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PTIME is closed under first-order reductions

Lax535992.PTIMEClosure · concepts/Lax535992/PTIMEClosure.lean · lax-535992

proven

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    Natural Language Statement

    Theorem

    Membership in PTIME travels backward along first-order reductions, along ordered first-order reductions and along relativized ordered first-order reductions: if a problem reduces to a problem of PTIME, it is in PTIME. The Horn shape of a definition survives the pullback along an interpretation, which rewrites the guards only. Membership reads a problem on its finite instances only: two problems with the same finite yes-instances are both in PTIME or both outside.

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

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 PTIME_mem_congr_finite proven

    2 PTIME_mem_of_foReduction proven

    3 PTIME_mem_of_orderedReduction proven

    4 PTIME_mem_of_relOrderedReduction 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 Lax485149.Problems
    9import Lax485149.Complement
    10import Lax485149.SecondOrderAtoms
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.ClassNL
    14import Lax485149.ClassL
    15import Lax535992.HornFragment
    16import Lax535992.LeastFixedPoint
    17import Lax535992.InflationaryFixedPoint
    18import Lax535992.HornSat
    19import Lax535992.CircuitValue
    20import Lax535992.Game
    21import Lax535992.DeterministicMachines
    22import Lax535992.ClassPTIME
    23
    24/-!
    25---
    26title: PTIME is closed under first-order reductions
    27type: theorem
    28---
    29Membership in PTIME travels backward along first-order reductions, along
    30ordered first-order reductions and along relativized ordered first-order
    31reductions: if a problem reduces to a problem of PTIME, it is in PTIME. The
    32Horn shape of a definition survives the pullback along an interpretation,
    33which rewrites the guards only. Membership reads a problem on its finite
    34instances only: two problems with the same finite yes-instances are both in
    35PTIME or both outside.
    36-/
    37
    38namespace Lax535992.PTIMEClosure
    39
    40open FirstOrder FirstOrder.Language
    41open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    42open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    43open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    44open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    45open Lax485149.ClassNL Lax485149.ClassL
    46open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    47open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    48open Lax535992.ClassPTIME
    49
    50/-- Membership in PTIME travels backward along first-order reductions. -/
    51axiom PTIME_mem_of_foReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    52 {P : DecisionProblem L} {Q : DecisionProblem L'}, FOReduction P Q → PTIME.Mem Q → PTIME.Mem P
    53
    54/-- Membership in PTIME travels backward along ordered first-order reductions. -/
    55axiom PTIME_mem_of_orderedReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    56 {P : DecisionProblem L} {Q : DecisionProblem L'},
    57 OrderedFOReduction P Q → PTIME.Mem Q → PTIME.Mem P
    58
    59/-- Membership in PTIME travels backward along relativized ordered first-order
    60reductions. -/
    61axiom PTIME_mem_of_relOrderedReduction :
    62 ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    63 {P : DecisionProblem L} {Q : DecisionProblem L'},
    64 RelOrderedFOReduction P Q → PTIME.Mem Q → PTIME.Mem P
    65
    66/-- Membership in PTIME only depends on the finite instances of a problem. -/
    67axiom PTIME_mem_congr_finite : ∀ {L : Language.{0, 0}} [L.IsRelational] {P Q : DecisionProblem L},
    68 (∀ (A : Type) [L.Structure A] [Finite A], P A ↔ Q A) → (PTIME.Mem P ↔ PTIME.Mem Q)
    69
    70end Lax535992.PTIMEClosure
    71
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