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diff --git a/docs/doc/shift.html b/docs/doc/shift.html index 012dfb0d..5b72e865 100644 --- a/docs/doc/shift.html +++ b/docs/doc/shift.html @@ -4,7 +4,7 @@ <title>BQN: Shift functions</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="shift-functions">Shift functions</h1> +<h1 id="shift-functions"><a class="header" href="#shift-functions">Shift functions</a></h1> <p>The shift functions <code><span class='Function'>«</span></code> and <code><span class='Function'>»</span></code> add major cells to one side an array, displacing cells on the opposite side and moving those in between. Shifts resemble but are more general than the bit-based shift operations used in low-level languages. They replace the APL pattern of a 2-wise reduction after appending or prepending an element (APL's <code><span class='Number'>2</span><span class='Function'>≠/</span><span class='Number'>0</span><span class='Separator'>,</span><span class='Value'>v</span></code> translates to <code><span class='Function'>»</span><span class='Modifier2'>⊸</span><span class='Function'>≠</span><span class='Value'>v</span></code>), one of the more common uses of 2-wise reduction.</p> <p>The result of a shift function always has the same shape as <code><span class='Value'>𝕩</span></code>. The function adds major cells to the beginning (<code><span class='Function'>»</span></code>) or end (<code><span class='Function'>«</span></code>) of <code><span class='Value'>𝕩</span></code>, moving the cells already in <code><span class='Value'>𝕩</span></code> to accomodate them. Some cells on the opposite side from those added will "fall off" and not be included in the result.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=MOKAvzAgwrsgM+KAvzLigL8xICAgICAgICAgICAgICMgU2hpZnQgQmVmb3JlCiJlbmQiIMKrICJhZGQgdG8gdGhlICIgICAjIFNoaWZ0IEFmdGVy">↗️</a><pre> <span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span> <span class='Function'>»</span> <span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span> <span class='Comment'># Shift Before @@ -19,7 +19,7 @@ </span>⟨ 2 3 0 ⟩ </pre> <p>If <code><span class='Value'>𝕨</span></code> is longer than <code><span class='Value'>𝕩</span></code>, some cells from <code><span class='Value'>𝕨</span></code> will be discarded, as well as all of <code><span class='Value'>𝕩</span></code>. In this case <code><span class='Value'>𝕨</span><span class='Function'>»</span><span class='Value'>𝕩</span></code> is <code><span class='Paren'>(</span><span class='Function'>≠</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>↑</span><span class='Value'>𝕨</span></code> and <code><span class='Value'>𝕨</span><span class='Function'>«</span><span class='Value'>𝕩</span></code> is <code><span class='Paren'>(</span><span class='Function'>-≠</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>↑</span><span class='Value'>𝕨</span></code>. For similar reasons, nudging an array of length 0 returns it unchanged.</p> -<h2 id="sequence-processing-with-shifts">Sequence processing with shifts</h2> +<h2 id="sequence-processing-with-shifts"><a class="header" href="#sequence-processing-with-shifts">Sequence processing with shifts</a></h2> <p>When working with a sequence of data such as text, daily measurements, or audio data, shift functions are generally the best way to handle the concept of "next" or "previous". In the following example <code><span class='Value'>s</span></code> is shown alongside the shifted-right data <code><span class='Function'>»</span><span class='Value'>s</span></code>, and each element is compared to the previous with <code><span class='Function'>-</span><span class='Modifier2'>⟜</span><span class='Function'>»</span></code>, which we see is the <a href="undo.html">inverse</a> of Plus <a href="scan.html">Scan</a> <code><span class='Function'>+</span><span class='Modifier'>`</span></code>.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=cyDihpAgMeKAvzLigL8y4oC/NOKAvzPigL814oC/NgpzIOKJjSDCu3MKLeKfnMK7IHMKCitgIC3in5zCuyBz">↗️</a><pre> <span class='Value'>s</span> <span class='Gets'>←</span> <span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</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'>5</span><span class='Ligature'>‿</span><span class='Number'>6</span> <span class='Value'>s</span> <span class='Function'>≍</span> <span class='Function'>»</span><span class='Value'>s</span> @@ -58,7 +58,7 @@ </span>⟨ ¯0.5 ¯0.5 ¯1 ¯0.5 ¯0.5 ¯1.5 ¯0.5 ⟩ </pre> <p>A feature these examples all share is that they maintain the length of <code><span class='Value'>s</span></code>. This is a common condition in sequence processing, particularly when the processed sequence needs to be combined or compared with the original in some way. However, it's not always the case. In some instances, for example when searching <code><span class='Value'>s</span></code> to see if there is any value less than the previous, the list should get shorter during processing. In these cases, <a href="windows.html">Windows</a> (<code><span class='Function'>↕</span></code>) is usually a better choice.</p> -<h2 id="arithmetic-and-logical-shifts">Arithmetic and logical shifts</h2> +<h2 id="arithmetic-and-logical-shifts"><a class="header" href="#arithmetic-and-logical-shifts">Arithmetic and logical shifts</a></h2> <p>The glyphs <code><span class='Function'>«</span></code> and <code><span class='Function'>»</span></code>, suggesting movement, were chosen for the same reasons as the digraphs <code><span class='Function'><<</span></code> and <code><span class='Function'>>></span></code>, and can be used to implement the same operations on boolean lists.