Codex maintained their focused, well-structured notes in
projects/jca/notes_undecidability_codex.md. Their approach
emphasizes precise formalization of the semantic gap between SREU’s
“rigid derivability under assumptions” and JCAQP’s “truth in the fixed
model T(F) with consistency requirement.” They made four main proof
attempts (direct SREU, internalize derivability, computation history
encoding, semi/higher-order unification) and documented each failure
point.
Clean formalization of the core lemma: The lemma
stating that in T(F), D ⊨ (s ≈ t) iff sσ and
tσ are syntactically identical (where σ is mgu of D) is
precisely stated and clearly distinguishes T(F) semantics from
equational derivability.
Concise and well-organized: 161 lines covering the essential analysis - no redundancy, clear section structure.
Good use of source material: References to Voronkov (1999) and Degtyarev-Voronkov (1996) are integrated to explain why SREU’s power comes from proof-theoretic entailment.
Speculative decidability angle: The “no-escaping as definable functions” observation is interesting - connecting dependent existentials to pattern unification is a novel angle I hadn’t explicitly considered.
Codex traded breadth for precision: Their notes are more focused on formalizing the exact barrier, while mine explored more alternative reduction sources (PCP, shifted pairing, jointness).
Less emphasis on the Maher papers: My notes now include the critical finding that Maher (2005) explicitly states JCA decidability is “unknown” - this literature confirmation seems absent from codex’s notes.
No jointness exploration: My notes explored whether JCA’s irreducibility to SCA (Maher Example 12) could be a source of complexity. Codex didn’t pursue this direction.
The “pattern unification” connection: If JCAQP’s no-escaping condition restricts answers to patterns (first-order definable functions), this could be a path to decidability. Higher-order pattern unification is decidable, so this analogy is worth exploring.
The “no-escaping as function variables” speculation: This reframing of the dependency constraint is valuable - it suggests looking at the problem through the lens of restricted higher-order unification rather than first-order constraint solving.
The “computation history” failure is fundamental: Their observation that “equations can only constrain finitely many positions” is essentially the same barrier I identified with universal quantification over subtrees.
Did codex see the Maher (2005) statement that JCA decidability is explicitly “unknown”? This literature confirmation seems important for establishing that the open problem is genuine.
The “pattern unification” angle seems promising for decidability - has codex considered how the quantifier prefix structure maps to the pattern restriction in higher-order unification?
Both approaches have now converged on the same fundamental conclusion: the consistency condition creates a semantic gap that blocks SREU-style undecidability proofs. The main divergence is in speculative directions: codex leans toward decidability via pattern unification analogies, while I’ve been more neutral (documenting barriers without committing to either direction).