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Symmetric ABPs for bounded-pathwidth patterns

Lax461425.ABPsForPatterns · concepts/Lax461425/ABPsForPatterns.lean · lax-461425

proven

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    Natural Language Statement

    Theorem

    For n≥1n ≥ 1, the homomorphism polynomial of a pattern of pathwidth at most kk on n×nn × n matrices is computed by a matrix-symmetric ABP whose vertex orbits have size at most n(k+1)n^(k+1).

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    Lean source view on GitHub

    1import Lax461425.SymmetricABPs
    2import Lax461425.PatternWidths
    3
    4/-!
    5---
    6title: Symmetric ABPs for bounded-pathwidth patterns
    7type: theorem
    8---
    9For `n ≥ 1`, the homomorphism polynomial of a pattern of pathwidth at most `k` on `n × n`
    10matrices is computed by a matrix-symmetric ABP whose vertex orbits have size at most
    11`n^(k+1)`.
    12-/
    13
    14namespace Lax461425.ABPsForPatterns
    15
    16open SymmetricABPs HomomorphismPolynomials PatternWidths
    17
    18/-- Homomorphism polynomials of patterns of pathwidth at most `k` have matrix-symmetric ABPs
    19with vertex orbits of size at most `n^(k+1)`. -/
    20axiom exists_abp {K : Type*} [CommSemiring K] {n k : ℕ} (hn : 1 ≤ n)
    21 (F : Pattern) (hF : PathwidthLE k F) :
    22 ∃ G : ABP (Fin n × Fin n) K,
    23 G.Computes (hom K F n) ∧ G.IsMatrixSymmetric ∧ G.orbitSize ≤ n ^ (k + 1)
    24
    25end Lax461425.ABPsForPatterns
    26
    Show Proof
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