Algorithmic Logic by Grazyna Mirkowska, Andrzej Salwicki PDF

By Grazyna Mirkowska, Andrzej Salwicki

ISBN-10: 9027719284

ISBN-13: 9789027719287

The aim of this e-book is manyfold. it really is meant either to give suggestions worthy in software program engineering and to reveal result of examine on houses of those techniques.

The significant aim of the ebook is to assist the reader in elaboration of his personal perspectives on foundations of computing. the current authors think that semantics of courses will continually be the required origin for each pupil of computing. in this starting place you can actually build next layers of ability and data in computing device technology. Later one discovers extra questions of a distinct nature, e.g. on rate and optimality of algorithms. This ebook will likely be typically interested in semantics.

Secondly, the booklet goals to provide a brand new set of logical axioms and inference principles applicable for reasoning concerning the houses of algorithms. Such instruments are worthwhile for formalizing the verification and research of algorithms. The instruments will be of quality—they could be constant and whole. those and comparable requisites lead us towards metamathematical questions about the constitution of algorithmic common sense.

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LANGUAGE 29 1. 1. Then sw is a term of the form (O' - (* + j))+ z )2. As a second example let us take the formula (1) to be w. The ex­ pression Jw is then a formula of the form (x = ((/• (x + y ))+ z )A (z < (x +; f) a O < /)). Obviously if x does not appear in the expression w then sw is ident­ ical to w. 3. Note also these negative examples. When w is the formula (2) or (3) then the expression sw is not a well-formed expression. □ The observations from the above examples can be summed up in the following lemma.

Hence for every valuation v 9 I ,^ i= M ~ a implies 91, a | Thus, every data structure 91 is a model for the set of formulas Z. For our next example let us take as Z the set (2) {while y do M od true, P|M y}, where y is an open formula, and M is a program. We shall prove that there is no model for the set Z. 52 II LOGIC OF DETERMINISTIC ITERATIVE PROGRAMS Let 21 be a data structure and let v be a fixed valuation. If 21, v jn P)My, then according to the definition of semantics (cf. § 2), every time we execute the program M, the obtained valuation i e N, satisfies the formula y.

Af* be an annotated program with the precondition oc{ and the postcon­ dition Pi. ). 7. The verification condition VC(M ) o f an annotated A A program M is proper in a data structure 91 if and only i f\ C ( M ) is valid in 91. 5. A. Let us consider the following annotated program: {n < 0} n n-n {n > 0}. Its verification condition is the formula (ft < 0 => (n : = n -ft) n > 0). This verification condition is proper in the structure of integers with the usual interpretation of the predicates < , > and functors • , 0.

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Algorithmic Logic by Grazyna Mirkowska, Andrzej Salwicki

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