On 7/13/25 11:22 AM, wij wrote:
On Sun, 2025-07-13 at 08:17 -0400, Richard Damon wrote:
On 7/12/25 6:53 AM, wij wrote:
This is my preliminary sumarization of 'computation' for review.
Quantum computing is not considered, but related commnents are also welcome.
The nature of computation - constructing functions The process of computing >>> functions Correct/clear concepts help understanding
Function::= f:A->B, function f is a 1-1 or multi-1 mapping from set A to set B
It should be noted that some branches of the field allow functions to
map input to multiple answers (and thus the result is a SET of elements
of set B. For instance the square root of 4 can be considered to be the
set { 2, -2 }. More basic branches only work with the "principle" answer
for this sort of case.
Mapping::= Let f(a)=b, 'a' is called the input, argument or parameter of f, and
the pair of a,b <a,b> is called a mapping of f.
Mapping pairs can express many things, such as stimulus-response records,
test-result records, input-output records, iff X then Y records...descriptions
of reasoning/association/connection/projection, etc. Moreover, a function can
also be defined by a set of mappings.
Computation::= The process of computing from the deterministic operation of >>> basic parts to the overall performance. (Non-parallel) computation is >>> step-by-step, deterministic operation process from the input of a model to the
output (solution). Input and output are composed of fixed and finite [parts].
[Parts] are things that can be objectively operated, usually referring to text
symbols. Each [part] has a location and value attribute.
[Note] Computation is related to theoretical reasoning in mathematics/logic.
If the [part] is discrete, it can be considered that the [part] can be
described/composed by "circuit logic". Circuit logic has the concept of
time sequence, input/output, and stable/non-stable state. Traditional
logic is less suitable for temporal propositional statements.
(Computer) Problem Statement::= Describes the input (known, conditions, etc.)
and required output data that the computer can process. 'Problems' must be
presented in a narrative, but 'problems' can be described in different ways.
Just like 'mathematical problems'-- they must eventually be converted into
narratives that can be processed in a 'mathematical' way. So, in short, the
final form of computer problem statements is the problem handled by >>> algorithms.
"Problems" in Computation theory are always about a Mapping to be
computed, or about families of mapping.
By 'mapping' I mean exactly xxx-yyy. Obvious not what you mean.
but xxx -> {yy1, yy2, yy3 } is just another form of that mapping.
They need not be presented as a
Narrative, but mappings defined by more direct means (like formulas)
tend to be easy to solve.
'narrative' is given by google translate, the original term reads not much different from 'description' or 'statement'. This time, no 'narrative' appears,
I am not good at controlling the output of google translate.
Then why do you use it?
Note, "Mathematics" isn't a process on Narratives, but on mathematical
objects.
But, what is "mathematical objects"?
Computers can only deal with input made of 0s and 1s.
Right, and so tend to only work on REPRESENTATIONS of things.
Note that "Computing" is by its nature, an activity based on
representations. We establish a way to represent our "Mathematical
Objects" in a symbolic way that the computer can handle, and then
interpret the symbolic output back to the Mathematical field.
Instructions::= A command statement for basic computer [part] operations, such
as basic instructions: write read/compare branch,.. move/convert, etc. The
basic [part] operation (i.e., instruction) function is deterministic and its
object is fixed.
Note, the "object" of an instruction need not be "fixed", just defined.
If not "fixed", nothing can be (computablly) certain, even in abstract theory.
My point is that the instruction can use indirection, we can access x[i]
or *p. There are machines with only direct accessing of memory with the
memory address as part of the instructions, but those need to be allowed
to use "self-modifying" code, like is typically done with the RASP model.
An instruction can take in a value to determine which object it is to
manipulate.
It is still a 'value', whose meaning is something else.
This is probably just a language issue, but when you say its "object is
fixed" that means that it can only access one given object.
That isn't the way most computers work.
Algorithm::= A sequence of instructions (similar to CPU instructions) that >>> describes the process of completing a [part] of a certain operation step by
step (or called instruction statement, flow statement, program, or simply
algorithm). In particular, the flow statement can contain conditional
branches, which can conditionally execute or repeat instructions. >>>
Algorithmic problem (category)::= Algorithmic problem (category) usually refers
to problems that contain substitute for an unlimited number of objects in the
problem statement (such as values/numbers, arrays, sets,.. variable-length
data, etc.).
Since the statement contains substitute for an unlimited number of objects,
the algorithm must use loops to repeat instructions accordingly, and must use
"pointers" or indirect addressing (or index addressing) type instructions to
achieve the ability to read and write ranges of variable as [parts] with fixed
instructions.
Discrete continuity::= If a function f:A->B, has a certain number of n mappings
<x1,f(x1)>, <x2,f(x2)>,...,<xn,f(xn)> are called minmax points or regional
minmax points, so that ∀a,b∈minmax point, ∀x,a<=x<=b<=n, f(a)<=f(x)<=f(b)
or f(a)>=f(x)>=f(b) holds.
One good thing is that the definitions shown can be immediately applied to POOP.
Take POO Halt for example:
Problem description: POOP (or POOH) problem is difficult to convert to a description
of algorithm (so far, at least, no one is sure what POO Halt really is, part of
the reason is it keeps changing).
Computation: POOH cannot be broken down to computable [part]
([part] must be fixed and computable), so the 'whole'.
So, I think, the first one is enough -- Problem description is not well defined to
be computable, all you are talking Your Own Problem.
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