Codex produced a well-structured analysis of the JCAQP undecidability
problem in projects/jca/notes_undecidability_codex.md. They
focused primarily on attempting an SREU reduction and systematically
documenting why it fails, then explored three alternative approaches
(internalizing derivability, computation history encoding, and
higher-order unification).
Clean formalization of the entailment lemma:
Codex explicitly stated that in T(F), D ⊨ (s ≈ t) iff
sσ and tσ are syntactically identical where σ
is an mgu of D. This is a crisp characterization I didn’t explicitly
formulate.
Fixed-signature observation: They correctly noted that the fixed signature constraint is “not the main obstacle” since symbols can be encoded as data (e.g., via a binary constructor). This is an important point that simplifies the problem space.
Systematic attempt organization: The notes clearly label each attempt (1-4) with the core idea and where it fails, making the analysis easy to follow.
Correct identification of the main obstruction: “SREU-style hardness uses rigid derivability under assumptions, but JCAQP uses truth in the fixed free term algebra plus consistency (nonvacuity).”
Codex focused depth on SREU: They gave more detailed treatment to the SREU encoding attempt and the specific role of the consistency condition in blocking it.
I focused more on semi-unification and structural encoding: My notes spent more time on semi-unification as an alternative source, and on the challenge of expressing “for all subtrees” verification.
Codex was more concise: Their notes are about 1/10th the length of mine, which has tradeoffs - less exploration but cleaner conclusions.
The “no-escaping-variables” direction: Codex’s suggestion to explore whether the dependency restriction could simulate function variables (treating dependent existentials as “function variables”) is intriguing. I didn’t explore this angle in depth.
The “always unifiable premises” framing: Looking for undecidable problems where premises are inherently unifiable (rather than trying to encode problems that use inconsistent premises) is a good reframing of the search space.
Did codex consider the role of jointness (multiple implications) as a potential source of complexity, independent of encoding computation?
The notes mention “rigid reachability” as a possible direction - is this a specific problem from the literature, or a general concept?
Both approaches reached the same fundamental conclusion: the pure equational language of JCAQP over T(F) appears too weak to encode unbounded verification. We both got stuck at the same barrier - the inability to express inductive/recursive checking over term structure using only equations and a fixed quantifier prefix. This convergence suggests the barrier is real, not an artifact of a particular approach.