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In contrast, we show that the smallest extension of the two-variable fragment (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{FO}^2 $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mtext>FO<\/mml:mtext>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), and of the forward fragment (FF) with Craig interpolation, is full first-order logic. Similarly, we also show that all extensions of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{FO}^2 $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mtext>FO<\/mml:mtext>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>and of the fluted fragment (FL) with Craig interpolation are undecidable.<\/jats:p>","DOI":"10.1007\/978-3-031-57231-9_7","type":"book-chapter","created":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T18:01:54Z","timestamp":1712340114000},"page":"137-159","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Craig Interpolation for Decidable First-Order Fragments"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2538-5846","authenticated-orcid":false,"given":"Balder","family":"ten Cate","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-9734-3457","authenticated-orcid":false,"given":"Jesse","family":"Comer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,4,6]]},"reference":[{"key":"7_CR1","doi-asserted-by":"publisher","unstructured":"Andr\u00e9ka, H., N\u00e9meti, I., van Benthem, J.: Modal languages and bounded fragments of predicate logic. 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