Order-Invariant CFO Contains Local-Order-Invariant FO

Abstract

Order-invariant cluster first-order logic (CFO) and local-order-invariant first-order logic are two fragments of order-invariant first-order logic, both motivated by the study of invariant logics on classes of bounded degree. Our main result is that order-invariant CFO contains local-order-invariant FO on arbitrary classes of finite structures, settling the conjecture of Ghasemi and Grange that order-invariant CFO contains plain FO. By combining this with a collapse result of Ghasemi and Grange, we obtain the equivalence of FO, local-order-invariant FO, and order-invariant CFO on classes of finite colored graphs of bounded degree. The proof makes use of Gaifman normal form, which allows our translation from local-order-invariant FO to order-invariant CFO to be effective. Combined with the fixed-parameter tractability of model-checking for order-invariant CFO on bounded-degree colored graph classes, this shows that model-checking for local-order-invariant FO on bounded-degree colored graph classes is fixed-parameter tractable. To generalize our collapse and model-checking results beyond colored graphs, we use a method based on first-order interpretations which also yields a collapse of order-invariant FO to FO when restricted to graph classes of bounded shrub-depth. Finally, we give an application to the database-motivated frozen-witness logic of Abiteboul, Simon, and Vianu by showing that local-order-invariant FO contains deterministic frozen-witness FO, and thus that deterministic frozen-witness FO collapses to FO on classes of finite structures of bounded degree.

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BibTeX RIS

BibTeX

@misc{aoki2026orderinvariant,
  title = {{Order-Invariant CFO Contains Local-Order-Invariant FO}},
  author = {Derek R. Aoki},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01501},
  eprint = {2611.01501},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01501},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Order-Invariant CFO Contains Local-Order-Invariant FO
AU  - Derek R. Aoki
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01501
UR  - https://arxiv.org/abs/2611.01501
AB  - Order-invariant cluster first-order logic (CFO) and local-order-invariant first-order logic are two fragments of order-invariant first-order logic, both motivated by the study of invariant logics on classes of bounded degree. Our main result is that order-invariant CFO contains local-order-invariant FO on arbitrary classes of finite structures, settling the conjecture of Ghasemi and Grange that order-invariant CFO contains plain FO. By combining this with a collapse result of Ghasemi and Grange, we obtain the equivalence of FO, local-order-invariant FO, and order-invariant CFO on classes of finite colored graphs of bounded degree. The proof makes use of Gaifman normal form, which allows our translation from local-order-invariant FO to order-invariant CFO to be effective. Combined with the fixed-parameter tractability of model-checking for order-invariant CFO on bounded-degree colored graph classes, this shows that model-checking for local-order-invariant FO on bounded-degree colored graph classes is fixed-parameter tractable. To generalize our collapse and model-checking results beyond colored graphs, we use a method based on first-order interpretations which also yields a collapse of order-invariant FO to FO when restricted to graph classes of bounded shrub-depth. Finally, we give an application to the database-motivated frozen-witness logic of Abiteboul, Simon, and Vianu by showing that local-order-invariant FO contains deterministic frozen-witness FO, and thus that deterministic frozen-witness FO collapses to FO on classes of finite structures of bounded degree.
ER  -

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