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

On 5/24/2025 12:25 PM, Mr Flibble wrote:

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

My SHD refutes the classical halting problem proof
that same way that ZFC refutes Russell's Paradox.
Instead of requiring a termination analyzer to report
on the behavior of its caller it reports on the actual
behavior actually specified by its actual input.
*This was probably the actual intent all along*
It merely took 90 years before anyone really
studied how pathological inputs actually do
change their own behavior by their pathology.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer