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FO(≤) ⊊ AC⁰: EVEN with arithmetic

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

proven

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    Natural Language Statement

    Theorem

    EVEN is AC⁰ definable: with the arithmetic of the order, the greatest element has rank one less than the size of the universe, and a sentence states its parity. Since EVEN is not FO(≤\le) definable, the inclusion of FO(≤\le) in AC⁰ is strict. This is the parity of the size of the input; the parity of a marked subset is PARITY, about which no bound is claimed here.

    Concept map
    27 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 even_ac0Definable proven

    2 exists_ac0Definable_not_foDefinable proven

    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: FO(≤) ⊊ AC⁰: EVEN with arithmetic
    28type: theorem
    29---
    30EVEN is AC⁰ definable: with the arithmetic of the order, the greatest
    31element has rank one less than the size of the universe, and a sentence
    32states its parity. Since EVEN is not FO(≤\le) definable, the inclusion of
    33FO(≤\le) in AC⁰ is strict. This is the parity of the size of the input;
    34the parity of a marked subset is PARITY, about which no bound is claimed
    35here.
    36-/
    37
    38namespace Lax945089.FirstOrderBelowACZero
    39
    40open FirstOrder FirstOrder.Language
    41open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    42open Lax904597.Classes
    43open Lax485149.Problems Lax485149.FirstOrderDefinability Lax485149.TransitiveClosure
    44open Lax485149.DeterministicTransitiveClosure Lax485149.ClassNL Lax485149.ClassL
    45open Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    46open Lax134656.PartialFixedPoint Lax895169.ArithmeticLogic
    47open Lax945089.OrderFreeFirstOrder Lax945089.EhrenfeuchtGames Lax945089.PebbleGames
    48open Lax945089.Even Lax945089.Parity
    49open Lax945089.TransitiveClosureReductions
    50
    51/-- EVEN is AC⁰ definable. -/
    52axiom even_ac0Definable : AC0Definable EVEN
    53
    54/-- Some AC⁰ definable problem is not FO(≤) definable. -/
    55axiom exists_ac0Definable_not_foDefinable :
    56 ∃ P : DecisionProblem Language.empty, AC0Definable P ∧ ¬FODefinable P
    57
    58end Lax945089.FirstOrderBelowACZero
    59
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