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Trakhtenbrot's theorem: finite satisfiability is RE-complete

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

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    Natural Language Statement

    Theorem

    FINSAT is RE-complete: it is definable in existential second-order logic with value invention, and every problem of RE reduces to it by an ordered first-order reduction, the generic reduction that writes the definition of a problem as a sentence whose finite models are its certificates. This is the logical form of Trakhtenbrot's theorem.

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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: Trakhtenbrot's theorem: finite satisfiability is RE-complete
    18type: theorem
    19---
    20FINSAT is RE-complete: it is definable in existential second-order logic
    21with value invention, and every problem of RE reduces to it by an ordered
    22first-order reduction, the generic reduction that writes the definition of a
    23problem as a sentence whose finite models are its certificates. This is the
    24logical form of Trakhtenbrot's theorem.
    25-/
    26
    27namespace Lax624099.FinsatREComplete
    28
    29open FirstOrder FirstOrder.Language
    30open Lax904597.Problems Lax904597.Interpretations Lax904597.Relativized Lax904597.Classes
    31open Lax904597.Machines Lax624099.Problems Lax624099.ValueInvention Lax624099.ClassRE
    32open Lax624099.FiniteSatisfiability
    33open Lax624099.Halting Lax624099.CodeHalting Lax624099.PostCorrespondence
    34open Lax624099.ConcreteInstances
    35
    36/-- FINSAT is RE-complete. -/
    37axiom finsat_RE_complete : RE.Complete FINSAT
    38
    39end Lax624099.FinsatREComplete
    40
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