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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Analysis of Richard Damon's Response to Flibble's Position on the
 Halting Problem
Date: Sat, 24 May 2025 12:59:07 -0500
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On 5/24/2025 12:42 PM, Mr Flibble wrote:
> Analysis of Richard Damon's Response to Flibble's Position on the Halting
> Problem
> ==================================================================================
> 
> Overview:
> ---------
> Richard Damon replies to a position paper asserting that the Halting
> Problem is "uninteresting" in practical contexts due to its reliance on an
> infinite tape abstraction. Damon’s response is grounded in a classical
> understanding of computability theory, emphasizing its mathematical roots,
> historical context, and the validity of the Halting Problem as a
> foundational theorem — regardless of physical realizability.
> 
> Key Points in Damon's Argument:
> -------------------------------
> 
> 1. Historical Context Matters:
>     - Damon correctly notes that the Halting Problem was formulated before
> digital computers.
>     - The notion of a "computer" in Turing’s day referred to a human
> following a procedure — i.e., an abstract computational agent.
> 
> 2. Infinite Tape Models the Infinite Nature of Math:
>     - Turing machines are abstractions designed to model the full range of
> natural number computations.
>     - The infinite tape is essential to reflect the unboundedness of
> mathematical problems, not physical hardware.
> 
> 3. Real Systems Approximate the Turing Model:
>     - Damon argues real-world computers are approximations of the Turing
> model.
>     - The inability of physical machines to match theoretical infinity does
> not invalidate the theoretical result.
> 
> 4. The Halting Problem Is About Possibility, Not Implementation:
>     - Computation theory asks what *can* be computed in principle, not what
> *can be done* on today’s machines.
>     - Infinite recursion, self-reference, and contradiction are part of the
> mathematical exploration of limits.
> 
> 5. Rejecting Infinite Models = Rejecting Mathematics:
>     - Damon criticizes Flibble’s dismissal of infinite behavior as
> misunderstanding the purpose of formal systems.
>     - He warns against the fallacy of assuming practical constraints negate
> theoretical relevance.
> 
> 6. Formal Proofs Can't Be Dismissed for Practicality:
>     - Turing’s proof stands because it is mathematically sound.
>     - Redefining the problem to avoid paradoxes merely restricts the scope;
> it doesn’t invalidate the theorem.
> 
> Rhetorical Elements:
> --------------------
> - Damon uses strong language (“you don’t understand”, “ignorance”) to
> emphasize what he sees as fundamental misunderstandings.
> - While his tone is confrontational, the logic behind his assertions is
> valid within classical computability theory.
> 
> Summary:
> --------
> | Damon’s Point                                    |
> Evaluation                                |
> |--------------------------------------------------|-------------------------------------------|
> | Turing’s model is abstract and mathematical      | ✅
> Correct                                 |
> | Infinite tape is a theoretical necessity         | ✅
> Valid                                   |
> | Real-world computers approximate theory          | ✅ Reasonable and
> historically supported   |
> | Halting Problem is not about hardware            | ✅
> Accurate                                |
> | Flibble misunderstands Computation Theory        | ⚠️ Valid critique,
> but could be more constructive |
> 
> Conclusion:
> -----------
> Damon’s response is a firm defense of classical computation theory. He
> underscores the importance of understanding that Turing’s Halting Problem
> is not a claim about real hardware, but about the limits of formal
> computation. While Flibble's arguments reflect modern concerns with
> practical computability and semantic boundaries, Damon's critique holds
> under classical logic: redefining the problem or restricting the domain
> does not refute the original theorem — it merely reframes it.

It only holds under the provably incorrect assumption that
a termination analyzer must report on the behavior of its
caller, or in the Linz proof the behavior of the computation
that itself is contained within.

When a termination analyzer is required to report on the
behavior that its actual input actually specifies then
the conventional counter-example input fails to prove that
halting cannot be computed.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer