Lax Lean Archive

REACHd is L-complete

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    REACHd, deterministic reachability, is L-complete under first-order reductions. It is FO(DTC) definable, by the determinization of the specification that defines REACH, and every FO(DTC) definable problem reduces to it by an ordered first-order reduction that builds the graph of the determinized specification.

    Concept map
    20 concepts
    100%
    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 reachd_dtcDefinable proven

    2 reachd_hard_of_dtcDefinable proven

    3 reachd_LOGSPACE_complete 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: REACHd is L-complete
    24type: theorem
    25---
    26REACHd, deterministic reachability, is L-complete under first-order
    27reductions. It is FO(DTC) definable, by the determinization of the
    28specification that defines REACH, and every FO(DTC) definable problem
    29reduces to it by an ordered first-order reduction that builds the graph of
    30the determinized specification.
    31-/
    32
    33namespace Lax485149.ReachdLComplete
    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/-- REACHd is a single deterministic transitive closure. -/
    44axiom reachd_dtcDefinable : DTCDefinable REACHd
    45
    46/-- Every FO(DTC) definable problem reduces to REACHd. -/
    47axiom reachd_hard_of_dtcDefinable :
    48 ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    49 DTCDefinable P → Nonempty (OrderedFOReduction P REACHd)
    50
    51/-- REACHd is L-complete. -/
    52axiom reachd_LOGSPACE_complete : LOGSPACE.Complete REACHd
    53
    54end Lax485149.ReachdLComplete
    55
    Show ProofShow ProofShow Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…