Lax Lean Archive

UNREACHd is L-complete

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

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    Natural Language Statement

    Theorem

    UNREACHd, the complement of REACHd, is L-complete under first-order reductions. A deterministic walk on a finite graph that does not arrive within as many steps as there are vertices never arrives, so non-arrival is itself witnessed by a deterministic walk that counts its steps; no analogue of inductive counting is needed.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    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: UNREACHd is L-complete
    24type: theorem
    25---
    26UNREACHd, the complement of REACHd, is L-complete under first-order
    27reductions. A deterministic walk on a finite graph that does not arrive
    28within as many steps as there are vertices never arrives, so non-arrival is
    29itself witnessed by a deterministic walk that counts its steps; no analogue
    30of inductive counting is needed.
    31-/
    32
    33namespace Lax485149.UnreachdLComplete
    34
    35open FirstOrder FirstOrder.Language
    36open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    37open Lax904597.Classes Lax904597.Sat
    38open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    39open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    40open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    41open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    42
    43/-- UNREACHd is L-complete. -/
    44axiom unreachd_LOGSPACE_complete : LOGSPACE.Complete UNREACHd
    45
    46end Lax485149.UnreachdLComplete
    47
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