span></code> under <code><span class='Function'>𝔽</span></code> or <code><span class='Value'>𝕨</span><span class='Function'>𝔽⊢</span></code>. Thus it satisfies the constraint <code><span class='Value'>𝕩</span> <span class='Function'>≡</span> <span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code> (<code><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Function'>⊢</span></code> is a <em>right inverse</em> of <code><span class='Value'>𝕨</span><span class='Function'>𝔽⊢</span></code>) provided <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span></code> and <code><span class='Function'>𝔽</span></code> both complete without error. <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span></code> should of course give an error if no inverse element exists, and can also fail if no inverse can be found. It is also preferred for <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span></code> to give an error if there are many choices of inverse with no clear way to choose one of them: for example, <code><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Function'>⍉</span><span class='Value'>m</span></code> returns the diagonal of matrix <code><span class='Value'>m</span></code>; <code><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Function'>⍉</span><span class='Modifier'>⁼</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span></code> requires values to be chosen for the off-diagonal elements in its result. It is better to give an error, encouraging the programmer to use a fully-specified approach like <code><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Modifier2'>⌾</span><span class='Paren'>(</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Modifier2'>⊸</span><span class='Function'>⍉</span><span class='Paren'>)</span></code> applied to a matrix of initial elements, than to return a result that could be very different from other implementations.</p>
<p>When working with limited-precision numbers, it may be difficult or impossible to exactly invert the operand function. Instead, it is generally acceptable to perform a computation that, if done with unlimited precision, would exactly invert <code><span class='Function'>𝔽</span></code> computed with unlimited precision. This principle is the basis for the numeric inverses specified below. It is also acceptable to find an inverse by numeric methods, provided that the error in the inverse value found relative to an unlimited-precision inverse can be kept close to the inherent error in the implementation's number format.</p>
<h3 id="required-functions">Required functions</h3>
<p>Function inverses are given for one or two arguments, with cases where inverse support is not required left blank.</p>
<p>For arithmetic functions the implementations below may in some cases not give the closest inverse (that is, there may be some other <code><span class='Value'>y</span></code> so that <code><span class='Function'>F</span> <span class='Value'>y</span></code> is closer to <code><span class='Value'>x</span></code> than <code><span class='Function'>F</span> <span class='Function'>F</span><span class='Modifier'>⁼</span><span class='Value'>x</span></code>). Even in these cases the exact functions given below must be used.</p>
<table>
<thead>
<tr>
<th>Fn</th>
<th>1</th>
<th>2</th>
</tr>
</thead>
<tbody>
<tr>
<td><code><span class='Function'>+</span></code></td>
<td><code><span class='Function'>+</span></code></td>
<td><code><span class='Function'>-</span><span class='Modifier'>˜</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>-</span></code></td>
<td><code><span class='Function'>-</span></code></td>
<td><code><span class='Function'>-</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>×</span></code></td>
<td></td>
<td><code><span class='Function'>÷</span><span class='Modifier'>˜</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>÷</span></code></td>
<td><code><span class='Function'>÷</span></code></td>
<td><code><span class='Function'>÷</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>√</span></code></td>
<td><code><span class='Function'>×</span><span class='Modifier'>˜</span></code></td>
<td><code><span class='Function'>⋆</span><span class='Modifier'>˜</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>∧</span></code></td>
<td></td>
<td><code><span class='Function'>÷</span><span class='Modifier'>˜</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>¬</span></code></td>
<td><code><span class='Function'>¬</span></code></td>
<td><code><span class='Function'>¬</span></code></td>
</tr>
</tbody>
</table>
<p>Unlike these inverses, the logarithm function—base <em>e</em> for <code><span class='Function'>⋆</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code> and base <code><span class='Value'>𝕨</span></code> for <code><span class='Value'>𝕨</span><span class='Function'>⋆</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code>—does not have any strict precision requirements.</p>
<table>
<thead>
<tr>
<th>Fn</th>
<th>1</th>
<th>2</th>
</tr>
</thead>
<tbody>
<tr>
<td><code><span class='Function'>⋆</span></code></td>
<td><code><span class='Function'>Log</span></code></td>
<td><code><span class='Function'>÷</span><span class='Modifier'>˜</span><span class='Modifier2'>○</span><span class='Function'>Log</span></code></td>
</tr>
</tbody>
</table>
<p>The following structural functions have unique inverses, except in a few cases. Dyadic <code><span class='Function'>⍉</span></code> with repeated axes is excluded, and monadic <code><span class='Function'><</span></code> can only be inverted on a rank-0 array. Dyadic <code><span class='Function'>⊣</span></code> is invertible only if the arguments match, and in this case any return value is valid, but in BQN the shared argument value is returned. For <code><span class='Function'>/</span><span class='Modifier'>⁼</span></code> the argument must be a list of non-descending natural numbers, and the result's fill element is 0.</p>
<table>
<thead>
<tr>
<th>Fn</th>
<th>1</th>
<th>2</th>
</tr>
</thead>
<tbody>
<tr>
<td><code><span class='Function'>⊢</span></code></td>
<td><code><span class='Function'>⊢</span></code></td>
<td><code><span class='Function'>⊢</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>⊣</span></code></td>
<td><code><span class='Function'>⊢</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Value'>𝕨</span><span class='Function'>≡</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
</tr>
<tr>
<td><code><span class='Function'><</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Number'>0</span><span class='Function'>==</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Function'>!</span><span class='Number'>0</span><span class='Function'><≡</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Function'>⊑</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>⌽</span></code></td>
<td><code><span class='Function'>⌽</span></code></td>
<td><code><span class='Paren'>(</span><span class='Function'>-</span><span class='Modifier2'>⊸</span><span class='Function'>⌽</span><span class='Paren'>)</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>⍉</span></code></td>
<td><code><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>⌽↕</span><span class='Modifier2'>∘</span><span class='Function'>=</span><span class='Paren'>)</span><span class='Modifier2'>⊸</span><span class='Function'>⍉</span><span class='Modifier2'>⍟</span><span class='Paren'>(</span><span class='Number'>0</span><span class='Function'><=</span><span class='Paren'>)</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>!∧</span><span class='Modifier'>´</span><span class='Function'>∊</span><span class='Value'>𝕨</span><span class='Separator'>⋄</span><span class='Value'>𝕨</span><span class='Function'>⍉</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Paren'>(</span><span class='Function'>⍋⍷</span><span class='Value'>𝕨</span><span class='Function'>∾↕=</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>⍉</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>/</span></code></td>
<td><code><span class='Function'>≠</span><span class='Modifier'>¨</span><span class='Modifier2'>∘</span><span class='Function'>⊔</span></code></td>
<td></td>
</tr>
</tbody>
</table>
<p>For a data value <code><span class='Value'>k</span></code>, the inverse <code><span class='Value'>𝕨k</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code> with or without a left argument is <code><span class='Value'>k</span><span class='Function'>⊣</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code>.</p>
<table>
<thead>
<tr>
<th>Fn</th>
<th>Inverse</th>
</tr>
</thead>
<tbody>
<tr>
<td><code><span class='Value'>k</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Value'>k</span><span class='Function'>≡</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
</tr>
</tbody>
</table>
<h3 id="optional-functions">Optional functions</h3>
