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

Thesis:
-------
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*.
- 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.
- The decider exists *outside* the execution environment.
- Deciders can analyze any encoded program as data.

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.
- 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.

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