From 538a6ab05b070080642e751175425f38f6b33d8e Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Tue, 16 Aug 2022 20:40:05 -0400 Subject: Better documentation for Span --- docs/doc/logic.html | 25 ++++++++++++++++++++++++- 1 file changed, 24 insertions(+), 1 deletion(-) (limited to 'docs') diff --git a/docs/doc/logic.html b/docs/doc/logic.html index 6a69d995..9b18b817 100644 --- a/docs/doc/logic.html +++ b/docs/doc/logic.html @@ -72,8 +72,31 @@ ∨` 0‿1‿0‿0‿1‿0‿1 ⟨ 0 1 1 1 1 1 1 ⟩ +

Not (¬) isn't complicated: for example ¬a=b indicates that a is not equal to b. Or a≠b, but you can't just put a slash through every symbol. One less obvious use is to convert a boolean to plus or minus 1, using the hook modifiers. b-¬b leaves 1 unchanged but subtracts 1 from 0, while (¬b)-b is the negation, converting 0 to 1 and 1 to ¯1.

+↗️
    -⟜¬ 0‿1
+⟨ ¯1 1 ⟩
+
+    ¬⊸- 0‿1
+⟨ 1 ¯1 ⟩
+
+

Span

+

Span isn't a logic function, given that 1¬0 is 2, not a boolean. It's defined to be 1+𝕨-𝕩 (I like to think of the line hanging off the right side as the 1 to be added). The reason it's called Span is that if the arguments are whole numbers with 𝕩≤𝕨, this is the length of the sequence 𝕩, 𝕩+1,… 𝕨.

+↗️
    4 ¬ 1   # 1, 2, 3, 4
+4
+
+    9 ¬ 7   # 7, 8, 9
+3
+
+

The fact that Not and Span share a glyph is no coincidence. ¬𝕩 is 0¬𝕩, because this is equal to 1-𝕩 and 1-0‿1 is 1‿0. And because ¬𝕩 is defined to be 1-𝕩 not just for booleans but for all numbers, it's also true that 𝕨¬𝕩 is 𝕨+¬𝕩.

+↗️
    5 + ¬ 0‿1‿1
+⟨ 6 5 5 ⟩
+
+    5 ¬ 0‿1‿1
+⟨ 6 5 5 ⟩
+
+

The identities ¬𝕩 ←→ 0¬𝕩 and 𝕨¬𝕩 ←→ 𝕨+¬𝕩 are also true with - in place of ¬! That's because ¬ is {1+𝕨-𝕩}, or 1+-, in either case. Changing between - and ¬ only adds or subtracts 1 from both sides of the identities.

Definitions

-

We define

+

The three logic functions can be defined easily in terms of other arithmetic. They're convenience functions in that sense.

Not ← 1+-  # also Span
 And ← ×
 Or  ← ×⌾¬
-- 
cgit v1.2.3