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqiIGkg4oaQICIxMDAxMTAxMSItJzAnCgrCq+KNnzMgaSAgICAgICAgIyBRdWljayBhbmQgZGlydHkgbGVmdCBzaGlmdAoKMyDipYrin5ww4oq4wqsgaSAgICAjIEFsdGVybmF0ZSBsZWZ0IHNoaWZ0">↗️</a><pre> <span class='Function'>⊢</span> <span class='Value'>i</span> <span class='Gets'>←</span> <span class='String'>"10011011"</span><span class='Function'>-</span><span class='String'>'0'</span> ⟨ 1 0 0 1 1 0 1 1 ⟩ @@ -76,7 +76,7 @@ <span class='Number'>3</span> <span class='Paren'>(</span><span class='Function'>⥊</span><span class='Modifier2'>⟜</span><span class='Function'>⊏»⊢</span><span class='Paren'>)</span> <span class='Value'>i</span> <span class='Comment'># Arithmetic right shift </span>⟨ 1 1 1 1 0 0 1 1 ⟩ </pre> -<h2 id="other-examples">Other examples</h2> +<h2 id="other-examples"><a class="header" href="#other-examples">Other examples</a></h2> <p>In <a href="take.html">Take</a> (<code><span class='Function'>↑</span></code>), there's no way to specify the fill element when the result is longer than the argument. To take along the first axis with a specified, constant fill value, you can use Shift Before instead, where the right argument is an array of fills with the desired final shape.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=ImFiYyIgwrsgNeKliidGJw==">↗️</a><pre> <span class='String'>"abc"</span> <span class='Function'>»</span> <span class='Number'>5</span><span class='Function'>⥊</span><span class='String'>'F'</span> "abcFF" @@ -95,7 +95,7 @@ <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=KMOXYDHCu+KKoinijL7ijL0gNeKAvzLigL804oC/Mw==">↗️</a><pre> <span class='Paren'>(</span><span class='Function'>×</span><span class='Modifier'>`</span><span class='Number'>1</span><span class='Function'>»⊢</span><span class='Paren'>)</span><span class='Modifier2'>⌾</span><span class='Function'>⌽</span> <span class='Number'>5</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>3</span> ⟨ 24 12 3 1 ⟩ </pre> -<h2 id="higher-rank">Higher rank</h2> +<h2 id="higher-rank"><a class="header" href="#higher-rank">Higher rank</a></h2> <p>Shifting always works on the first axis of <code><span class='Value'>𝕩</span></code> (which must have rank 1 or more), and shifts in major cells. A left argument can have rank equal to <code><span class='Value'>𝕩</span></code>, or one less than it, in which case it becomes a single cell of the result. With no left argument, a cell of fills <code><span class='Number'>1</span><span class='Function'>↑</span><span class='Number'>0</span><span class='Function'>↑</span><span class='Value'>𝕩</span></code> is nudged in.</p> <a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqiIGEg4oaQIOKliuKfnCjihpXDl8K0KSA04oC/MwoKwrsgYSAgICAgICAgICAgICAgICAjIE51ZGdlIGFkZHMgYSBjZWxsIG9mIGZpbGxzCgoib25lIiDCqyBhICAgICAgICAgICMgU2hpZnQgaW4gYSBjZWxsCgooInR3byLiiY0iY2VsIikgwqsgYSAgIyBTaGlmdCBpbiBtdWx0aXBsZSBjZWxscw==">↗️</a><pre> <span class='Function'>⊢</span> <span class='Value'>a</span> <span class='Gets'>←</span> <span class='Function'>⥊</span><span class='Modifier2'>⟜</span><span class='Paren'>(</span><span class='Function'>↕×</span><span class='Modifier'>´</span><span class='Paren'>)</span> <span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>3</span> ┌─ @@ -129,7 +129,7 @@ 'c' 'e' 'l' ┘ </pre> -<h2 id="definition">Definition</h2> +<h2 id="definition"><a class="header" href="#definition">Definition</a></h2> <p>In any instance of <code><span class='Function'>»</span></code> or <code><span class='Function'>«</span></code>, <code><span class='Value'>𝕩</span></code> must have rank at least 1.</p> <p>For a dyadic shift function, <code><span class='Value'>𝕨</span></code> must be <a href="join.html#join-to">Join</a>-compatible with <code><span class='Value'>𝕩</span></code> (that is, <code><span class='Value'>𝕨</span><span class='Function'>∾</span><span class='Value'>𝕩</span></code> completes without error) and cannot have greater rank than <code><span class='Value'>𝕩</span></code>. Then Shift Before (<code><span class='Function'>»</span></code>) is <code><span class='Brace'>{</span><span class='Paren'>(</span><span class='Function'>≠</span><span class='Value'>𝕩</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='Brace'>}</span></code> and Shift After (<code><span class='Function'>«</span></code>) is <code><span class='Brace'>{</span><span class='Paren'>(</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='Function'>∾</span><span class='Value'>𝕨</span><span class='Brace'>}</span></code></p> <p>When called monadically, the default argument is a cell of fills <code><span class='Number'>1</span><span class='Function'>↑</span><span class='Number'>0</span><span class='Function'>↑</span><span class='Value'>𝕩</span></code>. That is, Nudge (<code><span class='Function'>»</span></code>) is <code><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>↑</span><span class='Number'>0</span><span class='Function'>↑⊢</span><span class='Paren'>)</span><span class='Modifier2'>⊸</span><span class='Function'>»</span></code> and Nudge Back (<code><span class='Function'>«</span></code>) is <code><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>↑</span><span class='Number'>0</span><span class='Function'>↑⊢</span><span class='Paren'>)</span><span class='Modifier2'>⊸</span><span class='Function'>«</span></code>. This default argument always satisfies the compatibility requirement above and so the only conditions for nudge are that <code><span class='Value'>𝕩</span></code> has rank at least 1 and has a fill element.</p> |