Re: Position Paper: Why the Halting Problem Is Uninteresting Due to Infinite Tape Assumption

On 5/24/25 2:47 AM, Mr Flibble wrote:

Position Paper: Why the Halting Problem Is Uninteresting Due to Infinite
Tape Assumption
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 Thesis:
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The Halting Problem is uninteresting—at least in practical and engineering
contexts—because it depends on an idealized and physically unrealizable
abstraction: an infinite tape.

But that wasn't the context it was developed for. Remember, when it was created a "Computer" was a person who performed computations according to a written list of instructions.
The Halting Problem

 1. Infinite Tape Is Physically Unrealizable
-------------------------------------------
- Turing machines assume unbounded memory via an infinite tape.

Right, so it could handle the mathematics of the Natural Numbers.

- Real-world computers, compilers, and systems have finite RAM, stack, and
disk.

Which didn't exist at the time of the problem.
And

- The Halting Problem’s result does not apply directly to real machines.

It never did, and wasn't intended to.

 Conclusion: The Halting Problem is elegant in theory, but disconnected
from physical computation.

No, it deals with a different concept of computation, that applies to a theoretical bases that is the bases that physical machines are built to approximatedly model.

 2. Undecidability Hinges on Infinite Self-Simulation
-----------------------------------------------------
- The classic halting proof constructs a program that simulates itself
recursively.
- This simulation only "works" because the system permits infinite memory
and execution depth.

- Real systems crash or terminate when exceeding limits.

Which just shows that "Real System" are only approximate models of that which they are trying to model.

 Conclusion: The impossibility result emerges from allowing infinite
regress, not intrinsic complexity.

No, since the actual model allows for the infinte, since that is what actual numbers do, any finite model is itself inaccurate.
Your whole problem is you fail to understand what the field of Computation Theory is talking about, which is mathematics, and possibilitis in mathematics.
"Modern Computers" are largely designed to try to model that mathematics, but acknoledge their limitations.

 3. Real Programs Are Finite and Bounded
---------------------------------------
- Most software in practice uses:
   - Bounded recursion.
   - Static analysis.
   - Safe loops or iteration limits.
 - Many practical tools can reliably decide halting on real codebases.

PARTIALLY decide.

 Conclusion: The Halting Problem ignores how software is actually written
and validated.

The Halting Problem PREDATES that "SOFTWARE" and isn't about it.
All you are doing it proving you don't understand what it is talking about.
Note, there are plenty of other theories built for the systems that you are trying to reshape Computation Theory to handle, so maybe you should look at those and see what has already been done.

 4. Infinite Models Create Artificial Pathologies
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- The Halting Problem proof depends on paradoxical constructions like:
   - A program calling its own decider.

Nothing wrong with that.

   - Flipping the decision.

Nothing wrong with that.

   - Creating a contradiction.

No, the contradiction only occurs if you make the false assumption that the decider could have been correct in the first place.

 - This only happens because no semantic firewall exists (e.g., no type
system, no stratification).

Because such a firewall can't exist in the domain, as your system isn't powerful enough to express the mathematics needed.

 Conclusion: The paradox arises from theoretical permissiveness, not
meaningful computation.

No, it arises out of the needs of the system, and isn't actually a paradox. It shows that the power of mathematics to do things has exceeded its ability to prove things.

 5. Modern Computation Avoids This Entirely
------------------------------------------
- Typed languages, proof assistants, and total languages:
   - Forbid unrestricted recursion.
   - Disallow self-reference.
   - Provide guarantees of termination.
 - These tools prioritize safety, not expressiveness.

And when those limits are in place, you lose the ability to handle the original problem.

 Conclusion: Decidable systems are already widely adopted in safety-
critical software and logic.

But can't be used to solve the original problem that Computation Theory was designed to do, and thus fails.

 Rebuttal:
---------
"But the Halting Problem defines the limits of computation."
 Yes—but only in the same way that theoretical constructs define the limits
of idealized physical laws. In the real world, the constraints of
hardware, language, and verification matter more.

So, you think actual Mathematics are unimportant? That the tools used to understand the physical laws that make the universe run are worthless?
That the universe isn't actually running under some physical laws that we are working to understand better?
Sorry, you are just showing you don't understand how science works.

 Final Conclusion:
-----------------
The Halting Problem is foundational—but irrelevant to practical
computation for the same reason that faster-than-light travel is
irrelevant to car engineering.
 Its dependency on infinite memory and pathological self-reference make it
more of a mathematical curiosity than an obstacle to real-world software
development.

So, all you are doing is proving that you don't understand what you are talking about, and are just making FALSE CLAIMS about Computation Theory, as you don't even understand the purpose of the domain.
You see a hammer, and think that its user thinks everything is a nail, not noticing that he also has a screwdriver, drill, and saw.
Sorry, all you are doing is proving your ignorance.