Re: Position Paper: Why Simulating Halt Deciders (SHDs) Require a Reframing of the Halting Problem

On 5/24/25 1:25 PM, Mr Flibble wrote:

Position Paper: Why Simulating Halt Deciders (SHDs) Require a Reframing of
the Halting Problem
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 Thesis:
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Simulating Halt Deciders (SHDs), such as those proposed by Flibble and
Olcott, do not disprove the classical Halting Problem but instead operate
in a distinct computational framework. Therefore, they require a reframing
of the Halting Problem to accurately describe what is being solved.
 1. The Classical Halting Problem
--------------------------------
Definition:
Given a program P and input x, determine whether P(x) halts when executed.
 Requirements:
- The decider H must work for *all* valid programs P and inputs x.
- H must be total: it must give a correct answer for every input pair.
- The construction of D() = if H(D) then loop leads to contradiction.
 Implication:
- No universal halting decider exists.
 2. What SHDs Do Differently
---------------------------
SHDs simulate the behavior of P(x), often with the following properties:
- They detect infinite recursion or looping patterns *during simulation*.

And the queation becomes how they do this, when the imput program can contain a COPY of the algorithm of the SHD.
Part of your problem is that you have fallen for Olcotts very non-Turing Complete system which includes the stipulation that it is impossible to duplicate the code of his H, which he then violates when he creates his H1.
In actuality, both his H and D (and their variations) just fail to meet the actual definition of a Program from the basic theory, being a specific and complete set of instructions that the algorithm uses to process the data provided solely by the input to the program.

- They may abort early when a loop is structurally recognized.
- They return a result based on runtime behavior, not semantic derivation.
 In some models:
- SHDs may crash or refuse to decide on inputs that involve self-reference.
- They avoid the contradiction by forbidding or failing on such cases.
 3. Why This Requires Reframing
------------------------------
The classical Halting Problem assumes:
- All programs are valid analyzable inputs.

Which they are in the field.

- The decider exists *outside* the execution environment.

The decider is a seperate program. Note, in computation theory the programs *ARE* the execution environment. Programs are not run on computers, but ARE the computers, specially built for the task.
The equivalent of the modern computer would be the UTM, where we could prepare the "program" and its input as the input to the UTM.
And in this model, the UTM is given as the "Program" the decider, and the input is given as the input to that program.
And the input is defined in a way that it could also be run as an input to the UTM on its own, with its own input.
In others words, in te UTM model, if decider H is to decide on program P given input D, then we can say that
UTM <H> <P> <D> needs to answer about the behavior of UTM <P> <D>
Thus, H and P *ARE* defined in seperate execution environments, as we don't want to limit P to only be able to be run in the environment of H.

- Deciders can analyze any encoded program as data.

Which is the basic requirement.

 SHDs violate or restrict these assumptions:
- Some inputs are semantically ill-formed (e.g. self-referential
constructions).
- Deciders may exist *within* the runtime framework (e.g. as part of a
simulation).
- Programs are not just strings—they may include or exclude semantic
permissions.
 Reframed Problem:
"Given a well-formed, non-self-referential program P and input x, can an
SHD determine whether P(x) halts by simulation, within finite time?"
 This version:
- Recognizes semantic boundaries.
- Accepts partiality (SHDs may refuse to decide some inputs).
- Avoids contradiction by rejecting paradox-generating constructions.
 4. Analogy to Type Theory
--------------------------
In modern logic systems:
- Self-reference is blocked by stratified types or universe hierarchies.

but what stops the copying of the code from the other level of the hierarchy.
Thus FUNDAMENTALLY requries a limitation in the power of the layers.

- Proof assistants like Coq, Agda enforce termination through typing.
 Similarly, SHDs operate with:
- Semantic filters (not all programs are valid inputs).
- Safety over completeness.
 This makes them useful, but outside classical universality.
 5. Practical Implication
------------------------
Reframing the Halting Problem to fit SHDs allows:
- Simulation-based analyzers to be judged on their actual utility.
- SHDs to be integrated into verification pipelines (e.g. compilers,
interpreters).
- Clear boundaries between theoretical impossibility and practical
feasibility.
 Conclusion:
-----------
SHDs do not contradict the Halting Problem—they bypass it by refusing to
evaluate semantically ill-formed or reflexive inputs. This makes them
partial deciders operating in a semantically filtered domain.
 To evaluate them fairly, the Halting Problem must be reframed: not as a
universal challenge, but as a scoped analysis tool within a stratified or
simulation-based semantic model.

And, as pointed out, you need to answer the question of what you are doing to prevent the copying of algorithm from one level to another, or how to detect it with the ability to make unbounded variations when doing the copy.