{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T09:48:57Z","timestamp":1775641737277,"version":"3.50.1"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2026,3,3]]},"abstract":"<jats:p>Topology may be interpreted as the study of verifiability, where opens correspond to semi-decidable properties. In this paper we make a distinction between verifiable properties themselves and processes which carry out the verification procedure. The former are simply opens, while we call the latter \\emph{machines}. Given a frame presentation $\\mathcal{O} X = \\langle G \\mid R\\rangle$ we construct a space of machines $\u03a3^{\u03a3^G}$ whose points are given by formal combinations of basic machines corresponding to generators in $G$. This comes equipped with an `evaluation' map making it a weak exponential with base $\u03a3$ and exponent $X$. When it exists, the true exponential $\u03a3^X$ occurs as a retract of machine space. We argue this helps explain why some spaces are exponentiable and others not. We then use machine space to study compactness by giving a purely topological version of Escard\u00f3's algorithm for universal quantification over compact spaces in finite time. Finally, we relate our study of machine space to domain theory and domain embeddings.<\/jats:p>","DOI":"10.46298\/lmcs-22(2:2)2026","type":"journal-article","created":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T08:55:07Z","timestamp":1775638507000},"source":"Crossref","is-referenced-by-count":0,"title":["Machine Space I: Weak exponentials and quantification over compact spaces"],"prefix":"10.46298","volume":"Volume 22, Issue 2","author":[{"given":"Peter F.","family":"Faul","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Graham","family":"Manuell","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2026,4,7]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/2209.11339v6","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/2209.11339v6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T08:55:07Z","timestamp":1775638507000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/12327"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,7]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-22(2:2)2026","relation":{"has-preprint":[{"id-type":"arxiv","id":"2209.11339v5","asserted-by":"subject"},{"id-type":"arxiv","id":"2209.11339v4","asserted-by":"subject"},{"id-type":"arxiv","id":"2209.11339v3","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2209.11339","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2209.11339","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"value":"1860-5974","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,7]]},"article-number":"12327"}}