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However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called<jats:italic>open logical relation<\/jats:italic>, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03bb<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued first-order functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.<\/jats:p>","DOI":"10.1007\/978-3-030-44914-8_3","type":"book-chapter","created":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T10:02:53Z","timestamp":1587117773000},"page":"56-83","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["On the Versatility of Open Logical Relations"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3853-1777","authenticated-orcid":false,"given":"Gilles","family":"Barthe","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rapha\u00eblle","family":"Crubill\u00e9","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9200-070X","authenticated-orcid":false,"given":"Ugo Dal","family":"Lago","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2159-0615","authenticated-orcid":false,"given":"Francesco","family":"Gavazzo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,4,18]]},"reference":[{"key":"3_CR1","doi-asserted-by":"crossref","unstructured":"Abadi, M., Plotkin, G.D.: A simple differentiable programming language. 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