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Fagin’s theorem

lax-678846·formalized by Szymon Toruńczyk @szymtor·Codex 6·created 2026-09-09·GitHub @ce4e972·Lean v4.30.0 epoch · mathlib c5ea00351c28

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    Abstract

    This submission formalizes Fagin’s theorem: an isomorphism-invariant property of finite relational structures is definable in existential second-order logic if and only if its binary encoding language belongs to NP.

    The logic has no built-in order. NP is defined by polynomially bounded binary certificates checked by a concrete deterministic polynomial-time Turing machine. Empty universes and nullary relations are included.

    Both directions reuse the Immerman–Vardi formalization. To obtain an existential second-order definition, the proof guesses certificate relations and an order, simulates the concrete polynomial verifier, and replaces its least fixed point by exact iteration tables checked by first-order formulas. It then projects the certificate relations and removes the auxiliary order. In the other direction, a concrete polynomial-time verifier decodes the guessed relation tables and runs the existing first-order evaluator.

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    Cite this

    @misc{lax-678846,
  author = {Szymon Toruńczyk and Codex 6},
  title = {Fagin’s theorem},
  year = {2026},
  howpublished = {Lax Archive, lax-678846},
  url = {https://laxarchive.org/lax-678846/},
  note = {draft},
}

    References

    1. Ronald Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In Complexity of Computation 7:43–73, 1974. ronfagin.com/papers
    2. Leonid Libkin. Elements of Finite Model Theory. Springer, 2004. doi:10.1007/978-3-662-07003-1 · homepages.inf.ed.ac.uk/libkin/fmt/fmt.pdf

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