{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T07:40:35Z","timestamp":1684914035810},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2009,3,4]],"date-time":"2009-03-04T00:00:00Z","timestamp":1236124800000},"content-version":"unspecified","delay-in-days":5117,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:p>Inspired by the work of S. Kaplan on positive\/negative conditional rewriting, we investigate initial semantics for algebraic specifications with Gentzen formulas. Since the standard initial approach is limited to conditional equations (<jats:italic>i.e<\/jats:italic>. positive Horn formulas), the notion of semi-initial and quasi-initial algebras is introduced, and it is shown that all specifications with (positive) Gentzen formulas admit quasi-initial models.<\/jats:p><jats:p>The whole approach is generalized to the parametric case where quasi-initiality generalizes to quasi-freeness. Since quasi-free objects need not be isomorphic, the persistency requirement is added to obtain a<jats:italic>unique<\/jats:italic>semantics for many interesting practical examples. Unique persistent quasi-free semantics can be described as a free construction if the homomorphisms of the parameter category are suitably restricted. Furthermore, it turns out that unique persistent quasi-free semantics applies especially to specifications where the Gentzen formulas can be interpreted as hierarchical positive\/negative conditional equations. The data type constructor of finite function spaces is used as an example that does not admit a correct initial semantics, but does admit a correct unique persistent quasi-initial semantics. The example demonstrates that the concepts introduced in this paper might be of some importance in practical applications.<\/jats:p>","DOI":"10.1017\/s0960129500000645","type":"journal-article","created":{"date-parts":[[2009,3,4]],"date-time":"2009-03-04T09:00:42Z","timestamp":1236157242000},"page":"69-111","source":"Crossref","is-referenced-by-count":2,"title":["Parametric algebraic specifications with Gentzen formulas \u2013 from quasi-freeness to free functor semantics"],"prefix":"10.1017","volume":"5","author":[{"given":"Michael","family":"L\u00f6we","sequence":"first","affiliation":[]},{"given":"Uwe","family":"Wolter","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2009,3,4]]},"reference":[{"key":"S0960129500000645_ref025","first-page":"85","article-title":"Deduction in partial equational Horn theories","volume":"27","author":"Wolter","year":"1991","journal-title":"Journal of Information Processing and Cybernetics"},{"key":"S0960129500000645_ref023","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(86)90051-4"},{"key":"S0960129500000645_ref022","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-76771-5"},{"key":"S0960129500000645_ref020","doi-asserted-by":"crossref","DOI":"10.1515\/9783112573426","volume-title":"Initial Computability, Algebraic Specifications, and Partial Algebras","author":"Reichel","year":"1987"},{"key":"S0960129500000645_ref019","unstructured":"Reichel H. 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