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0.3/1 2024-10-28T19:47Z |
1. The Abstraction Idea
| Views on Abstraction |
|---|
| “This process [of extracting common features] can be regarded as a repeated abstraction … and under certain circumstances such abstracting processes can be repeatedly piled on top of one another. Here ‘abstract’ has to be understood in the literal meaning of the word as ‘removing’, as leaving aside everything inessential for the context in question or for a particular purpose … .” |
| – Hans Reichardt [VNR1977: Introduction,p.11] |
| “A high-level computer language abstracts away the machine [so that] the programmer need not be an expert in the machinations of computer hardware … in order to successfully program a computer. High-level languages (HLLs) automate, hide, or otherwise abstract away the underlying operations of the machine … .” |
| – Mark Jones Lorenzo [Lorenzo2019: Introduction, p.13] |
| “Abstraction is about digging deep into a situation to find out what is at its core making it tick. Another way to think of it is about stripping away irrelevant details, or rather, stripping away details that are irrelevant to what we’re thinking about.” |
| – Eugenia Cheng, The Joy of Abstraction [Cheng2023: 2.3 Forgetting Details] |
| “We call the symbol λx an abstraction operator, and speak of the function which is denoted by (λx M) as obtained from the expression M by abstraction.” |
| – Alonzo Church [Church1941: p.7] |
2. The Abstraction Challenge
The challenge at the oMiser/oFrugal level is the fact that there are only obs and every ob has an applicative interpretation; any ob can have an intended operational use as both data and script. The computational accomplishment of abstraction for oMiser straddles that ambiguity/flexibility of interpretation.
2.1 Pure abstraction
Pure abstraction operations on obs depend on an ob and its structure without
consideration of anything other than the ob “as-is.” The abstraction σ.s M
is a script for a function such that (σ.s M) N determines a form of M with
N substituted everywhere s occurs in M. This is based strictly on ob
structure and not on any applicative interpretation of M.
Although σ.s M determines an ob, that ob has no occurrences of s. In this
sense, σ.s M has s abstracted away; (σ.s M) s determines M as-is.
There are convenient companion functions for use in particular circumstances, often when one or more operands are known in advance.
δ(s, N) M = subst(N, s) M = (σ.s M) N = σ(s, M, N)
also illustrating various Frugalese forms for achieving the same result.
Employment of these and their intermediate variations, known as Curried forms, will be demonstrated in the development of oFrugal utility scripts.
2.2 Symbolic forms
The oMiser computational model’s applicative operations treat lindies as invariant individuals whose applicative interpretations are merely themselves. The model goes farther, preserving constructions with lindies as symbolic forms whose applicative interpretation is themselves.
For example, the oFrugal expression
(a x) b x ;
evaluates to
(a :: x) :: b :: x ;
and
!eval (a :: x) :: b :: x ;
the same. This hinges on the mathematical engineering by which scripts such
as !eval x :: y with lindies x and y are evaluated as obap.ap(x,y)
with result x :: y.
When application operator p and operand x are both symbolic forms, the
result of application is the composed symbolic form p :: x.
When only p is a symbolic form, the result of application is p :: ‵ x,
a (mixed) symbolic-form that evaluates to itself even though x may be an ob
of any form.
2.3 Functional abstraction
[Author’s Note]: Brief introduction to the two forms.
Catalog (Specimen, LYYMMNN Content TBD)
This placeholder links to raw materials and notes, including text files. There will be organized folios of content as consolidation of documentation on the web progresses.
| ID | Status | Started | Topic |
|---|---|---|---|
| sigma | 0.4.0 2024-06-24 | 2024-05-05 | σ.s M, subst(L, s), δ(s, L) definitions |
| lambda | 0.4.0 2024-06-24 | 2024-06-20 | λ.x & ρ.p Abstraction Operations |
| c000000 | undated | 2024-01-25 | Construction Diary & Job Jar |
I invite discussion about Miser Project topics in the Discussion section. Improvements and removal of defects in this particular documentation can be reported and addressed in the Issues section. There are also relevant projects from time to time.
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created 2024-01-25 by orcmid |