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Theorem · static normalization

FPRD-T131

Guarded folds compute the full bisimulation quotient

Exact statement

On every finite weakly acyclic deterministic labelled partial port graph, elementary guarded pair folds generate exactly the greatest bisimulation equivalence. Orienting those folds toward their quotients terminates and is confluent up to rooted labelled port-graph isomorphism, with the canonical minimal residual figure as normal form.

StatusProved, independently rederived, and exhaustively tested on the declared finite domain
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Two equally labelled states can be folded when every corresponding port is missing on both sides, already has the same target, or stays within the pair being identified. The last guarded case handles SELF-recursive behavior.

Hypotheses and scope

  • Finite weakly acyclic deterministic labelled partial port graph.
  • Each fold is applied after previously imposed sound folds, in the current quotient.

Proof or evidence

Soundness follows because the fold makes the one-step records equal. For completeness, choose a minimal unreduced bisimulation class and fold two sinks, or its unique sink with a minimal remaining state. Every fold decreases the state count and all maximal sequences end at the unique minimal residual figure. The replayed checker found zero normal-form mismatches across 235,160 chronological graphs and no missing guarded certificate among 31,944 equal extension pairs.

Guarded folds generate bisimilarity →

Verification notes

The minimal-class completeness argument, preservation of weak acyclicity, and object-confluence boundary were rechecked independently; the finite checker was rerun successfully.

Limitations

  • Acyclic automaton minimization and general term-graph bisimulation collapse are prior art comparators.
  • The theorem gives one object normal form, not a finite presentation of all fold derivations.
  • Static folding may remove a physical state and is not yet a time-respecting history rewrite.

Open work

Keep object-level fold normalization separate from a presentation of all paths between fold sequences.

Notes

Ordinary acyclic-automaton state merging and general term-graph bisimulation collapse are established prior art. The guarded rule adds the weakly acyclic SELF case used by the causal lift. This is an object-normal-form theorem, not a finite presentation of all fold paths. No novelty claim is made.