By Eric C.R. Hehner

ISBN-10: 0387941061

ISBN-13: 9780387941066

Figuring out programming and programming languages calls for wisdom of the underlying theoretical version. This publication explores features of programming which are amenable to mathematical facts. the writer describes a programming concept that is a lot easier and extra complete than the present theories thus far. within the theoretical version, a specification is simply a boolean expression and refinement is simply a regular implication. the writer develops a pragmatic and vast process for writing distinctive standards and designing courses whose executions most likely fulfill the requisites. starting with preparatory fabric in common sense, numbers, units, lists, capabilities and family, the ebook advances additional into software conception, the center of the publication. next chapters could be chosen or passed over in accordance with direction emphasis. The textual content can be worthwhile to scholars in classes on programming method or verification on the complex undergraduate or starting graduate point, in addition to for software program engineers within the box. All technical phrases are defined after which proven within the e-book anyplace attainable. No complex mathematical wisdom or programming language is believed. The ebook comprises quite a few routines and worked-out recommendations for particular routines. Transparency masters and suggestions for the rest routines can be found from the writer.

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The programmer then refines the specification to obtain a program, which a computer can execute. Sometimes the clearest, most understandable specification is already a program. When that is so, there is no need for any other specification, and no need for refinement. However, the programming notations are only part of the specification notations: those that happen to be implemented. Specifiers should use whatever notations help to make their specifications clear, including but not limited to programming notations.

For example, ¬ LIM n· 1/(n+1) = 0 Even though the limit of 1/(n+1) is 0 , the limit of 1/(n+1) = 0 is ⊥ . If, for some particular assignment of values to variables, the sequence never settles on one boolean value, then the axiom does not determine the value of LIM p for that assignment of values. End of Limits and Reals The purpose of a function is to introduce a local variable. But we must remember that any expression talks about its nonlocal variables. For example, ∃n: nat· x = 2×n says that x is an even natural.

And second, when S or R is an assignment, the assignment notation should be replaced by its equal using mathematical variables x , x′ , y , y′ , ... before substitution. Finally, when S or R is a dependent composition, the inner substitutions must be made first. Here is an example, again in integer variables x and y . x:= 3. y:= x+y eliminate assignments first = x′=3 ∧ y′=y. 1 Specification Laws We have seen some of the following laws before. For specifications P , Q , R , and S , and boolean b , Identity Law ok.

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