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FO(≤, IFP) = FO(LFP) = PTIME

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

proven

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    Natural Language Statement

    Theorem

    On ordered structures, a decision problem is FO(≤\le, IFP) definable if and only if it is FO(LFP) definable, hence if and only if it is in PTIME. A rule system is one simultaneous step, inflation supplying the monotonicity, which gives one direction. Conversely, an inflationary induction is compiled into FO(LFP) by walking its stages along the order, with an evaluator that derives positively both the truth and the falsity of the subformulas of the step formulas at each stage; inflation is what makes the complement of a stage advance positively.

    Concept map
    24 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 ifpDefinable_iff_mem_PTIME proven

    2 ifpDefinable_lfpDefinable proven

    3 lfpDefinable_ifpDefinable 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 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: FO(≤, IFP) = FO(LFP) = PTIME
    27type: theorem
    28---
    29On ordered structures, a decision problem is FO(≤\le, IFP) definable if
    30and only if it is FO(LFP) definable, hence if and only if it is in PTIME.
    31A rule system is one simultaneous step, inflation supplying the
    32monotonicity, which gives one direction. Conversely, an inflationary
    33induction is compiled into FO(LFP) by walking its stages along the order,
    34with an evaluator that derives positively both the truth and the falsity of
    35the subformulas of the step formulas at each stage; inflation is what makes
    36the complement of a stage advance positively.
    37-/
    38
    39namespace Lax535992.InflationaryIsLeastFixedPoint
    40
    41open FirstOrder FirstOrder.Language
    42open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    43open Lax904597.Classes Lax904597.Sat Lax904597.Machines
    44open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms
    45open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    46open Lax485149.ClassNL Lax485149.ClassL
    47open Lax535992.HornFragment Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint
    48open Lax535992.HornSat Lax535992.CircuitValue Lax535992.Game Lax535992.DeterministicMachines
    49open Lax535992.ClassPTIME
    50
    51/-- Every FO(LFP) definable problem is FO(≤, IFP) definable. -/
    52axiom lfpDefinable_ifpDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    53 LFPDefinable P → IFPDefinable P
    54
    55/-- Every FO(≤, IFP) definable problem is FO(LFP) definable. -/
    56axiom ifpDefinable_lfpDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    57 IFPDefinable P → LFPDefinable P
    58
    59/-- FO(≤, IFP) definability is membership in PTIME. -/
    60axiom ifpDefinable_iff_mem_PTIME : ∀ {L : Language.{0, 0}} [L.IsRelational] (P : DecisionProblem L),
    61 IFPDefinable P ↔ PTIME.Mem P
    62
    63end Lax535992.InflationaryIsLeastFixedPoint
    64
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