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Lax765601.AperiodicOfFlipFlops

Compositions of flip-flop machines are aperiodic

concepts/Lax765601/AperiodicOfFlipFlops.lean · lax-765601

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    In the paper

    Theorem

    Every composition of flip-flop Mealy machines computes an aperiodic function: the implication "composition of flip-flops ⇒\Rightarrow aperiodic" of Theorem A.2.8 of Transducers. A single flip-flop is aperiodic, because the last letter of f(uvnw)f(u v^n w) depends only on the last letter read and on the last letter of uvwu v w whose state transformation is a constant, neither of which depends on nn; and aperiodicity is preserved by composition, by the pumping form of aperiodicity (Claim A.2.9), in which the shifts kk of the two functions add up.

    Lean source view on GitHub

    1import Lax765601.PrimeMealyMachines
    2import Lax765601.Aperiodicity
    3
    4/-!
    5---
    6title: Compositions of flip-flop machines are aperiodic
    7type: theorem
    8---
    9Every composition of flip-flop Mealy machines computes an aperiodic function:
    10the implication "composition of flip-flops ⇒\Rightarrow aperiodic" of Theorem
    11A.2.8 of *Transducers*. A single flip-flop is aperiodic, because the last letter
    12of f(uvnw)f(u v^n w) depends only on the last letter read and on the last letter of
    13uvwu v w whose state transformation is a constant, neither of which depends on
    14nn; and aperiodicity is preserved by composition, by the pumping form of
    15aperiodicity (Claim A.2.9), in which the shifts kk of the two functions add up.
    16
    17# Formalization notes
    18
    19The hypothesis is membership in the composition closure `CompClosure FlipFlopFam`
    20of the family of flip-flop machines. No finiteness of the alphabets is needed:
    21every member of the closure is a Mealy function, and the argument only uses its
    22state space.
    23-/
    24
    25namespace Lax765601.AperiodicOfFlipFlops
    26
    27open Lax765601.CompositionClosure Lax765601.PrimeMealyMachines Lax765601.Aperiodicity
    28
    29/-- A composition of flip-flop Mealy machines is aperiodic. -/
    30axiom aperiodic_of_compClosure_flipFlop {A B : Type} {f : List A → List B}
    31 (hf : CompClosure FlipFlopFam A B f) : Aperiodic f
    32
    33end Lax765601.AperiodicOfFlipFlops
    34
    Show Proof

    Formalization notes

    The hypothesis is membership in the composition closure CompClosureFlipFlopFamCompClosure FlipFlopFam of the family of flip-flop machines. No finiteness of the alphabets is needed: every member of the closure is a Mealy function, and the argument only uses its state space.

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