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Definition

FPRD-D21

Controlled-Dyck future quotients

Exact statement

A controlled-Dyck future-quotient datum combines a generalized-Dyck stack alphabet with a finite total controller, the start-reachable stack/controller dynamic and absorbing failure state, bounded-height carriers with a tail point, at-most-length and exact-length Boolean future signatures, reachable graded summaries preserving readout, bonding, and named actions, and the corresponding fixed-length Hankel cuts and layered recognizers.

StatusPrecisely stated with reachable and relative-minimality boundaries
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The finite controller restricts which typed stacks and controller states are actually reachable. Quotients are therefore built from the reachable dynamic rather than from every syntactically possible stack/controller pair.

Definitions

  • A raw live state is (s,q)(s,q), with opener stack ss and controller state qq; mismatched closing or underflow enters an absorbing failure state ⊥\bot.
  • The bounded-height carrier CNC_N retains reachable live states of height at most NN and sends failure and deeper stacks to one tail point.
  • The bounded signature BN(x)=(o(x⋅w))∣w∣≤NB_N(x)=(o(x\cdot w))_{|w|\le N} records all Boolean futures through horizon NN; the exact signature Ej(x)E_j(x) records only suffixes of length jj.
  • A valid graded summary is a reachable quotient preserving current output, bonding maps, and the named degree-lowering letter actions.

Hypotheses and scope

  • The controller and generalized-Dyck alphabet are finite, and controller transitions are total.
  • Only states reachable from the empty stack and initial controller state enter the carriers.
  • Minimum and uniqueness are relative to the declared category of surjective reachable graded summaries.

Proof or evidence

The specification fixes the raw carrier, reachability, finite truncations, Boolean observations, quotient category, and two observation scales used by the later theorems.

Controlled-Dyck quotient specification →

Verification notes

The definition was reconciled with the M24 proof and closure audit. In particular, raw all-controller-state counts are only upper carrier bounds, and exact-length partitions are coarser than bounded-horizon partitions at the same horizon.

Limitations

  • The definition does not claim unrestricted visibly-pushdown control-state minimality.
  • A redundant refinement is still a valid summary; reducedness is required for canonical uniqueness.
  • The tail point is a finite-observation device and is not automatically identical to semantic failure at every horizon.

Open work

Add a two-state controller example in which distinct raw configurations merge under a bounded future signature.

Notes

The raw all-controller-state geometric count is only an upper carrier bound. The definition fixes Boolean future observation and a relative graded-summary category; it does not define unrestricted VPA control-state minimization.