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Lax916827.ConfigurationGraphRational

Computing the reachable configuration graph is rational

concepts/Lax916827/ConfigurationGraphRational.lean · lax-916827

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    In the paper

    Theorem

    The function which maps an input string to the string representation of its reachable configuration graph is rational (Lemma C.2.3 of Transducers). The main observation is that the representations of reachable configuration graphs form a regular language — reachability is checked locally, slice by slice — so that an automaton with output can guess the representation and verify it.

    Lean source view on GitHub

    1import Lax132576.RationalFunctions
    2import Lax916827.ConfigurationGraphs
    3
    4/-!
    5---
    6title: Computing the reachable configuration graph is rational
    7type: theorem
    8---
    9The function which maps an input string to the string representation of its
    10reachable configuration graph is rational (Lemma C.2.3 of *Transducers*). The
    11main observation is that the representations of reachable configuration graphs
    12form a regular language — reachability is checked locally, slice by slice — so
    13that an automaton with output can guess the representation and verify it.
    14
    15# Formalization notes
    16
    17The alphabet and the state space are assumed finite, so that the alphabet `C` is
    18finite; the output alphabet may be arbitrary. `enc M` is the representation of
    19`ConfigurationGraphs`.
    20-/
    21
    22namespace Lax916827.ConfigurationGraphRational
    23
    24open Lax132576.RationalFunctions Lax916827.TwoWayTransducers Lax916827.ConfigurationGraphs
    25
    26/-- The string representation of the reachable configuration graph is a rational
    27function of the input. -/
    28axiom isRationalFun_enc {A B Q : Type} [Finite A] [Finite Q] (M : TwoWay A B Q) :
    29 IsRationalFun (TwoWay.enc M)
    30
    31end Lax916827.ConfigurationGraphRational
    32
    Show Proof

    Formalization notes

    The alphabet and the state space are assumed finite, so that the alphabet CC is finite; the output alphabet may be arbitrary. encMenc M is the representation of ConfigurationGraphsConfigurationGraphs.

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