Review of claude’s Approach (Round 1)

Summary

Claude compiled a comprehensive notes file in prove_JCA_undecidable.md, restating the JCAQP definition, constraints (fixed signature, existence problem), and surveying candidate undecidable sources (semi-unification, SREU, second-order unification, PCP). They explored reductions mainly from semi-unification and from computation/PCP encodings using term-structured traces, then documented why each path hits expressivity barriers in the pure equational language over T(F).

Strengths

• Clear structure and coverage of multiple candidate source problems.
• Explicit discussion of the “assumption gap” for SREU under fixed-model semantics.
• Concrete worked example for semi-unification, which helps show why a naive encoding is too strong.
• Good articulation of the core obstacle: equational constraints cannot enforce recursive structure or “for all subpositions” checks.

Different Tradeoffs

• Claude prioritized breadth: surveying multiple undecidable sources and outlining several partial encodings, trading off depth on any single reduction.
• My approach focused more on formalizing the semantic gap (entailment in T(F) vs. rigid derivability) and on fixed-signature notes, with less exploration of PCP/TM encodings.

Ideas Worth Noting

• The semi-unification angle might be the most promising among classic undecidable problems because it is closer to pure unification/matching than SREU.
• The trace-encoding attempt underlines a key constraint: without pattern matching or recursive predicates, verifying “all steps” seems infeasible—this could be formalized as a limitation lemma.

Questions

• Did you find any specific semi-unification variant (e.g., one-step semi-unification or a restricted fragment) that seems closer to expressible in JCAQP?
• Would it help to formalize an impossibility result for expressing “all subterms satisfy property P” using only equations and finite prefixes, to justify why TM/PCP encodings fail?