{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R37",
  "content_hash": null,
  "slug": "asser-claim-c2-semilinear-complement-closure",
  "type": "claim",
  "title": "The two-variable counting fragment is closed under complement",
  "summary": "The spectra of two-variable first-order logic with counting are exactly the semilinear subsets of the natural numbers, so this fragment has spectrum complements.",
  "relevance": "For Asser's complement problem for first-order spectra, record asser-claim-c2-semilinear-complement-closure (“The two-variable counting fragment is closed under complement”) records a bound, answer, status fact, or structural consequence. The record states: The spectra of two-variable first-order logic with counting are exactly the semilinear subsets of the natural numbers, so this fragment has spectrum complements.",
  "relevance_source": "recorded",
  "body": "Kopczyński and Tan translate the existence of finite models of a \\(C^2\\) sentence into Presburger conditions on the sizes of vertex classes in regular and biregular graphs. This proves that every \\(C^2\\) spectrum is semilinear. Their converse construction represents every semilinear set as the spectrum of a \\(C^2\\) sentence. Semilinear sets are closed under complement, which gives the fragment-level result. Restricting their natural-number convention to the positive cardinalities used by this problem preserves the conclusion.\n\nThis theorem permits arbitrary finite relational vocabularies inside \\(C^2\\) and allows counting quantifiers. The checked variable-hierarchy reduction places the unresolved case at three variables.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "spectra of first-order sentences using two variables and counting quantifiers over arbitrary finite relational vocabularies",
    "family": "two-variable first-order logic with counting, C2"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1137/130943625",
      "locator": "Theorem 2.1 through Corollary 2.4, pp. 4–5"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1137/130943625",
    "locator": "Theorem 2.1 through Corollary 2.4, pp. 4–5"
  },
  "relations": [
    {
      "slug": "R38",
      "title": "Asser's complement problem remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R34",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "first-order-spectra-complement-closure",
      "title": "first order spectra complement closure",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}