{"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/combinatorics-of-explicit-substitutions","title":"Combinatorics of explicit substitutions","arxiv_id":"1804.03862","date":"2018-04-11","proceeding":null,"authors":["Maciej Bendkowski","Pierre Lescanne"],"abstract":"$\\lambda\\upsilon$ is an extension of the $\\lambda$-calculus which internalises the calculus of substitutions. In the current paper, we investigate the combinatorial properties of $\\lambda\\upsilon$ focusing on the quantitative aspects of substitution resolution. We exhibit an unexpected correspondence between the counting sequence for $\\lambda\\upsilon$-terms and famous Catalan numbers. As a by-product, we establish effective sampling schemes for random $\\lambda\\upsilon$-terms. We show that typical $\\lambda\\upsilon$-terms represent, in a strong sense, non-strict computations in the classic $\\lambda$-calculus. Moreover, typically almost all substitutions are in fact suspended, i.e. unevaluated, under closures. Consequently, we argue that $\\lambda\\upsilon$ is an intrinsically non-strict calculus of explicit substitutions. Finally, we investigate the distribution of various redexes governing the substitution resolution in $\\lambda\\upsilon$ and investigate the quantitative contribution of various substitution primitives.","url_abs":"http://arxiv.org/abs/1804.03862v1","url_pdf":"http://arxiv.org/pdf/1804.03862v1.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":"combinatorics-of-explicit-substitutions","repo_url":"https://github.com/maciej-bendkowski/combinatorics-of-explicit-substitutions","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}