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Values of the decision versions and of #SelSAT

Lax175070.CountingValues · concepts/Lax175070/CountingValues.lean · lax-175070

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    Natural Language Statement

    Lemma

    The decision version of a counting problem by a property of numbers holds on a structure exactly when its count has the property. The number of models of an instance with a selected variable in which the selected variables have a given value is invariant under isomorphism, so it is the value of #SelSATb_b; SelMajSAT and SelEqSAT hold exactly when these counts compare as they ask.

    Concept map
    23 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 decide_iff proven

    2 selEqSat_iff proven

    3 selMajSat_iff proven

    4 sharpSelSat_count_iso proven

    5 sharpSelSat_eq proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax485149.Problems
    3import Lax485149.Complement
    4import Lax904597.Classes
    5import Lax904597.Sat
    6import Lax904597.Interpretations
    7import Lax904597.Machines
    8import Lax564036.Hierarchy
    9import Lax366625.CountingProblems
    10import Lax366625.CountingClasses
    11import Lax366625.WitnessCounting
    12import Lax366625.CountingSat
    13import Lax366625.CountingRuns
    14import Lax175070.CountDefinability
    15import Lax175070.SelectedSat
    16
    17/-!
    18---
    19title: Values of the decision versions and of #SelSAT
    20type: lemma
    21---
    22The decision version of a counting problem by a property of numbers holds on
    23a structure exactly when its count has the property. The number of models of
    24an instance with a selected variable in which the selected variables have a
    25given value is invariant under isomorphism, so it is the value of
    26#SelSATb_b; SelMajSAT and SelEqSAT hold exactly when these counts compare
    27as they ask.
    28-/
    29
    30namespace Lax175070.CountingValues
    31
    32open FirstOrder FirstOrder.Language
    33open Lax904597.Problems Lax904597.Classes Lax904597.Sat Lax904597.Interpretations
    34 Lax904597.Machines Lax564036.Hierarchy
    35open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    36 Lax366625.CountingSat
    37open Lax366625.CountingRuns Lax175070.CountDefinability Lax175070.SelectedSat Lax485149.Problems
    38open Lax485149.Complement
    39
    40/-- The decision version holds when the count has the property. -/
    41axiom decide_iff :
    42 ∀ {L : Language.{0, 0}} [L.IsRelational] (R : ℕ → Prop) (C : CountingProblem L)
    43 (A : Type) [L.Structure A], decide R C A ↔ R (C A)
    44
    45/-- The number of models with the selected variables at `b` is isomorphism-invariant. -/
    46axiom sharpSelSat_count_iso :
    47 ∀ (b : Bool) {A B : Type} [satSel.Structure A] [satSel.Structure B],
    48 (A ≃[satSel] B) → Nat.card {ν : A → Prop // SelModel A b ν} = Nat.card {ν : B → Prop //
    49 SelModel B b ν}
    50
    51/-- The value of #SelSAT is the number it counts. -/
    52axiom sharpSelSat_eq :
    53 ∀ (b : Bool) (A : Type) [satSel.Structure A],
    54 SharpSelSAT b A = Nat.card {ν : A → Prop // SelModel A b ν}
    55
    56/-- SelMajSAT compares the two counts. -/
    57axiom selMajSat_iff :
    58 ∀ (A : Type) [satSel.Structure A], SelMajSAT A ↔ SharpSelSAT false A < SharpSelSAT true A
    59
    60/-- SelEqSAT compares the two counts. -/
    61axiom selEqSat_iff :
    62 ∀ (A : Type) [satSel.Structure A], SelEqSAT A ↔ SharpSelSAT true A = SharpSelSAT false A
    63
    64end Lax175070.CountingValues
    65
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