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NL is closed under first-order reductions

Lax485149.NLClosure · concepts/Lax485149/NLClosure.lean · lax-485149

proven

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    Natural Language Statement

    Theorem

    Membership in NL travels backward along first-order reductions and along ordered first-order reductions: if a problem reduces to a problem of NL, it is in NL. Membership reads a problem on its finite instances only: two problems with the same finite yes-instances are both in NL or both outside.

    The Krom shape of a definition survives the pullback along an interpretation.

    Concept map
    20 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 NL_mem_congr_finite proven

    2 NL_mem_of_foReduction proven

    3 NL_mem_of_orderedReduction 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 Lax485149.Problems
    8import Lax485149.Complement
    9import Lax485149.SecondOrderAtoms
    10import Lax485149.KromFragment
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.FirstOrderDefinability
    14import Lax485149.HeadAutomata
    15import Lax485149.Reachability
    16import Lax485149.DeterministicReachability
    17import Lax485149.TwoSat
    18import Lax485149.ClassNL
    19import Lax485149.ClassL
    20
    21/-!
    22---
    23title: NL is closed under first-order reductions
    24type: theorem
    25---
    26Membership in NL travels backward along first-order reductions and along
    27ordered first-order reductions: if a problem reduces to a problem of NL, it
    28is in NL. Membership reads a problem on its finite instances only: two
    29problems with the same finite yes-instances are both in NL or both
    30outside.
    31
    32The Krom shape of a definition survives the pullback along an interpretation.
    33-/
    34
    35namespace Lax485149.NLClosure
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    39open Lax904597.Classes Lax904597.Sat
    40open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    41open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    42open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    43open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    44
    45/-- Membership in NL travels backward along first-order reductions. -/
    46axiom NL_mem_of_foReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    47 {P : DecisionProblem L} {Q : DecisionProblem L'}, FOReduction P Q → NL.Mem Q → NL.Mem P
    48
    49/-- Membership in NL travels backward along ordered first-order reductions. -/
    50axiom NL_mem_of_orderedReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    51 {P : DecisionProblem L} {Q : DecisionProblem L'}, OrderedFOReduction P Q → NL.Mem Q → NL.Mem P
    52
    53/-- Membership in NL only depends on the finite instances of a problem. -/
    54axiom NL_mem_congr_finite : ∀ {L : Language.{0, 0}} [L.IsRelational] {P Q : DecisionProblem L},
    55 (∀ (A : Type) [L.Structure A] [Finite A], P A ↔ Q A) → (NL.Mem P ↔ NL.Mem Q)
    56
    57end Lax485149.NLClosure
    58
    Show ProofShow ProofShow Proof
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