Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

First-order reductions are strictly weaker than logarithmic-space reductions

Lax945089.ReductionsBelowLogSpace · concepts/Lax945089/ReductionsBelowLogSpace.lean · lax-945089

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

    There is a problem QQ to which EVEN reduces by an FO(DTC) reduction but by no ordered first-order reduction. The first-order reductions under which the complete problems of these submissions are complete are therefore strictly weaker than deterministic logarithmic-space reductions, in their logical form.

    Concept map
    27 concepts
    100%
    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 Mathlib.ModelTheory.Order
    2import Mathlib.Order.Defs.LinearOrder
    3import Lax904597.Problems
    4import Lax904597.Interpretations
    5import Lax904597.Relativized
    6import Lax904597.SecondOrder
    7import Lax904597.Classes
    8import Lax485149.Problems
    9import Lax485149.FirstOrderDefinability
    10import Lax485149.TransitiveClosure
    11import Lax485149.DeterministicTransitiveClosure
    12import Lax485149.ClassNL
    13import Lax485149.ClassL
    14import Lax535992.InflationaryFixedPoint
    15import Lax535992.ClassPTIME
    16import Lax134656.PartialFixedPoint
    17import Lax895169.ArithmeticLogic
    18import Lax945089.OrderFreeFirstOrder
    19import Lax945089.EhrenfeuchtGames
    20import Lax945089.PebbleGames
    21import Lax945089.Even
    22import Lax945089.Parity
    23import Lax945089.TransitiveClosureReductions
    24
    25/-!
    26---
    27title: First-order reductions are strictly weaker than logarithmic-space reductions
    28type: theorem
    29---
    30There is a problem QQ to which EVEN reduces by an FO(DTC) reduction but by
    31no ordered first-order reduction. The first-order reductions under which
    32the complete problems of these submissions are complete are therefore
    33strictly weaker than deterministic logarithmic-space reductions, in their
    34logical form.
    35-/
    36
    37namespace Lax945089.ReductionsBelowLogSpace
    38
    39open FirstOrder FirstOrder.Language
    40open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    41open Lax904597.Classes
    42open Lax485149.Problems Lax485149.FirstOrderDefinability Lax485149.TransitiveClosure
    43open Lax485149.DeterministicTransitiveClosure Lax485149.ClassNL Lax485149.ClassL
    44open Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    45open Lax134656.PartialFixedPoint Lax895169.ArithmeticLogic
    46open Lax945089.OrderFreeFirstOrder Lax945089.EhrenfeuchtGames Lax945089.PebbleGames
    47open Lax945089.Even Lax945089.Parity
    48open Lax945089.TransitiveClosureReductions
    49
    50/-- Some problem receives EVEN by an FO(DTC) reduction and by no ordered
    51first-order reduction. -/
    52axiom exists_dtcReduction_not_orderedReduction :
    53 ∃ (L : Language.{0, 0}) (_ : L.IsRelational) (Q : DecisionProblem L),
    54 Nonempty (DTCReduction EVEN Q) ∧ IsEmpty (OrderedFOReduction EVEN Q)
    55
    56end Lax945089.ReductionsBelowLogSpace
    57
    Show Proof
    Builds on
    Used by

    none

    From Mathlib

    Discussion

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

    Loading discussion…