{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/recursive-first-order-syntactic-unification","title":"Schematic Unification","arxiv_id":"2306.09152","date":"2023-06-15","proceeding":null,"authors":["David M. Cerna"],"abstract":"We present a generalization of first-order unification to a term algebra where variable indexing is part of the object language. We exploit variable indexing by associating some sequences of variables ($X_0,\\ X_1,\\ X_2,\\dots$) with a mapping $\\sigma$ whose domain is the variable sequence and whose range consist of terms that may contain variables from the sequence. From a given term $t$, an infinite sequence of terms may be produced by iterative application of $\\sigma$. Given a unification problem $U$ and mapping $\\sigma$, the \\textit{schematic unification problem} asks whether all unification problems $U$, $\\sigma(U)$, $\\sigma(\\sigma(U))$, $\\dots$ are unifiable. We provide a terminating and sound algorithm. Our algorithm is \\textit{complete} if we further restrict ourselves to so-called $\\infty$-stable problems. We conjecture that this additional requirement is unnecessary for completeness. Schematic unification is related to methods of inductive proof transformation by resolution and inductive reasoning.","url_abs":"https://arxiv.org/abs/2306.09152v7","url_pdf":"https://arxiv.org/pdf/2306.09152v7.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"recursive-first-order-syntactic-unification","repo_url":"https://github.com/ermine516/schematic-unification","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}