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FP is closed under parsimonious reductions

Lax366625.FPClosure · concepts/Lax366625/FPClosure.lean · lax-366625

proven

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    Natural Language Statement

    Theorem

    Membership in FP travels backward along ordered parsimonious reductions and along relativized ordered parsimonious reductions, and reads a problem on its finite instances only.

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

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 FP_mem_congr_finite proven

    2 FP_mem_of_orderedParsimonious proven

    3 FP_mem_of_relOrderedParsimonious proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Sat
    7import Lax904597.Machines
    8import Lax535992.HornSat
    9import Lax535992.CircuitValue
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax366625.CountingProblems
    13import Lax366625.CountingClasses
    14import Lax366625.WitnessCounting
    15import Lax366625.SecondOrderCounting
    16import Lax366625.QuantitativeLogic
    17import Lax366625.CountingSat
    18import Lax366625.MachineNumbers
    19import Lax366625.CountingRuns
    20import Lax366625.NumberedCircuits
    21import Lax366625.HornNumbers
    22
    23/-!
    24---
    25title: FP is closed under parsimonious reductions
    26type: theorem
    27---
    28Membership in FP travels backward along ordered parsimonious reductions and
    29along relativized ordered parsimonious reductions, and reads a problem on its
    30finite instances only.
    31-/
    32
    33namespace Lax366625.FPClosure
    34
    35open FirstOrder FirstOrder.Language
    36open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    37open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    38open Lax535992.ClassPTIME
    39open Lax366625.CountingProblems Lax366625.CountingClasses Lax366625.WitnessCounting
    40open Lax366625.SecondOrderCounting
    41open Lax366625.QuantitativeLogic Lax366625.CountingSat Lax366625.MachineNumbers
    42open Lax366625.CountingRuns
    43open Lax366625.NumberedCircuits Lax366625.HornNumbers
    44
    45/-- Membership in FP travels backward along ordered parsimonious reductions. -/
    46axiom FP_mem_of_orderedParsimonious : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    47 {C : CountingProblem L} {D : CountingProblem L'},
    48 OrderedParsimoniousReduction C D → FP.Mem D → FP.Mem C
    49
    50/-- Membership in FP travels backward along relativized ordered
    51parsimonious reductions. -/
    52axiom FP_mem_of_relOrderedParsimonious :
    53 ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    54 {C : CountingProblem L} {D : CountingProblem L'},
    55 RelOrderedParsimoniousReduction C D → FP.Mem D → FP.Mem C
    56
    57/-- Membership in FP only depends on the finite instances of a problem. -/
    58axiom FP_mem_congr_finite : ∀ {L : Language.{0, 0}} [L.IsRelational] {C D : CountingProblem L},
    59 (∀ (A : Type) [L.Structure A] [Finite A], C A = D A) → (FP.Mem C ↔ FP.Mem D)
    60
    61end Lax366625.FPClosure
    62
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