By Grazyna Mirkowska, Andrzej Salwicki

The aim of this e-book is manyfold. it really is meant either to offer options worthwhile in software program engineering and to reveal result of study on homes of those techniques.

The significant target of the e-book is to assist the reader in elaboration of his personal perspectives on foundations of computing. the current authors think that semantics of courses will regularly be the mandatory origin for each scholar of computing. in this starting place you could build next layers of ability and information in computing device technology. Later one discovers extra questions of a distinct nature, e.g. on price and optimality of algorithms. This e-book will probably be often excited about semantics.

Secondly, the ebook goals to provide a brand new set of logical axioms and inference ideas acceptable for reasoning in regards to the houses of algorithms. Such instruments are precious for formalizing the verification and research of algorithms. The instruments could be of quality—they might be constant and whole. those and comparable requisites lead us towards metamathematical questions in regards to the constitution of algorithmic common sense.

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1. According to the definitions given above, the mappings oc®, M% depend on the finite set of variables that occur in the for mula a, the term r or the program M . Hence only a finite part of the arguments described by the valuation is used in order to establish the values ocn(v), r%(v), In order to simplify our definitions we shall treat these mappings as defined on the set W. □ For a given data structure 31 and valuation v, oc<%(v) will be called the value o f the formula a in the structure 31 at the valuation v.

D is the weakest precondition). □ Obviously, the formula Ma satisfies both conditions (i) and (ii) and therefore Ma is the weakest precondition. 3. EXPRESSIVENESS 45 The notion of weakest precondition is dual to the notion of the strong est postcondition, since the formula Mac describes the maximal set of (data) valuations for which the program M has a finite computation with result satisfying the formula a. Below, we shall mention some of the properties of the weakest pre condition. 4. ) = (M oca Mf$).

The occurrence o f an individual variable x in a for mula a is bounded by a classical quantifier iff x occurs in a part o f a o f the form (3x)/J or (Vx)/J for some formula [3. In the opposite case an occurrence o f x is called free. 6. The occurrence of z in formula (3) is free; the occur rence of y in this formula is bounded by the existential quantifiers (3y). In the formula ((3v)x < y v x = y ) both occurrences of x are free; the first occurrence of y is bounded and the second, free. □ We write a(x) indicating that the variable x is free in a.