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Diffstat (limited to 'docs/doc/fold.html')
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1 files changed, 7 insertions, 7 deletions
diff --git a/docs/doc/fold.html b/docs/doc/fold.html index 2890eb4d..b9dddf8b 100644 --- a/docs/doc/fold.html +++ b/docs/doc/fold.html @@ -4,7 +4,7 @@ <title>BQN: Fold and Insert</title> </head> <div class="nav">(<a href="https://github.com/mlochbaum/BQN">github</a>) / <a href="../index.html">BQN</a> / <a href="index.html">doc</a></div> -<h1 id="fold-and-insert">Fold and Insert</h1> +<h1 id="fold-and-insert"><a class="header" href="#fold-and-insert">Fold and Insert</a></h1> <svg viewBox='-184.8 -12 588 288'> <g font-size='21px' fill='currentColor' stroke-linecap='round' text-anchor='middle' font-family='BQN,monospace'> <rect class='code' stroke-width='1.5' rx='12' x='-128.8' y='0' width='476' height='264'/> @@ -43,7 +43,7 @@ <p>The closely related 1-modifiers Fold (<code><span class='Modifier'>´</span></code>) and Insert (<code><span class='Modifier'>˝</span></code>) apply a dyadic operand function <code><span class='Function'>𝔽</span></code> repeatedly between elements or major cells of <code><span class='Value'>𝕩</span></code>. Neither is quite like the APL2-style Reduce operator (<code><span class='Function'>/</span></code> or <code><span class='Value'>⌿</span></code> in APL), although I sometimes use the term "reduction" to mean either Fold or Insert. There are a bunch of other names like "accumulate" and "aggregate" for this class of calculations—I don't know which is best but I know "catamorphism" is worst.</p> <p>A distinguishing feature of APL-family reductions is that they don't use an initial value, and try to derive an "identity element" from the operand if the argument array is empty. BQN retains this capability but also allows the programmer to supply an initial value as <code><span class='Value'>𝕨</span></code>.</p> -<h2 id="fold">Fold</h2> +<h2 id="fold"><a class="header" href="#fold">Fold</a></h2> <p>As its glyph suggests, Fold is slightly simpler than Insert. The argument <code><span class='Value'>𝕩</span></code> must always be a list, and Fold applies <code><span class='Function'>𝔽</span></code> between elements—always two at a time—of the list to yield a single result value. In this sense, <code><span class='Function'>𝔽</span><span class='Modifier'>´</span></code> removes a layer of <a href="depth.html">depth</a> from <code><span class='Value'>𝕩</span></code>, although it's not necessarily true that the depth of <code><span class='Function'>𝔽</span><span class='Modifier'>´</span><span class='Value'>𝕩</span></code> is less than that of <code><span class='Value'>𝕩</span></code> because the function <code><span class='Function'>𝔽</span></code> might increase depth.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=K8K0IDLigL804oC/M+KAvzEKK8K0IOKfqDLigL80LCAz4oC/MeKfqQ==">↗️</a><pre> <span class='Function'>+</span><span class='Modifier'>´</span> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>1</span> 10 @@ -64,7 +64,7 @@ <span class='Function'>∨</span><span class='Modifier'>´</span> <span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>0</span> 1 </pre> -<h3 id="identity-values">Identity values</h3> +<h3 id="identity-values"><a class="header" href="#identity-values">Identity values</a></h3> <p>Folding over a list of length 1 never calls the operand function: it returns the lone element unchanged.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=IcK0IOKfqOKOiuKfqQ==">↗️</a><pre> <span class='Function'>!</span><span class='Modifier'>´</span> <span class='Bracket'>⟨</span><span class='Modifier2'>⎊</span><span class='Bracket'>⟩</span> ⎊ @@ -132,7 +132,7 @@ </tr> </tbody> </table> -<h3 id="right-to-left">Right-to-left</h3> +<h3 id="right-to-left"><a class="header" href="#right-to-left">Right-to-left</a></h3> <p>The functions we've shown so far are associative (ignoring floating point imprecision), meaning it's equally valid to combine elements of the argument list in any order. But it can be useful to fold using a non-associative function. In this case you must know that Fold performs a <em>right fold</em>, starting from the end of the array and working towards the beginning.