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FO(≤, PFP) definability is closed under first-order reductions

Lax134656.PartialFixedPointClosure · concepts/Lax134656/PartialFixedPointClosure.lean · lax-134656

proven

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    Natural Language Statement

    Theorem

    FO(≤\le, PFP) definability travels backward along first-order reductions, ordered first-order reductions and relativized ordered first-order reductions, and reads a problem on its finite instances only. For a relativized reduction the block is pulled back onto the definable domain, the transfer of assignments being a bijection onto the assignments inside the domain, which a deterministic iteration needs.

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

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 pfpDefinable_congr_finite proven

    2 pfpDefinable_of_foReduction proven

    3 pfpDefinable_of_orderedReduction proven

    4 pfpDefinable_of_relOrderedReduction proven

    Lean source view on GitHub

    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.SecondOrder
    5import Lax904597.Classes
    6import Lax904597.Machines
    7import Lax485149.Problems
    8import Lax485149.Complement
    9import Lax535992.InflationaryFixedPoint
    10import Lax535992.DeterministicMachines
    11import Lax535992.ClassPTIME
    12import Lax564036.Hierarchy
    13import Lax134656.SecondOrderTransitiveClosure
    14import Lax134656.OrderFreeTransitiveClosure
    15import Lax134656.PartialFixedPoint
    16import Lax134656.Qsat
    17import Lax134656.SuccinctReach
    18import Lax134656.SpaceBoundedMachines
    19import Lax134656.ClassPSPACE
    20
    21/-!
    22---
    23title: FO(≤, PFP) definability is closed under first-order reductions
    24type: theorem
    25---
    26FO(≤\le, PFP) definability travels backward along first-order reductions,
    27ordered first-order reductions and relativized ordered first-order
    28reductions, and reads a problem on its finite instances only. For a
    29relativized reduction the block is pulled back onto the definable domain,
    30the transfer of assignments being a bijection onto the assignments inside
    31the domain, which a deterministic iteration needs.
    32-/
    33
    34namespace Lax134656.PartialFixedPointClosure
    35
    36open FirstOrder FirstOrder.Language
    37open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.SecondOrder
    38open Lax904597.Classes Lax904597.Machines
    39open Lax485149.Problems Lax485149.Complement
    40open Lax535992.InflationaryFixedPoint Lax535992.DeterministicMachines Lax535992.ClassPTIME
    41open Lax564036.Hierarchy
    42open Lax134656.SecondOrderTransitiveClosure Lax134656.OrderFreeTransitiveClosure
    43open Lax134656.PartialFixedPoint
    44open Lax134656.Qsat Lax134656.SuccinctReach Lax134656.SpaceBoundedMachines Lax134656.ClassPSPACE
    45
    46/-- FO(≤, PFP) definability travels backward along first-order reductions. -/
    47axiom pfpDefinable_of_foReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    48 {P : DecisionProblem L} {Q : DecisionProblem L'},
    49 FOReduction P Q → PFPDefinable Q → PFPDefinable P
    50
    51/-- FO(≤, PFP) definability travels backward along ordered first-order
    52reductions. -/
    53axiom pfpDefinable_of_orderedReduction :
    54 ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    55 {P : DecisionProblem L} {Q : DecisionProblem L'},
    56 OrderedFOReduction P Q → PFPDefinable Q → PFPDefinable P
    57
    58/-- FO(≤, PFP) definability travels backward along relativized ordered
    59first-order reductions. -/
    60axiom pfpDefinable_of_relOrderedReduction :
    61 ∀ {L L' : Language.{0, 0}} [L.IsRelational] [L'.IsRelational]
    62 {P : DecisionProblem L} {Q : DecisionProblem L'},
    63 RelOrderedFOReduction P Q → PFPDefinable Q → PFPDefinable P
    64
    65/-- FO(≤, PFP) definability only depends on the finite instances of a problem. -/
    66axiom pfpDefinable_congr_finite :
    67 ∀ {L : Language.{0, 0}} [L.IsRelational] {P Q : DecisionProblem L},
    68 (∀ (A : Type) [L.Structure A] [Finite A], P A ↔ Q A) → (PFPDefinable P ↔ PFPDefinable Q)
    69
    70end Lax134656.PartialFixedPointClosure
    71
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