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FO(≤) ⊆ FO(DTC) ⊆ FO(TC)

Lax485149.FirstOrderInTransitiveClosure · concepts/Lax485149/FirstOrderInTransitiveClosure.lean · lax-485149

proven

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    Natural Language Statement

    Theorem

    Every FO(≤\le) definable problem is FO(DTC) definable, by a walk that takes no step, and every FO(DTC) definable problem is FO(TC) definable, the determinization of a specification being a specification. Hence FO(≤)⊆FO(DTC)⊆FO(TC)\mathrm{FO}(\le) \subseteq \mathrm{FO(DTC)} \subseteq \mathrm{FO(TC)} as classes of problems on finite structures.

    Concept map
    20 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 dtcDefinable_tcDefinable proven

    2 foDefinable_dtcDefinable proven

    3 foDefinable_tcDefinable 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 Lax485149.Problems
    8import Lax485149.Complement
    9import Lax485149.SecondOrderAtoms
    10import Lax485149.KromFragment
    11import Lax485149.TransitiveClosure
    12import Lax485149.DeterministicTransitiveClosure
    13import Lax485149.FirstOrderDefinability
    14import Lax485149.HeadAutomata
    15import Lax485149.Reachability
    16import Lax485149.DeterministicReachability
    17import Lax485149.TwoSat
    18import Lax485149.ClassNL
    19import Lax485149.ClassL
    20
    21/-!
    22---
    23title: FO(≤) ⊆ FO(DTC) ⊆ FO(TC)
    24type: theorem
    25---
    26Every FO(≤\le) definable problem is FO(DTC) definable, by a walk that
    27takes no step, and every FO(DTC) definable problem is FO(TC) definable, the
    28determinization of a specification being a specification. Hence
    29FO(≤)⊆FO(DTC)⊆FO(TC)\mathrm{FO}(\le) \subseteq \mathrm{FO(DTC)} \subseteq \mathrm{FO(TC)}
    30as classes of problems on finite structures.
    31-/
    32
    33namespace Lax485149.FirstOrderInTransitiveClosure
    34
    35open FirstOrder FirstOrder.Language
    36open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    37open Lax904597.Classes Lax904597.Sat
    38open Lax485149.Problems Lax485149.Complement Lax485149.SecondOrderAtoms Lax485149.KromFragment
    39open Lax485149.TransitiveClosure Lax485149.DeterministicTransitiveClosure
    40open Lax485149.FirstOrderDefinability Lax485149.HeadAutomata Lax485149.Reachability
    41open Lax485149.DeterministicReachability Lax485149.TwoSat Lax485149.ClassNL Lax485149.ClassL
    42
    43/-- Every FO(≤) definable problem is FO(DTC) definable. -/
    44axiom foDefinable_dtcDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    45 FODefinable P → DTCDefinable P
    46
    47/-- Every FO(DTC) definable problem is FO(TC) definable. -/
    48axiom dtcDefinable_tcDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    49 DTCDefinable P → TCDefinable P
    50
    51/-- Every FO(≤) definable problem is FO(TC) definable. -/
    52axiom foDefinable_tcDefinable : ∀ {L : Language.{0, 0}} [L.IsRelational] {P : DecisionProblem L},
    53 FODefinable P → TCDefinable P
    54
    55end Lax485149.FirstOrderInTransitiveClosure
    56
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