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We present several implicit characterizations of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathtt{BFF}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                <mml:mi>BFF<\/mml:mi>\n              <\/mml:math><\/jats:alternatives><\/jats:inline-formula> based on a typed programming language of terms. These terms may perform calls to imperative procedures, which are not recursive. The type discipline has two layers: the terms follow a standard simply-typed discipline and the procedures follow a standard tier-based type discipline. <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathtt{BFF}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                <mml:mi>BFF<\/mml:mi>\n              <\/mml:math><\/jats:alternatives><\/jats:inline-formula> consists exactly of the second-order functionals that are computed by typable and terminating programs. The completeness of this characterization surprisingly still holds in the absence of lambda-abstraction. Moreover, the termination requirement can be specified as a completeness-preserving instance, which can be decided in time quadratic in the size of the program. As typing is decidable in polynomial time, we obtain the first tractable (<jats:italic>i.e.<\/jats:italic>, decidable in polynomial time), sound, complete, and implicit characterization of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathtt{BFF}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                <mml:mi>BFF<\/mml:mi>\n              <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, thus solving a problem opened for more than 20 years.<\/jats:p>","DOI":"10.1007\/978-3-030-99253-8_19","type":"book-chapter","created":{"date-parts":[[2022,3,28]],"date-time":"2022-03-28T20:02:48Z","timestamp":1648497768000},"page":"368-388","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Complete and tractable machine-independent characterizations of second-order polytime"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9750-0460","authenticated-orcid":false,"given":"Emmanuel","family":"Hainry","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3295-543X","authenticated-orcid":false,"given":"Bruce M.","family":"Kapron","sequence":"additional","affiliation":[]},{"given":"Jean-Yves","family":"Marion","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0601-5425","authenticated-orcid":false,"given":"Romain","family":"P\u00e9choux","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,29]]},"reference":[{"key":"19_CR1","doi-asserted-by":"publisher","unstructured":"Avery, J.: Size-change termination and bound analysis. 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