{
  "annotation_id": "f998fe8d-12ff-48a7-a1a1-caf2685271a7",
  "date_created": "2026-03-02T18:02:34.378000Z",
  "date_modified": "2026-03-02T18:02:34.378000Z",
  "file_hash": "c42969ddaaf162ef465b9a47bc0c0aa765b9c001bc71ed07e7b95486f9eb7b4b",
  "private": false,
  "record": {
    "abstract": "We provide a computational definition of the notions of vector space and\nbilinear functions. We use this result to introduce a minimal language\ncombining higher-order computation and linear algebra. This language extends\nthe Lambda-calculus with the possibility to make arbitrary linear combinations\nof terms alpha.t + beta.u. We describe how to \"execute\" this language in terms\nof a few rewrite rules, and justify them through the two fundamental\nrequirements that the language be a language of linear operators, and that it\nbe higher-order. We mention the perspectives of this work in the field of\nquantum computation, whose circuits we show can be easily encoded in the\ncalculus. Finally, we prove the confluence of the entire calculus.",
    "arxiv_id": "quant-ph/0612199",
    "authors": [
      "Pablo Arrighi",
      "Gilles Dowek"
    ],
    "categories": [
      "quant-ph",
      "cs.LO",
      "cs.PL"
    ],
    "doi": "10.23638/LMCS-13(1:8)2017",
    "journal_ref": "Logical Methods in Computer Science, Volume 13, Issue 1 (March 17,\n  2017) lmcs:3203",
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Lineal: A linear-algebraic Lambda-calculus",
    "url": "https://arxiv.org/abs/quant-ph/0612199"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "f08b86d2-32a9-4e93-a623-74c6dcb60260",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}