Re: Refutation of Linz's Halting Problem Proof

On 4/20/25 6:15 PM, Mr Flibble wrote:

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                 Refutation of Linz's Halting Problem Proof
                   Based on Type Error Assumption
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Date: April 20, 2025
 This document refutes Peter Linz's proof of the undecidability of the
halting
problem, as presented in "An Introduction to Formal Languages and
Automata," under
the assumption that self-referential input conflating deciders with
programs
constitutes a category (type) error. By enforcing a type system that
separates
deciders from programs, the diagonalization argument central to Linz's
proof is
invalidated, challenging the claim of universal undecidability.
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1. Background: Linz's Proof
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Linz demonstrates the halting problem's undecidability using a
diagonalization

This is not a "diagonalizeation" argument. There is no need to order the possible deciders into a list.

argument. He assumes a hypothetical halting oracle H(P, I), which decides
whether
program P halts on input I. A program D is constructed such that:
   - D uses H to check if D(D) halts.
   - If H(D, D) says D(D) halts, D loops; otherwise, it halts.
This creates a contradiction: D(D) halts if and only if it does not,
implying no
such H exists. The proof relies on self-reference, where D is both the
program and
input to H.

The proof does NOT use "self-referece", as nothing needs to "refer" to itself, and in fact, "reference" is a concept foreign to the domain of Turing Machines.

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2. Assumption: Self-Reference as a Type Error
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The refutation assumes that self-referential input, where a decider (e.g.,
H) is
conflated with the program being decided (e.g., D), is a category error in
a typed
computational model. We define:
   - Type Program: The type of executable programs (e.g., Turing machines).
   - Type Decider: The type of decision procedures (e.g., oracles deciding
     properties of programs).

Or problem has no "oracles" in it. Just Programs.

   - Type Input: The type of inputs to programs, which may include encoded
programs
     but excludes deciders.

Why does it exclude deciders? Why can't we give the encoding of a decider to a decider to decide on?
Isn't that a violation of the defintion of the decider to be able to decide on ALL inputs.

In this system, the halting oracle has signature H: Program x Input ->
{halt, loop}.
Self-referential cases, where P = H or I encodes H, are ill-typed because
H is of
type Decider, not Program.

But we can't have a halting oracle, as that isn't in the domain of valid answers.
If you are starting by assuming H is an oracle machine, then you have left the domain of the problem.
Yes, Inputs can't be based on Oracle Machies, but Oracle Machines are not answers to the problem, which must be an actual program.

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3. Refutation of Linz's Proof
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The following arguments demonstrate that Linz's proof fails under the type
error
assumption:
 3.1 Type Restriction Invalidates Self-Reference
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Constructing D such that D(D) queries H(D, D) commits a type error. Since
H is of
type Decider, it cannot be passed as a Program to itself, nor can D
(derived from H)
be a valid input to H. The type system enforces that inputs to H are of
type Input,
excluding Decider or programs embedding H. Thus, H(D, D) is undefined, and
D's
construction is invalid.

And this is just repeating your error, H is of type PROGRAM, as your type "decider" is outside the bounds of the theory, allowing non-program oracle machines.
This just breaks the rest of your logic, and shows that you just don't understand what you are talking about.

 3.2 Collapse of Diagonalization Argument
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Linz's proof depends on H being applicable to all pairs (P, I), including
(D, D).
By disallowing self-referential inputs, the diagonalization step—where
D(D) halts
if and only if it does not—cannot be derived. Without this contradiction,
the proof
fails to show that H cannot exist.
 3.3 Decidability in a Typed Model
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In a typed system, a halting oracle H could decide halting for well-typed
inputs
(where P ≠ H and I does not encode H). Linz's proof assumes a type-free
model, but
in a typed model, the halting problem may be decidable for restricted
cases,
undermining the claim of universal undecidability.