<p>Several primitives are easily undone, but doing so is not important for BQN programming. These primitives are listed below along with suggested algorithms to undo them. Unlike the implementations above, these functions are not valid in all cases, and the inputs must be validated or the results checked in order to use them.</p>
<table>
<thead>
<tr>
<th>Fn</th>
<th>1</th>
<th>2</th>
</tr>
</thead>
<tbody>
<tr>
<td><code><span class='Function'>×</span></code></td>
<td><code><span class='Function'>⊢</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>∧</span></code></td>
<td><code><span class='Function'>⊢</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>∨</span></code></td>
<td><code><span class='Function'>⊢</span></code></td>
<td><code><span class='Function'>-</span><span class='Modifier'>˜</span><span class='Function'>÷</span><span class='Number'>1</span><span class='Function'>-⊣</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>∾</span></code></td>
<td></td>
<td><code><span class='Brace'>{</span><span class='Paren'>(</span><span class='Function'>=</span><span class='Modifier2'>○</span><span class='Function'>=</span><span class='Modifier2'>⟜</span><span class='Value'>𝕩</span><span class='Modifier2'>◶</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Function'>≠</span><span class='Value'>𝕨</span><span class='Paren'>)</span><span class='Function'>↓</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>≍</span></code></td>
<td><code><span class='Function'>⊏</span></code></td>
<td><code><span class='Number'>¯1</span><span class='Modifier2'>⊸</span><span class='Function'>⊏</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>↑</span></code></td>
<td><code><span class='Number'>¯1</span><span class='Modifier2'>⊸</span><span class='Function'>⊑</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>↓</span></code></td>
<td><code><span class='Function'>⊑</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>↕</span></code></td>
<td><code><span class='Function'>≢</span></code></td>
<td></td>
</tr>
</tbody>
</table>
<h3 id="required-modifiers">Required modifiers</h3>
<p>The following cases of Self/Swap must be supported.</p>
<table>
<thead>
<tr>
<th>Fn</th>
<th>1</th>
<th>2</th>
</tr>
</thead>
<tbody>
<tr>
<td><code><span class='Function'>+</span><span class='Modifier'>˜</span></code></td>
<td><code><span class='Function'>÷</span><span class='Modifier2'>⟜</span><span class='Number'>2</span></code></td>
<td><code><span class='Function'>+</span><span class='Modifier'>⁼</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>-</span><span class='Modifier'>˜</span></code></td>
<td></td>
<td><code><span class='Function'>+</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>×</span><span class='Modifier'>˜</span></code></td>
<td><code><span class='Function'>√</span></code></td>
<td><code><span class='Function'>×</span><span class='Modifier'>⁼</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>÷</span><span class='Modifier'>˜</span></code></td>
<td></td>
<td><code><span class='Function'>×</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>⋆</span><span class='Modifier'>˜</span></code></td>
<td></td>
<td><code><span class='Function'>√</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>√</span><span class='Modifier'>˜</span></code></td>
<td></td>
<td><code><span class='Function'>÷⋆</span><span class='Modifier'>⁼</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>∧</span><span class='Modifier'>˜</span></code></td>
<td><code><span class='Function'>√</span></code></td>
<td><code><span class='Function'>∧</span><span class='Modifier'>⁼</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>∨</span><span class='Modifier'>˜</span></code></td>
<td><code><span class='Function'>√</span><span class='Modifier2'>⌾</span><span class='Function'>¬</span></code></td>
<td><code><span class='Function'>∨</span><span class='Modifier'>⁼</span></code></td>
</tr>
<tr>
<td><code><span class='Function'>¬</span><span class='Modifier'>˜</span></code></td>
<td></td>
<td><code><span class='Function'>+-</span><span class='Number'>1</span><span class='Modifier'>˙</span></code></td>
</tr>
</tbody>
</table>
<p>Inverses of other modifiers and derived functions or modifiers obtained from them are given below. Here the "inverse" of a modifier is another modifier that, if applied to the same operands as the original operator, gives its inverse function. A constant is either a data value or <code><span class='Function'>𝔽</span><span class='Modifier'>˙</span></code> for an arbitrary value <code><span class='Function'>𝔽</span></code>.</p>
