Lax Lean Archive

L ⊆ NL

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

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    Natural Language Statement

    Theorem

    Every problem of L is in NL: an FO(DTC) definable problem is FO(TC) definable, and FO(TC) definability is membership in NL.

    Concept map
    20 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    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: L ⊆ NL
    24type: theorem
    25---
    26Every problem of L is in NL: an FO(DTC) definable problem is FO(TC)
    27definable, and FO(TC) definability is membership in NL.
    28-/
    29
    30namespace Lax485149.LSubsetNL
    31
    32open FirstOrder FirstOrder.Language
    33open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    34open Lax904597.Classes Lax904597.Sat
    35open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    36open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    37open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    38open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    39
    40/-- L is contained in NL. -/
    41axiom LOGSPACE_subset_NL : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    42 LOGSPACE.Mem P → NL.Mem P
    43
    44end Lax485149.LSubsetNL
    45
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