Lax Lean Archive

The halting problem is RE-complete

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

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    Natural Language Statement

    Theorem

    HALT is RE-complete: acceptance on an unbounded tape is recognized by guessing the run in existential second-order logic with value invention, and every problem of RE reduces to HALT by an ordered first-order reduction, the machine that runs a semi-decision procedure for the problem on its input.

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

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    Lean source view on GitHub

    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: The halting problem is RE-complete
    18type: theorem
    19---
    20HALT is RE-complete: acceptance on an unbounded tape is recognized by
    21guessing the run in existential second-order logic with value invention, and
    22every problem of RE reduces to HALT by an ordered first-order reduction, the
    23machine that runs a semi-decision procedure for the problem on its input.
    24-/
    25
    26namespace Lax624099.HaltREComplete
    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/-- HALT is RE-complete. -/
    36axiom halt_RE_complete : RE.Complete HALT
    37
    38end Lax624099.HaltREComplete
    39
    Show Proof
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