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The Abiteboul–Vianu theorem on ordered structures

Lax134656.AbiteboulVianuOrdered · concepts/Lax134656/AbiteboulVianuOrdered.lean · lax-134656

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    Natural Language Statement

    Theorem

    On ordered structures, the inflationary and the partial fixed-point logics define the same problems if and only if PTIME = PSPACE: FO(≤\le, IFP) captures PTIME and FO(≤\le, PFP) captures PSPACE.

    Concept map
    22 concepts
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    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 Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Machines
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax535992.InflationaryFixedPoint
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax564036.Hierarchy
    13import Lax134656.SecondOrderTransitiveClosure
    14import Lax134656.OrderFreeTransitiveClosure
    15import Lax134656.PartialFixedPoint
    16import Lax134656.Qsat
    17import Lax134656.SuccinctReach
    18import Lax134656.SpaceBoundedMachines
    19import Lax134656.ClassPSPACE
    20
    21/-!
    22---
    23title: The Abiteboul–Vianu theorem on ordered structures
    24type: theorem
    25---
    26On ordered structures, the inflationary and the partial fixed-point logics
    27define the same problems if and only if PTIME = PSPACE: FO(≤\le, IFP)
    28captures PTIME and FO(≤\le, PFP) captures PSPACE.
    29-/
    30
    31namespace Lax134656.AbiteboulVianuOrdered
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    35open Lax904597.Classes Lax904597.Machines
    36open Lax485149.Problems Lax485149.Complement
    37open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    38open Lax564036.Hierarchy
    39open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    40open Lax134656.PartialFixedPoint
    41open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    42
    43/-- FO(≤, IFP) = FO(≤, PFP) exactly when PTIME = PSPACE. -/
    44axiom ifpDefinable_eq_pfpDefinable_iff_ptime_eq_pspace :
    45 (∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    46 IFPDefinable P ↔ PFPDefinable P) ↔ PTIME = PSPACE
    47
    48end Lax134656.AbiteboulVianuOrdered
    49
    Show Proof
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