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NL = FO(TC)

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

proven

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    Natural Language Statement

    Theorem

    A decision problem is in NL, that is, SO-Krom definable, if and only if it is FO(TC) definable. Equivalently, since FO(TC) is closed under complement, a problem is in NL if and only if its complement is FO(TC) definable.

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

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 mem_NL_iff_tcDefinable_compl proven

    2 tcDefinable_iff_mem_NL 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 = FO(TC)
    24type: theorem
    25---
    26A decision problem is in NL, that is, SO-Krom definable, if and only if it
    27is FO(TC) definable. Equivalently, since FO(TC) is closed under complement,
    28a problem is in NL if and only if its complement is FO(TC) definable.
    29-/
    30
    31namespace Lax485149.NLIsTransitiveClosure
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    35open Lax904597.Classes Lax904597.Sat
    36open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    37open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    38open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    39open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    40
    41/-- FO(TC) definability is membership in NL. -/
    42axiom tcDefinable_iff_mem_NL : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    43 TCDefinable P ↔ NL.Mem P
    44
    45/-- A problem is in NL exactly when its complement is FO(TC) definable. -/
    46axiom mem_NL_iff_tcDefinable_compl :
    47 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    48 NL.Mem P ↔ TCDefinable (DecisionProblem.compl P)
    49
    50end Lax485149.NLIsTransitiveClosure
    51
    Show ProofShow Proof
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