Lax Lean Archive

Code halting is RE-complete

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

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    Natural Language Statement

    Theorem

    CODEHALT is RE-complete: a drawn code halting on zero is recognized by guessing its computation in existential second-order logic with value invention, and every problem of RE reduces to CODEHALT by an ordered first-order reduction that draws the instance as the program running a semi-decision procedure for the problem on it.

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

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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: Code halting is RE-complete
    18type: theorem
    19---
    20CODEHALT is RE-complete: a drawn code halting on zero is recognized by
    21guessing its computation in existential second-order logic with value
    22invention, and every problem of RE reduces to CODEHALT by an ordered
    23first-order reduction that draws the instance as the program running a
    24semi-decision procedure for the problem on it.
    25-/
    26
    27namespace Lax624099.CodehaltREComplete
    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/-- CODEHALT is RE-complete. -/
    37axiom codehalt_RE_complete : RE.Complete CODEHALT
    38
    39end Lax624099.CodehaltREComplete
    40
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