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FO(TC) definability is closed under first-order reductions

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

proven

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    Natural Language Statement

    Theorem

    FO(TC) definability travels backward along first-order reductions and along ordered first-order reductions: if PP reduces to an FO(TC) definable problem, then PP is FO(TC) definable, a walk on the interpreted structure being a walk on the base structure with the tags carried in the modes. It reads a problem on its finite instances only.

    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 tcDefinable_congr_finite proven

    2 tcDefinable_of_foReduction proven

    3 tcDefinable_of_orderedReduction 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(TC) definability is closed under first-order reductions
    24type: theorem
    25---
    26FO(TC) definability travels backward along first-order reductions and along
    27ordered first-order reductions: if PP reduces to an FO(TC) definable
    28problem, then PP is FO(TC) definable, a walk on the interpreted structure
    29being a walk on the base structure with the tags carried in the modes. It
    30reads a problem on its finite instances only.
    31-/
    32
    33namespace Lax485149.TransitiveClosureClosure
    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/-- FO(TC) definability travels backward along first-order reductions. -/
    44axiom tcDefinable_of_foReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    45 {P : DecisionProblem L} {Q : DecisionProblem L'}, FOReduction P Q → TCDefinable Q → TCDefinable P
    46
    47/-- FO(TC) definability travels backward along ordered first-order
    48reductions. -/
    49axiom tcDefinable_of_orderedReduction :
    50 ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    51 {P : DecisionProblem L} {Q : DecisionProblem L'},
    52 OrderedFOReduction P Q → TCDefinable Q → TCDefinable P
    53
    54/-- FO(TC) definability only depends on the finite instances of a problem. -/
    55axiom tcDefinable_congr_finite : ∀ {L : Language.{0, 0}} [L.IsRelational] {P Q : DecisionProblem L},
    56 (∀ (A : Type) [L.Structure A] [Finite A], P A ↔ Q A) → (TCDefinable P ↔ TCDefinable Q)
    57
    58end Lax485149.TransitiveClosureClosure
    59
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