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First-order reductions are computable

Lax624099.ReductionsComputable · concepts/Lax624099/ReductionsComputable.lean · lax-624099

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    Natural Language Statement

    Theorem

    Undecidability travels forward along relativized ordered first-order reductions, the most general reductions of the NP core: if a problem reduces to another and the first is not decidable on concrete instances, neither is the second, over any presentations of the two vocabularies.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax904597.Problems
    2import Lax904597.Interpretations
    3import Lax904597.Relativized
    4import Lax904597.Classes
    5import Lax624099.Problems
    6import Lax624099.ValueInvention
    7import Lax624099.ClassRE
    8import Lax624099.FiniteSatisfiability
    9import Lax624099.Halting
    10import Lax624099.CodeHalting
    11import Lax624099.PostCorrespondence
    12import Lax624099.ConcreteInstances
    13import Lax904597.Machines
    14
    15/-!
    16---
    17title: First-order reductions are computable
    18type: theorem
    19---
    20Undecidability travels forward along relativized ordered first-order
    21reductions, the most general reductions of the NP core: if a problem reduces
    22to another and the first is not decidable on concrete instances, neither is
    23the second, over any presentations of the two vocabularies.
    24-/
    25
    26namespace Lax624099.ReductionsComputable
    27
    28open FirstOrder FirstOrder.Language
    29open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.Classes
    30open Lax904597.Machines Lax624099.Problems Lax624099.ValueInvention Lax624099.ClassRE
    31open Lax624099.FiniteSatisfiability
    32open Lax624099.Halting Lax624099.CodeHalting Lax624099.PostCorrespondence
    33open Lax624099.ConcreteInstances
    34
    35/-- Undecidability transfers along relativized ordered first-order
    36reductions, because they are computable. -/
    37axiom not_computablePred_of_relOrderedReduction : ∀ {L L' : Language.{0, 0}} [L.IsRelational]
    38 [L'.IsRelational] {P : DecisionProblem L} {Q : DecisionProblem L'},
    39 RelOrderedFOReduction P Q → ∀ (V : FinVocab L) (V' : FinVocab L'),
    40 ¬ComputablePred (DecisionProblem.toPred P V) → ¬ComputablePred (DecisionProblem.toPred Q V')
    41
    42end Lax624099.ReductionsComputable
    43
    Show Proof
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