<table>
<thead>
<tr>
<th>Mod</th>
<th>Inverse</th>
<th>Requirements</th>
</tr>
</thead>
<tbody>
<tr>
<td><code><span class='Modifier'>¨</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Number'>0</span><span class='Function'><≡</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier'>⁼¨</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Modifier'>⌜</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Number'>0</span><span class='Function'><≡</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span> <span class='Function'>𝔽</span><span class='Modifier'>⁼⌜</span><span class='Value'>𝕩;</span><span class='Brace'>}</span></code></td>
<td>Monadic case only</td>
</tr>
<tr>
<td><code><span class='Modifier'>˘</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Number'>0</span><span class='Function'><=</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier'>⁼˘</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Modifier'>`</span></code></td>
<td><code><span class='Paren'>(</span><span class='Function'>⊏∾</span><span class='Number'>2</span><span class='Function'>𝔽</span><span class='Modifier'>⁼˝˘</span><span class='Modifier2'>∘</span><span class='Function'>↕⊢</span><span class='Paren'>)</span><span class='Modifier2'>⍟</span><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'><≠</span><span class='Paren'>)</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>F</span><span class='Modifier2'>∘</span><span class='Function'>G</span></code></td>
<td><code><span class='Brace'>{</span><span class='Value'>𝕨</span><span class='Function'>G</span><span class='Modifier'>⁼</span><span class='Function'>F</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>F</span> <span class='Function'>G</span></code></td>
<td></td>
<td></td>
</tr>
<tr>
<td><code><span class='Nothing'>·</span><span class='Function'>F</span> <span class='Function'>G</span></code></td>
<td></td>
<td></td>
</tr>
<tr>
<td><code><span class='Modifier2'>○</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>𝔾</span><span class='Modifier'>⁼</span><span class='Paren'>(</span><span class='Function'>𝔾</span><span class='Value'>𝕨</span><span class='Paren'>)</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Modifier'>⁼</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Modifier2'>⊸</span><span class='Function'>⊢</span><span class='Modifier2'>∘</span><span class='Function'>𝔽</span><span class='Brace'>}</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Modifier2'>⌾</span></code></td>
<td><code><span class='Brace'>{</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Modifier2'>⌾</span><span class='Function'>𝔾</span><span class='Brace'>}</span></code></td>
<td>Verify result for computational Under</td>
</tr>
<tr>
<td><code><span class='Modifier2'>⍟</span><span class='Value'>n</span></code></td>
<td><code><span class='Modifier2'>⍟</span><span class='Paren'>(</span><span class='Function'>-</span><span class='Value'>n</span><span class='Paren'>)</span></code></td>
<td>Atomic number n</td>
</tr>
<tr>
<td><code><span class='Modifier2'>⊘</span></code></td>
<td><code><span class='Brace'>{</span><span class='Paren'>(</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Paren'>)</span><span class='Modifier2'>⊘</span><span class='Paren'>(</span><span class='Function'>𝔾</span><span class='Modifier'>⁼</span><span class='Paren'>)</span><span class='Brace'>}</span></code></td>
<td></td>
</tr>
<tr>
<td><code><span class='Value'>k</span><span class='Modifier2'>⊸</span><span class='Function'>𝔽</span></code></td>
<td><code><span class='Value'>k</span><span class='Modifier2'>⊸</span><span class='Paren'>(</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Paren'>)</span></code></td>
<td>Constant k</td>
</tr>
<tr>
<td><code><span class='Value'>k</span><span class='Function'>𝔽⊢</span></code></td>
<td></td>
<td></td>
</tr>
<tr>
<td><code><span class='Function'>𝔽</span><span class='Modifier2'>⟜</span><span class='Value'>k𝕩</span></code></td>
<td><code><span class='Value'>k</span><span class='Function'>𝔽</span><span class='Modifier'>˜⁼</span><span class='Value'>𝕩</span></code></td>
<td>Constant k</td>
</tr>
<tr>
<td><code><span class='Function'>⊢𝔽</span><span class='Value'>k</span><span class='Modifier'>˙</span></code></td>
<td></td>
<td>Arbitrary k</td>
</tr>
</tbody>
</table>
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