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| author | Marshall Lochbaum <mwlochbaum@gmail.com> | 2020-07-20 14:05:36 -0400 |
|---|---|---|
| committer | Marshall Lochbaum <mwlochbaum@gmail.com> | 2020-07-20 14:05:36 -0400 |
| commit | e52d50ed594dd5626523ca7931315e47bde8c9d1 (patch) | |
| tree | d355718d8f81ea0773cd177e5b5eb17e043a5da3 /docs/doc/logic.html | |
| parent | 883eda3df8e162e2d8a62b4d1ec03eadcf8b8069 (diff) | |
Make header id slugs match Github's
Diffstat (limited to 'docs/doc/logic.html')
| -rw-r--r-- | docs/doc/logic.html | 4 |
1 files changed, 2 insertions, 2 deletions
diff --git a/docs/doc/logic.html b/docs/doc/logic.html index 7ac951d8..d9d8a405 100644 --- a/docs/doc/logic.html +++ b/docs/doc/logic.html @@ -1,6 +1,6 @@ <head><link href="../style.css" rel="stylesheet"/></head> <div class="nav"><a href="https://github.com/mlochbaum/BQN">BQN</a></div> -<h1 id="logic-functions--and--or--not--also-span-">Logic functions: And, Or, Not (also Span)</h1> +<h1 id="logic-functions-and-or-not-also-span">Logic functions: And, Or, Not (also Span)</h1> <p>BQN retains the APL symbols <code><span class='Function'>∧</span></code> and <code><span class='Function'>∨</span></code> for logical <em>and</em> and <em>or</em>, and changed APL's <code><span class='Value'>~</span></code> to <code><span class='Function'>¬</span></code> for <em>not</em>, since <code><span class='Value'>~</span></code> looks too much like <code><span class='Modifier'>˜</span></code> and <code><span class='Function'>¬</span></code> is more common in mathematics today. Like J, BQN extends Not to the linear function <code><span class='Number'>1</span><span class='Modifier2'>⊸</span><span class='Function'>-</span></code>. However, it discards <a href="https://aplwiki.com/wiki/GCD">GCD</a> and <a href="https://aplwiki.com/wiki/LCM">LCM</a> as extensions of And and Or, and instead uses bilinear extensions: And is identical to Times (<code><span class='Function'>×</span></code>), while Or is <code><span class='Function'>×</span><span class='Modifier2'>⌾</span><span class='Function'>¬</span></code>, following De Morgan's laws (other ways of obtaining a function for Or give an equivalent result—there is only one bilinear extension).</p> <p>If the arguments are probabilities of independent events, then an extended function gives the probability of the boolean function on their outcomes (for example, if <em>A</em> occurs with probability <code><span class='Value'>a</span></code> and <em>B</em> with probability <code><span class='Value'>b</span></code> independent of <em>A</em>, then <em>A</em> or <em>B</em> occurs with probability <code><span class='Value'>a</span><span class='Function'>∨</span><span class='Value'>b</span></code>). These extensions have also been used in complexity theory, because they allow mathematicians to transfer a logical circuit from the discrete to the continuous domain in order to use calculus on it.</p> <p>Both valences of <code><span class='Function'>¬</span></code> are equivalent to the fork <code><span class='Number'>1</span><span class='Function'>+-</span></code>. The dyadic valence, called "Span", computes the number of integers in the range from <code><span class='Value'>𝕩</span></code> to <code><span class='Value'>𝕨</span></code>, inclusive, when both arguments are integers and <code><span class='Value'>𝕩</span><span class='Function'>≤</span><span class='Value'>𝕨</span></code> (note the reversed order, which is used for consistency with subtraction). This function has many uses, and in particular is relevant to the <a href="windows.html">Windows</a> function.</p> @@ -29,7 +29,7 @@ 1 1 1 </pre> <p>As with logical And and Or, any value and 0 is 0, while any value or 1 is 1. The other boolean values give the identity elements for the two functions: 1 and any value gives that value, as does 0 or the value.</p> -<h2 id="why-not-gcd-and-lcm-">Why not GCD and LCM?</h2> +<h2 id="why-not-gcd-and-lcm">Why not GCD and LCM?</h2> <p>The main reason for omitting these functions is that they are complicated and, when applied to real or complex numbers, require a significant number of design decisions where there is no obvious choice (for example, whether to use comparison tolerance). On the other hand, these functions are fairly easy to implement, which allows the programmer to control the details, and also add functionality such as the extended GCD.</p> <p>A secondary reason is that the GCD falls short as an extension of Or, because its identity element 0 is not total. <code><span class='Number'>0</span><span class='Function'>∨</span><span class='Value'>x</span></code>, for a real number <code><span class='Value'>x</span></code>, is actually equal to <code><span class='Function'>|</span><span class='Value'>x</span></code> and not <code><span class='Value'>x</span></code>: for example, <code><span class='Number'>0</span><span class='Function'>∨</span><span class='Number'>¯2</span></code> is <code><span class='Number'>2</span></code> in APL. This means the identity <code><span class='Number'>0</span><span class='Function'>∨</span><span class='Value'>x</span> <span class='Gets'>←→</span> <span class='Value'>x</span></code> isn't reliable in APL.</p> <h2 id="identity-elements">Identity elements</h2> |