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Definition · observation models

FPRD-LRN-D01

Exact prefixes and revisable snapshots

Exact statement

For F(q)=∑n≥0anqn∈Q[[q]]F(q)=\sum_{n\ge0}a_nq^n\in\mathbb Q[[q]], an exact-prefix learner receives a0,…,aN−1a_0,\ldots,a_{N-1}, while a revisable-snapshot learner receives polynomials PtP_t satisfying only ∀n ∃Tn ∀t≥Tn:[qn]Pt=an\forall n\,\exists T_n\,\forall t\ge T_n:[q^n]P_t=a_n.

StatusPrecisely stated; the two models have different learnability
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

An exact polynomial prefix fixes zeros as well as nonzero coefficients below its frontier. A revisable snapshot supplies no detectable frontier of final coefficients.

Definitions

  • Identification in the limit means that the learner's hypotheses eventually equal the target forever.
  • A rational Q\mathbb Q-series is equivalently a rational linear recurrence sequence, a finite-rank Hankel sequence, or a unary weighted-automaton behavior.

Hypotheses and scope

  • Coefficients are exact rationals, not noisy numerical estimates.

Proof or evidence

The distinction makes the positive reconstruction theorem and negative diagonal theorem mathematically compatible rather than contradictory.

Observation models →

Verification notes

Exact prefixes, arbitrary positive texts, active query models, and pointwise-stabilizing snapshots were separated before importing learning results.

Limitations

  • Other coefficient domains, noise models, and probabilistic success criteria require separate statements.

Open work

Determine which observation model is supplied by the probabilistic-program application motivating the revisable examples.