Efficient learning beyond PTLD

Identify semantic conditions strictly weaker than prime-target left-division determinism (PTLD) but stronger than bare FSRP that yield an effective a priori bound or a polynomial-time procedure for recovering the minimal canonical residual controller.

Background

The paper provides polynomial-time strong structural reconstruction for the PTLD branch, while the more general FSRP branch is handled by a computable limit reconstruction based on enumeration of finite automata.

The unresolved learning problem is to find intermediate semantic hypotheses that retain efficient canonical controller recovery without requiring PTLD. The authors specifically mention bounded residual splitting, bounded exact-factor multiplicity, and bounded defect-state parameters as possible substitutes for prime-target thinness.

References

Which semantic conditions, weaker than PTLD but stronger than bare FSRP, provide an effective a priori bound or a polynomial procedure for recovering the minimal controller? In particular, can bounded residual splitting, bounded exact-factor multiplicity, or a bounded defect-state parameter replace prime-target thinness in an efficient strong learner?

— Relative Prime Factorization and Finite-State Presentations under Fixed Finite-Monoid Observation  (2609.03643 - Kuriyama, 3 Sep 2026) in Section 13, “Open problems and next steps,” Question: Efficient learning beyond PTLD