[CG] Re: Cactus Language • Preliminaries

Cactus Language • Preliminaries 3 • https://inquiryintoinquiry.com/2025/04/09/cactus-language-preliminaries-3/ A few definitions from formal language theory are required at this point. Note. Please see the blog post linked above for the proper mathematical formats. The transcript below uses ‡A‡ and ‡a‡ for Fraktur A and a, respectively. • An “alphabet” is a finite set of signs, typically, ‡A‡ = {‡a‡₁, …, ‡a‡ₙ}. • A “string” over an alphabet ‡A‡ is a finite sequence of signs from ‡A‡. • The “length” of a string is just its length as a sequence of signs. • The “empty string” is the unique sequence of length 0. It is sometimes denoted by an empty pair of quotation marks (“”) but more often by the Greek symbols epsilon or lambda. • A sequence of length k > 0 is typically presented in the following concatenated forms. s₁ s₂ … sₖ₋₁ sₖ or s₁ ∙ s₂ ∙ … ∙ sₖ₋₁ ∙ sₖ with sₘ ∈ ‡A‡ for all m = 1 … k. The following notations provide useful alternatives. • ε = “” = the empty string. • η = {ε} = the language consisting of a single empty string. Several operations on strings find sufficient application to motivate the following definitions. • To “erase” an appearance of a sign is to replace it with an appearance of the blank symbol “ ”. • To “delete” an appearance of a sign is to replace it with an appearance of the empty string “”. • If s is a string which ends with a sign t then s ∙ t⁻¹ is the string which results by deleting the terminal t from s. • A “token” is a particular appearance of a sign. Finally — • The “kleene star” ‡A‡* of alphabet ‡A‡ is the set of all strings over ‡A‡. In particular, ‡A‡* includes among its elements the empty string ε. • The “kleene plus” ‡A‡⁺ of an alphabet ‡A‡ is the set of all positive length strings over ‡A‡, in other words, everything in ‡A‡* but the empty string. • A “formal language” ‡L‡ over an alphabet ‡A‡ is a subset of ‡A‡*. In brief, ‡L‡ ⊆ ‡A‡*. If s is a string over ‡A‡ and s is an element of ‡L‡ then it is customary to call s a “sentence” of ‡L‡. Thus, a formal language ‡L‡ is defined by specifying its elements, which amounts to saying what it means to be a sentence of ‡L‡. Resources — Cactus Language • Preliminaries • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.… Survey of Animated Logical Graphs • https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs… Survey of Theme One Program • https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/ Regards, Jon