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Lax251941.AcceptanceUndecidable

The acceptance problem is undecidable

concepts/Lax251941/AcceptanceUndecidable.lean · lax-251941

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    Theorem

    No machine decides the acceptance problem ATMA_{TM} (Sipser, Theorem 4.11). The proof is the diagonalisation argument: a decider HH for ATMA_{TM} would give a machine DD that, on the description ⟨N⟩\langle N\rangle of a machine NN, runs HH on ⟨N,⟨N⟩⟩\langle N, \langle N\rangle\rangle and answers the opposite; running DD on its own description is then a contradiction.

    Lean source view on GitHub

    1import Lax251941.Acceptance
    2
    3/-!
    4---
    5title: The acceptance problem is undecidable
    6type: theorem
    7---
    8No machine decides the acceptance problem ATMA_{TM} (Sipser, Theorem 4.11). The
    9proof is the diagonalisation argument: a decider HH for ATMA_{TM} would give a
    10machine DD that, on the description ⟨N⟩\langle N\rangle of a machine NN, runs
    11HH on ⟨N,⟨N⟩⟩\langle N, \langle N\rangle\rangle and answers the opposite; running DD
    12on its own description is then a contradiction.
    13
    14# Formalization notes
    15
    16The statement is `¬ TuringDecidable ATM` for the machines and the language of
    17`Acceptance`. It is equivalent to the membership predicate of `ATM` not being
    18mathlib's `ComputablePred`, decidability of a language and computability of its
    19membership predicate being the same thing for these machines.
    20-/
    21
    22namespace Lax251941.AcceptanceUndecidable
    23
    24open Lax251941.Acceptance
    25
    26/-- Sipser's Theorem 4.11: the acceptance problem is undecidable. -/
    27axiom not_turingDecidable_ATM : ¬ TuringDecidable ATM
    28
    29end Lax251941.AcceptanceUndecidable
    30
    Show Proof

    Formalization notes

    The statement is ¬TuringDecidableATM¬ TuringDecidable ATM for the machines and the language of AcceptanceAcceptance. It is equivalent to the membership predicate of ATMATM not being mathlib's ComputablePredComputablePred, decidability of a language and computability of its membership predicate being the same thing for these machines.

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