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REACH is NL-complete

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

proven

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    Natural Language Statement

    Theorem

    REACH is NL-complete under first-order reductions. It is FO(TC) definable, by a specification of arity one with a single mode that walks along the edges from the marked sources to the marked targets, and every FO(TC) definable problem reduces to it by an ordered first-order reduction that builds the graph of the specification, its nodes the tagged tuples.

    Concept map
    20 concepts
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    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 reach_hard_of_tcDefinable proven

    2 reach_NL_complete proven

    3 reach_tcDefinable 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: REACH is NL-complete
    24type: theorem
    25---
    26REACH is NL-complete under first-order reductions. It is FO(TC) definable,
    27by a specification of arity one with a single mode that walks along the
    28edges from the marked sources to the marked targets, and every FO(TC)
    29definable problem reduces to it by an ordered first-order reduction that
    30builds the graph of the specification, its nodes the tagged tuples.
    31-/
    32
    33namespace Lax485149.ReachNLComplete
    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/-- REACH is a single transitive closure. -/
    44axiom reach_tcDefinable : TCDefinable REACH
    45
    46/-- Every FO(TC) definable problem reduces to REACH. -/
    47axiom reach_hard_of_tcDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    48 TCDefinable P → Nonempty (OrderedFOReduction P REACH)
    49
    50/-- REACH is NL-complete. -/
    51axiom reach_NL_complete : NL.Complete REACH
    52
    53end Lax485149.ReachNLComplete
    54
    Show ProofShow ProofShow Proof
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