Lax Lean Archive

AC⁰ ⊆ L

Lax895169.ACZeroInLogSpace · concepts/Lax895169/ACZeroInLogSpace.lean · lax-895169

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    Natural Language Statement

    Theorem

    Every AC⁰ definable problem is FO(DTC) definable, hence in L and in NL. The sentence is evaluated by a deterministic multihead automaton whose number of heads is the quantifier depth plus a constant; addition and multiplication of ranks are not relations of the instance, so they are computed by walking heads along the order.

    Concept map
    22 concepts
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    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 ac0Definable_dtcDefinable proven

    2 ac0Definable_mem_LOGSPACE proven

    3 ac0Definable_mem_NL proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Classes
    3import Lax485149.Complement
    4import Lax485149.FirstOrderDefinability
    5import Lax485149.DeterministicTransitiveClosure
    6import Lax485149.ClassNL
    7import Lax485149.ClassL
    8import Lax535992.LeastFixedPoint
    9import Lax535992.InflationaryFixedPoint
    10import Lax535992.ClassPTIME
    11import Lax895169.BitPredicate
    12import Lax895169.ArithmeticLogic
    13import Lax895169.BitLogic
    14import Lax895169.LogTimeMachines
    15
    16/-!
    17---
    18title: AC⁰ ⊆ L
    19type: theorem
    20---
    21Every AC⁰ definable problem is FO(DTC) definable, hence in L and in NL. The
    22sentence is evaluated by a deterministic multihead automaton whose number of
    23heads is the quantifier depth plus a constant; addition and multiplication
    24of ranks are not relations of the instance, so they are computed by walking
    25heads along the order.
    26-/
    27
    28namespace Lax895169.ACZeroInLogSpace
    29
    30open FirstOrder FirstOrder.Language
    31open Lax904597.Problems Lax904597.Classes
    32open Lax485149.Complement Lax485149.FirstOrderDefinability Lax485149.DeterministicTransitiveClosure
    33open Lax485149.ClassNL Lax485149.ClassL
    34open Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    35open Lax895169.BitPredicate Lax895169.ArithmeticLogic Lax895169.BitLogic Lax895169.LogTimeMachines
    36
    37/-- Every AC⁰ definable problem is FO(DTC) definable. -/
    38axiom ac0Definable_dtcDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    39 AC0Definable P → DTCDefinable P
    40
    41/-- Every AC⁰ definable problem is in L. -/
    42axiom ac0Definable_mem_LOGSPACE : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    43 AC0Definable P → LOGSPACE.Mem P
    44
    45/-- Every AC⁰ definable problem is in NL. -/
    46axiom ac0Definable_mem_NL : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    47 AC0Definable P → NL.Mem P
    48
    49end Lax895169.ACZeroInLogSpace
    50
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