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4omN4peLPMK0ICJhYmNkIgoKJ2EnIOKJjeKXizwgJ2InIOKJjeKXizwgJ2MnIOKJjeKXizwgJ2QnICAjIEV4cGFuZGVkIGZvcm0=">↗️</a><pre> <span class='Function'>≍</span><span class='Modifier2'>○</span><span class='Function'><</span><span class='Modifier'>´</span> <span class='String'>"abcd"</span> ⟨ 'a' ⟨ 'b' "cd" ⟩ ⟩ @@ -153,7 +153,7 @@ <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=K+KfnMO3wrQgMuKAvzHigL8y4oC/MeKAvzHigL804oC/MeKAvzE=">↗️</a><pre> <span class='Function'>+</span><span class='Modifier2'>⟜</span><span class='Function'>÷</span><span class='Modifier'>´</span> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>1</span> 2.7183098591549295 </pre> -<h3 id="initial-element">Initial element</h3> +<h3 id="initial-element"><a class="header" href="#initial-element">Initial element</a></h3> <p>When the operand isn't just an arithmetic primitive, folding with no initial element can be dangerous. Even if you know <code><span class='Value'>𝕩</span></code> isn't empty, saving you from an "Identity not found" error, the case with only one element can easily violate expectations. Here's a somewhat silly example of a function meant to merge elements of the argument into a single list (<code><span class='Function'>∾⥊</span><span class='Modifier'>¨</span></code> is a much better way to do this):</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oi+4peL4qWKwrQg4p+oMuKAvzTiiY024oC/OCwiYWJjZCIsMOKfqQoK4oi+4peL4qWKwrQg4p+oMuKAvzTiiY024oC/OCwiYWJjZCLin6kKCuKIvuKXi+KlisK0IOKfqDLigL804omNNuKAvzjin6k=">↗️</a><pre> <span class='Function'>∾</span><span class='Modifier2'>○</span><span class='Function'>⥊</span><span class='Modifier'>´</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Function'>≍</span><span class='Number'>6</span><span class='Ligature'>‿</span><span class='Number'>8</span><span class='Separator'>,</span><span class='String'>"abcd"</span><span class='Separator'>,</span><span class='Number'>0</span><span class='Bracket'>⟩</span> ⟨ 2 4 6 8 'a' 'b' 'c' 'd' 0 ⟩ @@ -180,7 +180,7 @@ <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=IlNUT1AiIOKMveKKuOKIvsK0ICJBQkNERSLigL8iMDEyIuKAvyJhYmNkIg==">↗️</a><pre> <span class='String'>"STOP"</span> <span class='Function'>⌽</span><span class='Modifier2'>⊸</span><span class='Function'>∾</span><span class='Modifier'>´</span> <span class='String'>"ABCDE"</span><span class='Ligature'>‿</span><span class='String'>"012"</span><span class='Ligature'>‿</span><span class='String'>"abcd"</span> "EDCBA210dcbaSTOP" </pre> -<h2 id="insert">Insert</h2> +<h2 id="insert"><a class="header" href="#insert">Insert</a></h2> <p>Fold only works on lists. What if you want to, say, sum the columns of a table?</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqiIHRhYiDihpAgKDIr4oaVNSkgfOKMnCA5K+KGlTMKCivLnSB0YWI=">↗️</a><pre> <span class='Function'>⊢</span> <span class='Value'>tab</span> <span class='Gets'>←</span> <span class='Paren'>(</span><span class='Number'>2</span><span class='Function'>+↕</span><span class='Number'>5</span><span class='Paren'>)</span> <span class='Function'>|</span><span class='Modifier'>⌜</span> <span class='Number'>9</span><span class='Function'>+↕</span><span class='Number'>3</span> ┌─ @@ -241,7 +241,7 @@ </span>⟨ 0 4 ⟩ </pre> <p>As a historical note, Insert is named after J's adverb <code><span class='Function'>/</span></code>, which comes from SHARP APL's <code><span class='Value'>⌿</span></code>, reduce-down. In the original APL, only arithmetic reductions were defined, and nested arrays didn't exist—arrays were either all characters or all numbers. SHARP extended them by splitting the array into cells as we've shown. However, there's another interpretation, which is what you'll find in mainstream APLs today…</p> -<h2 id="apl2-reduction">APL2 reduction?</h2> +<h2 id="apl2-reduction"><a class="header" href="#apl2-reduction">APL2 reduction?</a></h2> <p>If you try an expression like <code><span class='Value'>⍪⌿</span></code> in Dyalog APL, you'll get results very different from BQN's <code><span class='Function'>∾</span><span class='Modifier'>˝</span></code>. Instead of combining the cells like we see above, APL applies the function on pairs of <em>elements</em> much like Fold. The difference is that, because reduction happens only along one axis but an array might have other axes, there can be multiple values in the result, so that it will always be an array like the argument. BQN can perform this operation as well: <code><span class='Value'>⍪⌿</span></code> is written <code><span class='Function'>∾</span><span class='Modifier'>¨˝</span></code> in BQN.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oi+wqjLnSB0YWI=">↗️</a><pre> <span class='Function'>∾</span><span class='Modifier'>¨˝</span> <span class='Value'>tab</span> ⟨ ⟨ 1 0 1 4 3 ⟩ ⟨ 0 1 2 0 4 ⟩ ⟨ 1 2 3 1 5 ⟩ ⟩ |