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The Immerman–Vardi theorem: PTIME = FO(LFP)

Lax535992.ImmermanVardi · concepts/Lax535992/ImmermanVardi.lean · lax-535992

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    Natural Language Statement

    Theorem

    A decision problem is in PTIME if and only if it is FO(LFP) definable, on ordered structures. With PTIME defined by the Horn fragment, this is the Immerman–Vardi theorem read through the equivalence of SO-Horn and FO(LFP).

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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.Sat
    7import Lax904597.Machines
    8import Lax485149.Problems
    9import Lax485149.Complement
    10import Lax485149.SecondOrderAtoms
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.ClassNL
    14import Lax485149.ClassL
    15import Lax535992.HornFragment
    16import Lax535992.LeastFixedPoint
    17import Lax535992.InflationaryFixedPoint
    18import Lax535992.HornSat
    19import Lax535992.CircuitValue
    20import Lax535992.Game
    21import Lax535992.DeterministicMachines
    22import Lax535992.ClassPTIME
    23
    24/-!
    25---
    26title: The Immerman–Vardi theorem: PTIME = FO(LFP)
    27type: theorem
    28---
    29A decision problem is in PTIME if and only if it is FO(LFP) definable, on
    30ordered structures. With PTIME defined by the Horn fragment, this is the
    31Immerman–Vardi theorem read through the equivalence of SO-Horn and
    32FO(LFP).
    33-/
    34
    35namespace Lax535992.ImmermanVardi
    36
    37open FirstOrder FirstOrder.Language
    38open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    39open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    40open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    41open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    42open Lax485149.ClassNL Lax485149.ClassL
    43open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    44open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    45open Lax535992.ClassPTIME
    46
    47/-- FO(LFP) definability is membership in PTIME. -/
    48axiom lfpDefinable_iff_mem_PTIME : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    49 LFPDefinable P ↔ PTIME.Mem P
    50
    51end Lax535992.ImmermanVardi
    52
    Show Proof
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