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Representation obstruction

CRD-RW-F2

Word sequences erase redex identity

Exact statement

For aa→aaa\to a, the sequence of underlying words is too coarse to recover redex independence or overlap: every complete occurrence-labelled reduction of ana^n projects to the same chain an→an−1→⋯→aa^n\to a^{n-1}\to\cdots\to a.

StatusRepresentation doctrine refuted for history classification
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The same semantic state sequence can arise from different choices of occurrences. A history representation must therefore be chosen to match the observation being studied.

Definitions

  • The word-sequence projection forgets the contracted occurrence and retains only the successive underlying words.
  • Residual provenance tracks how original positions survive contractions; it is a history annotation, not part of the semantic endpoint.

Hypotheses and scope

  • Complete occurrence-labelled reductions in R={aa→a}R=\{aa\to a\}.

Proof or evidence

Every step shortens aka^k to ak−1a^{k-1}, regardless of which adjacent pair is contracted. Thus all complete histories project to the identical word chain. Already at a3a^3, the two overlapping first choices become indistinguishable after projection.

Projection obstruction and replacement representation →

Verification notes

The projection argument is direct and was checked against CRD-F-6 and the canonical CRD-RW-F2 row. The repair is stated narrowly: occurrence-labelled edges are required for this independence question, not for every rewrite-theoretic task.

Limitations

  • Word sequences remain adequate for ordinary term-level reachability and confluence questions.
  • Putting provenance into endpoint equality would change the compared semantics by definition.

Open work

Keep occurrence, redex-position, and residual-provenance data whenever the question concerns independence of histories.