J.A. Bergstra and J.V. Tucker. The completeness of the algebraic specification methods for computable data types. Information and Control, 12:186--200, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....useful in existing algebraic frameworks. It is well known that, if equipped with hiding mechanisms, all the common algebraic specification methods are adequate for computable 3 data types, which are obviously the most important ones in the context of software development. Bergstra Tucker [BT82, BT87] proved that any computable algebra A possesses an equational specification involving at most 3(n 1) hidden functions (where n is the number of sorts in A) which defines A under both its initial and final algebra semantics. i.e. allowing hidden mechanisms a monomorphic (equational) ....

....analogously jtj pred denotes the number of pred symbols in t. 7 Monomorphic specifications of integers In theorem 2 we have shown that without the use of hidden symbols the data type of integers cannot be monomorphically specified. However, due to the completeness theorem of Bergstra Tucker [BT82] a monomorphic specification of integers is enabled if only the signature is appropriately enriched. For example, adding the less predicate ( which should more accurately be written as boolean valued function) to the signature, with the axioms succ(x) y x = pred(y) x succ(x) x x x ....

J.A. Bergstra and J.V. Tucker. The completeness of the algebraic specification methods for computable data types. Information and Control, 54:186-- 200, 1982.

....theorems have the following form: Let A be a many sorted algebra. The following are equivalent: i) A is definable by algorithms; and (ii) A is uniquely definable by a finite set of equations. Here is one such theorem that we will apply to dynamical systems shortly, in Section 3. 3: Theorem [BT82]. Let A be a many sorted minimal Sigma algebra with n sorts. The following are equivalent: 1. A is computable. 2. There is an equational specification ( Sigma 0 ; E 0 ) such that (i) Sorts( Sigma 0 ) Sorts( Sigma) ii) Sigma 0 Gamma Sigma contains 3(n 1) hidden functions; ....

J A Bergstra and J V Tucker. The completeness of the algebraic specification methods for computable data types. Information and Control, 54:186--200, 1982.

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Bergstra, J A and J V Tucker, The completeness of the algebraic specification methods for computable data types, Information and Control, 54 (1983) 186-200.

.... first example of a data type whose algebraic specification requires hidden (auxiliary) functions was given by Majster [9] It was subsequently shown that allowing hidden functions is sufficient in the sense that every computable data type has an initial algebra specification with hidden functions [5]. The initial algebra of a hidden function specification is obtained by first taking the initial algebra of the specification as if none of its functions were hidden, and then restricting this algebra to the functions that are actually visible. More generally, a model of a hidden function ....

....homomorphic images of the data type are no longer models (in the above sense) of the specification. The reason is that, whereas the homomorphic images of the data type are independent of the hidden functions of the specification, the models of the specification are not. In fact, it was shown in [5] that every computable data type has an algebraic specification with hidden functions that defines it under initial and final algebra semantics simultaneously. All non trivial minimal models of such a specification are isomorphic to the data type, so none of the non trivial homomorphic images of ....

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J.A. Bergstra and J.V. Tucker, The completeness of the algebraic specification methods for computable data types, Information & Control, 54 (1982) 186--200.

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J.A. Bergstra and J.V. Tucker. The completeness of the algebraic specification methods for computable data types. Information and Control, 12:186--200, 1982.

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