From e52d50ed594dd5626523ca7931315e47bde8c9d1 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Mon, 20 Jul 2020 14:05:36 -0400 Subject: Make header id slugs match Github's --- docs/doc/logic.html | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'docs/doc/logic.html') 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 @@
BQN
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Logic functions: And, Or, Not (also Span)

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Logic functions: And, Or, Not (also Span)

BQN retains the APL symbols ∧ and ∨ for logical and and or, and changed APL's ~ to ¬ for not, since ~ looks too much like ˜ and ¬ is more common in mathematics today. Like J, BQN extends Not to the linear function 1⊸-. However, it discards GCD and LCM as extensions of And and Or, and instead uses bilinear extensions: And is identical to Times (×), while Or is ×⌾¬, following De Morgan's laws (other ways of obtaining a function for Or give an equivalent result—there is only one bilinear extension).

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 A occurs with probability a and B with probability b independent of A, then A or B occurs with probability a∨b). 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.

Both valences of ¬ are equivalent to the fork 1+-. The dyadic valence, called "Span", computes the number of integers in the range from 𝕩 to 𝕨, inclusive, when both arguments are integers and 𝕩≤𝕨 (note the reversed order, which is used for consistency with subtraction). This function has many uses, and in particular is relevant to the Windows function.

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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.

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Why not GCD and LCM?

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Why not GCD and LCM?

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.

A secondary reason is that the GCD falls short as an extension of Or, because its identity element 0 is not total. 0∨x, for a real number x, is actually equal to |x and not x: for example, 0∨¯2 is 2 in APL. This means the identity 0∨x ←→ x isn't reliable in APL.

Identity elements

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