Generalized Post Embedding Problems

The Regular Post Embedding Problem extended with partial (co)directness is shown decidable. This extends to universal and/or counting versions. It is also shown that combining directness and codirectness in Post Embedding problems leads to undecidability.

💡 Research Summary

The paper investigates extensions of the Regular Post Embedding Problem (PEP), a classic decision problem that asks, given two morphisms u, v : Σ* → Γ* and a regular language R ⊆ Σ*, whether there exists a word σ ∈ R such that u(σ) is a scattered subword of v(σ) (denoted u(σ) ⊑ v(σ)). PEP is known to be decidable but extremely hard (F_{ω^ω}‑complete).

The authors introduce two new variants that incorporate partial directness or partial codirectness. In PEP partial dir, a second regular language R′ ⊆ Σ* specifies which prefixes of a candidate solution must satisfy the embedding condition; all other prefixes are unrestricted. Dually, PEP partial co‑dir requires the condition only for suffixes belonging to R′. Setting R′ = ∅ recovers the original PEP, while R′ = Σ* yields the previously studied PEP dir (every prefix must embed). Thus the new problems strictly generalize both PEP and PEP dir.

The core technical contribution is a decidability proof for both PEP partial dir and PEP partial co‑dir. The proof proceeds by showing that any solution can be transformed into a short solution whose length is bounded by a computable function of the input. The bound is obtained through a combination of three ingredients:

1. Myhill‑congruence indices µ(R) and µ(R′). Since R and R′ are regular, their Myhill‑congruences have finite index; the product µ(R)·µ(R′) bounds the number of equivalence classes that suffixes of a candidate solution can fall into.

2. Higman’s Lemma (the well‑quasi‑ordering of words under the subword relation). This guarantees that any infinite sequence of suffixes must contain an increasing pair, which in turn yields a “cutting point” where a portion of the solution can be removed without breaking the embedding property.

3. Length‑Function Theorem for controlled bad sequences. By defining a k‑controlled sequence (where the length of the i‑th word is at most i·k) and an n‑bad sequence (one that contains no increasing subsequence of length n), the theorem provides a computable function H(n, k, Γ) that bounds the maximal length of an n‑bad k‑controlled sequence.

The authors set k to the maximal expansion factor K_u = max_{a∈Σ}|u(a)| and n to µ(R)·µ(R′)+1. The resulting bound L = H(n, K_u, Γ) is effectively computable. They then prove two “cutting lemmas”:

• Lemma 6 (blue indices) – If two positions a < b are congruent, both satisfy u(i,N) ⊑ v(i,N) (called blue), and the left‑margin string at a is a subword of the left‑margin at b, then the segment σ