Download e-book for kindle: Algorithmic Logic by Grazyna Mirkowska, Andrzej Salwicki

By Grazyna Mirkowska, Andrzej Salwicki

ISBN-10: 9027719284

ISBN-13: 9789027719287

The aim of this e-book is manyfold. it's meant either to give ideas worthy in software program engineering and to show result of examine on homes of those techniques.

The significant target of the publication 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 mandatory origin for each pupil of computing. in this origin you can still build next layers of ability and information in computing device technological know-how. Later one discovers extra questions of a distinct nature, e.g. on fee and optimality of algorithms. This ebook will likely be frequently excited about semantics.

Secondly, the ebook goals to provide a brand new set of logical axioms and inference ideas applicable for reasoning concerning the houses of algorithms. Such instruments are priceless for formalizing the verification and research of algorithms. The instruments might be of quality—they will be constant and entire. those and comparable standards lead us towards metamathematical questions in regards to the constitution of algorithmic good judgment.

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Example text

1. e. the semantic consequence operation has no compactness property. P roof. T o prove the theorem it is sufficient to consider the set Z u {~ a} from the above example. □ Another difference between semantic consequence operation defined here and the classical one is the upward Lowenheim-Skolem Theorem (cf. Rasiowa and Sikorski, 1968). This states that if a set of statements has an infinite model, then it has models of any infinite cardinality. The following theorem shows that the last sentence fails in the algorithmic case.

Let Z be the set of formulas Z = {(x := 0)((x := succ(x))*‘ 0 < x},ev and let a be the formula ( x : = 0) Q ( x : = succ(x)) 0 ^ x, where 0 is a constant, succ is one-argument functor and 0 ^ is a one-argument predicate. We shall prove that Z j=: a, but that there is no finite subset Z 0 ci Z such that Z 0 a. Let 21 be a model for the set Z, it then follows that for every valu­ ation v and every natural number z, 21, v' [=1 (x : = succ(x))‘ 0 ^ x, where v' = Vq. (((x := succ(x))* 0 < x)^(^') = 1.

Algorithmic properties have equal rights with first-order properties: they may simplify rea­ soning about data structures. ” We start with the definition of a class o f algorithmic languages which admit iterative programs. Iterative programs are built from assignment statements by means of program connectives like composition, branch­ ing and iteration. Each program is interpreted as a binary input-output relation in the set of all computer memory states. We then define the notion of computation.

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Algorithmic Logic by Grazyna Mirkowska, Andrzej Salwicki

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