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AC⁰ is the logarithmic-time hierarchy

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

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    Natural Language Statement

    Theorem

    A decision problem is AC⁰ definable if and only if it is decidable in logarithmic time with constantly many alternations, and if and only if it is bit-definable. A sentence of the bit-level logic is compiled into a machine atom by atom, the order and the addition being decided by sweeps. Conversely, a sweep carries a constant number of state bits past each of the log⁡n\log n positions, so its whole history is a constant number of bit vectors over the positions, each of which is an element of the universe: the sentence guesses those elements and checks the transitions.

    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_iff_ltDecidable proven

    2 bitDefinable_ltDecidable proven

    3 ltDecidable_bitDefinable 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 is the logarithmic-time hierarchy
    19type: theorem
    20---
    21A decision problem is AC⁰ definable if and only if it is decidable in
    22logarithmic time with constantly many alternations, and if and only if it
    23is bit-definable. A sentence of the bit-level logic is compiled into a
    24machine atom by atom, the order and the addition being decided by sweeps.
    25Conversely, a sweep carries a constant number of state bits past each of
    26the log⁡n\log n positions, so its whole history is a constant number of bit
    27vectors over the positions, each of which is an element of the universe:
    28the sentence guesses those elements and checks the transitions.
    29-/
    30
    31namespace Lax895169.ACZeroIsLogTime
    32
    33open FirstOrder FirstOrder.Language
    34open Lax904597.Problems Lax904597.Classes
    35open Lax485149.Complement Lax485149.FirstOrderDefinability Lax485149.DeterministicTransitiveClosure
    36open Lax485149.ClassNL Lax485149.ClassL
    37open Lax535992.LeastFixedPoint Lax535992.InflationaryFixedPoint Lax535992.ClassPTIME
    38open Lax895169.BitPredicate Lax895169.ArithmeticLogic Lax895169.BitLogic Lax895169.LogTimeMachines
    39
    40/-- Every problem decidable in logarithmic time is bit-definable. -/
    41axiom ltDecidable_bitDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    42 LTDecidable P → BitDefinable P
    43
    44/-- Every bit-definable problem is decidable in logarithmic time. -/
    45axiom bitDefinable_ltDecidable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    46 BitDefinable P → LTDecidable P
    47
    48/-- AC⁰ definability is decidability in logarithmic time. -/
    49axiom ac0Definable_iff_ltDecidable :
    50 ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    51 AC0Definable P ↔ LTDecidable P
    52
    53end Lax895169.ACZeroIsLogTime
    54
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