Jacques Loeckx and Kurt Sieber. The Foundations of Program Verification. Wiley-Teubner, Second edition, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....w and all A assignments ff 2 Gamma A Gamma B : A; ff j= w iff B; ff j= w. This is a stronger notion than beeing a submodel, especially elementary submodels have the same first order theory ( CK90] We use some basic notions from the theory of the semantics of programs, see for instance [LS87]. 3 Modules 3.1 Syntax The syntax of modules as given in this section closely resembles the syntax of Algorithmic Specifications as given in [Loe87] Before we give the exact syntax of a module we define the notion of a standard signature. A signature will be called standard if some ....

....x i bool if x = false otherwise select c x j if x = c(x 1 ; x j ; x n ) for some x i 6= i = 1 : n s j otherwise Lemma 1 A is a standard algebra. Proof: This follows from the continuity of the sequential if Gamma then Gamma else function ([LS87]) and the strictness of the remaining functions. 2 In order to define the semantics of the recursive function symbols we need the notion of iteration. Intuitively, the n times iteration of a term t is obtained by n times simultaneously unfolding all occurrences of recursive function symbols in t. ....

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Jacques Loeckx and Kurt Sieber. The Foundations of Program Verification. Wiley /Teubner, 2nd edition, 1987.

....theorem, but it seems now more convenient to take the development the other way round. A similar principle appears in Suppes [16] chapter 7, Theorem Schema 4 as The Principle of Transfinite Induction and in Enderton [8] chapter 9 similarly. Cohn [7] Burstall [6] Huet [9] and Loeckx and Sieber [10] refer to it as Noetherian Induction . The name transfinite reminds us that there is no restriction requiring the image set of any element through the parent relation to be finite. In common with other more recent writings, such as those of Paulson [13] however, we make use of a definition of ....

Jacques Loeckx and Kurt Sieber. The Foundations of Program Verification. Wiley - Teubner, second edition, 1987.

....due to Hennessy and Plotkin[HP79, Plo81] See also [Ast91] which contains discussions on terminology, with suggestion for the term structural inductive semantics. 12 The work on denotational semantics may be traced back to [Fre92, Tar56, Car48, Car47] The subject is covered in [Sto77, Sch86, LS87, Ten91, Win93] 13 Formally, a semantic definition is said to be compositional if and only if it is a homomorphism from the syntactic algebra into a semantic algebra (i.e. any algebra over the same signature whose carriers are the sets of elements in given semantic domains associated to the ....

....(X i ; v i ) i = 1; n, n 1, be partial orders which have only finite chains. Then (X 1 Theta : Theta Xn ; v) has only finite chains. 22 The results are presented in this section in the form of propositions, for which proofs can be found or are analogous to those in e.g. dB80, LS87, Win93] 23 v is reflexive if, for all x 2 X, x v x; it is transitive if, for all x 1 ; x 2 ; x3 2 X, x 1 v x 3 if x 1 v x 2 and x 2 v x3 ; antisymmetric if, for all x 1 ; x 2 2 X, x1 = x 2 if x1 v x 2 and x2 v x1 . 24 In this case, if X 0 = X, l is also called the least element of the ....

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Jacques Loeckx and Kurt Sieber. The Foundations of Program Verification. John Wiley & Sons, 1987.

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Jacques Loeckx and Kurt Sieber. The Foundations of Program Verification. Wiley-Teubner, Second edition, 1987.

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LS87. Jacques Loeckx and Kurt Sieber. The Foundations of Program Verification (Second edition). John Wiley and Sons, New York, N.Y., 1987.

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