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diff --git a/docs/spec/inferred.html b/docs/spec/inferred.html index d4d22610..e1d08a02 100644 --- a/docs/spec/inferred.html +++ b/docs/spec/inferred.html @@ -66,7 +66,7 @@ </tr> </tbody> </table> -<p>Additionally, the identity of <code><span class='Function'>∾</span><span class='Modifier'>˝</span></code> must be recognized: if <code><span class='Number'>0</span><span class='Function'>=≠</span><span class='Value'>𝕩</span></code>, then <code><span class='Function'>∾</span><span class='Modifier'>˝</span><span class='Value'>𝕩</span></code> is <code><span class='Paren'>(</span><span class='Number'>0</span><span class='Function'>∾</span><span class='Number'>2</span><span class='Function'>↓≢</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>⥊</span><span class='Value'>𝕩</span></code>.</p> +<p>Additionally, the identity of <code><span class='Function'>∾</span><span class='Modifier'>˝</span></code> must be recognized: if <code><span class='Number'>0</span><span class='Function'>=≠</span><span class='Value'>𝕩</span></code> and <code><span class='Number'>1</span><span class='Function'><=</span><span class='Value'>𝕩</span></code>, then <code><span class='Function'>∾</span><span class='Modifier'>˝</span><span class='Value'>𝕩</span></code> is <code><span class='Paren'>(</span><span class='Number'>0</span><span class='Function'>∾</span><span class='Number'>2</span><span class='Function'>↓≢</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>⥊</span><span class='Value'>𝕩</span></code>. If <code><span class='Number'>1</span><span class='Function'>==</span><span class='Value'>𝕩</span></code>, then there is no identity element, as the result of <code><span class='Function'>∾</span></code> always has rank at least 1, but the cell rank is 0.</p> <h2 id="undo">Undo</h2> <p>The Undo 1-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> |