Re: Flibble’s Leap: Why Behavioral Divergence Implies a Type Distinction in the Halting Problem

On 2025-05-11 13:21:31 +0000, Mr Flibble said:

Flibble’s Leap: Why Behavioral Divergence Implies a Type Distinction in
the Halting Problem
===========================================================================================   Summary
-------
Flibble argues that the Halting Problem's undecidability proof is built on
a category (type) error: it assumes a program and its own representation
(as a finite string) are interchangeable. This assumption fails under
simulating deciders, revealing a type distinction through behavioral
divergence. As such, all deciders must respect this boundary, and
diagonalization becomes ill-formed. This reframing dissolves the paradox
by making the Halting Problem itself an ill-posed question.
 1. Operational Evidence of Type Distinction
-------------------------------------------
- When a program (e.g., `DD()`) is passed to a simulating halt decider
(`HHH`), it leads to infinite recursion.
- This behavior differs from direct execution (e.g., a crash due to a
stack overflow).
- This divergence shows that the program (as code) and its representation
(as data) are operationally distinct.
- Therefore, treating them as the same "type" (as Turing's proof does)
leads to inconsistency.
 2. Generalization to All Deciders
---------------------------------
- If simulation reveals this type mismatch, then no valid decider can rely
on conflating program and representation.
- Diagonalization (e.g., defining D(x) = if H(x,x) then loop else halt)
necessarily crosses the type boundary.
- The contradiction in the Halting Problem arises from this type error,
not from inherent undecidability.
- Hence, the Halting Problem is ill-defined for self-referential input.
 3. Comparisons to Other Formal Systems
--------------------------------------
- In type-theoretic systems (like Coq or Agda), such self-reference is
disallowed:
  - Programs must be well-typed.
  - Self-referential constructs like `H(x, x)` are unrepresentable if they
would lead to paradox.
- In category theory, morphisms must respect domain/codomain boundaries:
  - Reflexive constructions require stratification to avoid logical
inconsistency.
 4. Conclusion
-------------
- The Halting Problem depends on self-reference and diagonalization.
- If these constructs are blocked due to type/category errors, the proof
breaks down.
- Thus, the Halting Problem isn’t undecidable—it’s malformed when it
crosses type boundaries.
- This is not a refutation of Turing, but a reformulation of the problem
under more disciplined typing rules.
 This model mirrors how Russell’s Paradox was avoided in modern logic: not
by solving the contradiction, but by redefining the terms that made the
contradiction possible.

The halting problem has very simple types: the input is two strings
and the output is a bit. The same input can be given to a UTM, so
the input type of the halting problem is the input type of a UTM.
There is no divergence of behaviour: a UTM has only one behaviour for
each input and no other UTM needs be considered.
--
Mikko