Lax Lean Archive

NL = coNL

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

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    Natural Language Statement

    Theorem

    The classes NL and coNL are equal: a problem is SO-Krom definable if and only if its complement is. This is the Immerman–Szelepcsényi theorem for the class, obtained from the closure of FO(TC) under complement and the equality NL = FO(TC).

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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: NL = coNL
    24type: theorem
    25---
    26The classes NL and coNL are equal: a problem is SO-Krom definable if and
    27only if its complement is. This is the Immerman–Szelepcsényi theorem for the
    28class, obtained from the closure of FO(TC) under complement and the equality
    29NL = FO(TC).
    30-/
    31
    32namespace Lax485149.NLEqCoNL
    33
    34open FirstOrder FirstOrder.Language
    35open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    36open Lax904597.Classes Lax904597.Sat
    37open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    38open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    39open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    40open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    41
    42/-- NL is closed under complement. -/
    43axiom NL_eq_coNL : NL = coNL
    44
    45end Lax485149.NLEqCoNL
    46
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