Lax Lean Archive

Mortality reduces to positive first-order definability

Lax503819.Reduction · concepts/Lax503819/Reduction.lean · lax-503819

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    There is a computable map from deterministic Turing-machine tables to finite automata on powerset alphabets, such that the output language is FO⁺-definable if and only if the input machine is mortal. This is the reduction established in Section 6 of Kuperberg (2023), including the powerset extension in Remark 6.25.

    The finite alphabets may depend on the input machine. The quantifier order requires a single computable transformation, not a separate existence result for each machine. The configuration alphabet, superposition construction and integer-game argument will be implemented in the proof package; they are not assumed in this statement.

    Concept map
    5 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax503819.Automata
    2import Lax503819.Mortality
    3
    4/-!
    5---
    6title: Mortality reduces to positive first-order definability
    7type: theorem
    8---
    9There is a computable map from deterministic Turing-machine tables to finite
    10automata on powerset alphabets, such that the output language is FO⁺-definable
    11if and only if the input machine is mortal. This is the reduction established
    12in Section 6 of Kuperberg (2023), including the powerset extension in Remark 6.25.
    13
    14The finite alphabets may depend on the input machine. The quantifier order
    15requires a single computable transformation, not a separate existence result
    16for each machine. The configuration alphabet, superposition construction and
    17integer-game argument will be implemented in the proof package; they are not
    18assumed in this statement.
    19-/
    20
    21namespace Lax503819.Reduction
    22
    23open Lax503819.Automata Lax503819.Mortality
    24
    25axiom mortality_reduces :
    26 ∃ f : Machine → Input, Computable f ∧
    27 ∀ m, Mortal m ↔ PositiveDefinable (f m)
    28
    29end Lax503819.Reduction
    30
    Show Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…