By Tamas Gergely, Laszlo Ury

This paintings offers a in simple terms classical first-order logical method of the sector of analysis in theoretical desktop technology occasionally often called the speculation of courses, or programming conception. This box basically makes an attempt to supply an actual mathematical foundation for the typical actions considering reasoning approximately laptop courses and programming languages, and it additionally makes an attempt to discover useful functions within the components of software specification, verification and programming language layout. many alternative ways with varied mathematical frameworks were proposed as a foundation for programming concept. They range within the mathe matical equipment they use to outline and examine courses and software houses and so they range within the thoughts they take care of to appreciate the programming paradigm. varied methods use assorted instruments and viewpoints to represent the information surroundings of courses. lots of the ways are concerning mathe matical common sense and so they supply their very own good judgment. those logics, in spite of the fact that, are very eclectic given that they use specific entities to mirror a unique international of courses, and likewise, they're often incomparable with one another. This Babel's mess annoyed us and we made up our minds to peel off the eclectic com ponents and check out to reply to all of the questions by utilizing classical first-order logic.

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By a conservative extension of a theory T in FormO' we mean a theory T' ;2 T in FormO'. iff every model A of T has an extension A' to a model of T'. r 28 1. 4 Relations in Many-Sorted Structures Let us fix a a-type model A. Let W S;;; V be a finite set of variables with IWI = n. , w n } of W. , k(w n )). Therefore a relation on n~l A(sort(Wi)) can be identified with a relation on ValA[W]. A W-type relation on A is a subset P of V alA [W]. , W n ) instead of WI U ... U W n . If Wi contains only one variable symbol, say x, we simply write x instead of {x}.

E. a theory. e. open) formula1/; such that T F= ¢> iff T F= 1/;. Let the similarity type a' be an extension of the similarity type a. By a conservative extension of a theory T in FormO' we mean a theory T' ;2 T in FormO'. iff every model A of T has an extension A' to a model of T'. r 28 1. 4 Relations in Many-Sorted Structures Let us fix a a-type model A. Let W S;;; V be a finite set of variables with IWI = n. , w n } of W. , k(w n )). Therefore a relation on n~l A(sort(Wi)) can be identified with a relation on ValA[W].

The syntax is defined as a set of program schemas constructed with a given set of program constructs by using the terms and the open formulas of a fixed alphabet (similarity type). This ensures that the function and relation symbols from which the terms and formulas are built up are included in the program schemas. We take the traditional view of a programming language as syntactic: in this case the syntax is the set of program schemas of the given similarity type. The semantics of the program schemas is defined by the characterization of the changes caused by the execution.