{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T22:42:12Z","timestamp":1757630532698,"version":"3.44.0"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T00:00:00Z","timestamp":1757030400000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T00:00:00Z","timestamp":1757030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T00:00:00Z","timestamp":1757030400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,6,16]]},"abstract":"<jats:p>Following Milner's seminal paper, the representation of functions as processes has received considerable attention. For pure $\u03bb$-calculus, the process representations yield (at best) non-extensional $\u03bb$-theories (i.e., $\u03b2$ rule holds, whereas $\u03b7$ does not).  In the paper, we study how to obtain extensional representations, and how to move between extensional and non-extensional representations. Using Internal $\u03c0$, $\\mathrm{I}\u03c0$ (a subset of the $\u03c0$-calculus in which all outputs are bound), we develop a refinement of Milner's original encoding of functions as processes that is parametric on certain abstract components called wires. These are, intuitively, processes whose task is to connect two end-point channels. We show that when a few algebraic properties of wires hold, the encoding yields a $\u03bb$-theory. Exploiting the symmetries and dualities of $\\mathrm{I}\u03c0$, we isolate three main classes of wires. The first two have a sequential behaviour and are dual of each other; the third has a parallel behaviour and is the dual of itself. We show the adoption of the parallel wires yields an extensional $\u03bb$-theory; in fact, it yields an equality that coincides with that of B\u00f6hm trees with infinite $\u03b7$. In contrast, the other two classes of wires yield non-extensional $\u03bb$-theories whose equalities are those of the L\u00e9vy-Longo and B\u00f6hm trees.<\/jats:p>","DOI":"10.46298\/lmcs-21(3:25)2025","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T18:46:32Z","timestamp":1757529992000},"source":"Crossref","is-referenced-by-count":0,"title":["Extensional and Non-extensional Functions as Processes"],"prefix":"10.46298","volume":"Volume 21, Issue 3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3238-9279","authenticated-orcid":false,"given":"Ken","family":"Sakayori","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5823-3235","authenticated-orcid":false,"given":"Davide","family":"Sangiorgi","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2025,9,5]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/2405.03536v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/2405.03536v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T18:46:32Z","timestamp":1757529992000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/13557"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,5]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-21(3:25)2025","relation":{"has-preprint":[{"id-type":"arxiv","id":"2405.03536v2","asserted-by":"subject"},{"id-type":"arxiv","id":"2405.03536v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2405.03536","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2405.03536","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,5]]},"article-number":"13557"}}