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Binary encodings of graphs and positive CNF formulas

Lax783278.Encoding · concepts/Lax783278/Encoding.lean · lax-783278

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    Natural Language Statement

    Definition

    A graph on nn labeled vertices is encoded by 1n01^n0 followed by its n×nn\times n adjacency matrix in row order. A formula on nn variables with mm clauses is encoded by 1n0 1m01^n0\,1^m0 followed by the m×nm\times n clause-variable incidence matrix in row order. These encodings retain isolated vertices, unused variables, empty clauses, and repeated clauses. Malformed strings are excluded from the associated languages.

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    1import Lax783278.ArcKayles
    2import Lax783278.PositiveCNF
    3import Lax434930.PolynomialTime
    4
    5/-!
    6---
    7title: Binary encodings of graphs and positive CNF formulas
    8type: definition
    9---
    10A graph on nn labeled vertices is encoded by 1n01^n0 followed by its
    11n×nn\times n adjacency matrix in row order. A formula on nn variables
    12with mm clauses is encoded by 1n0 1m01^n0\,1^m0 followed by the m×nm\times n
    13clause-variable incidence matrix in row order. These encodings retain
    14isolated vertices, unused variables, empty clauses, and repeated clauses.
    15Malformed strings are excluded from the associated languages.
    16-/
    17
    18namespace Lax783278.Encoding
    19
    20open scoped Classical
    21
    22open Lax434930.PolynomialTime
    23
    24structure Graph where
    25 vertices : ℕ
    26 graph : SimpleGraph (Fin vertices)
    27
    28noncomputable def graphWord (G : Graph) : Word :=
    29 List.replicate G.vertices true ++ [false] ++
    30 (List.finRange G.vertices).flatMap fun u =>
    31 (List.finRange G.vertices).map fun v => decide (G.graph.Adj u v)
    32
    33def formulaWord (φ : PositiveCNF.Formula) : Word :=
    34 List.replicate φ.nvars true ++ [false] ++
    35 List.replicate φ.clauses.length true ++ [false] ++
    36 φ.clauses.flatMap fun C =>
    37 (List.finRange φ.nvars).map fun x => decide (x ∈ C)
    38
    39def arcKayles : Language :=
    40 {w | ∃ G : Graph, graphWord G = w ∧ ArcKayles.Winning G.graph Finset.univ}
    41
    42def positiveCNF : Language :=
    43 {w | ∃ φ : PositiveCNF.Formula, formulaWord φ = w ∧ PositiveCNF.FirstWins φ}
    44
    45end Lax783278.Encoding
    46
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