By Eric C.R. Hehner

ISBN-10: 0387941061

ISBN-13: 9780387941066

Realizing programming and programming languages calls for wisdom of the underlying theoretical version. This ebook explores features of programming which are amenable to mathematical facts. the writer describes a programming idea that is a lot easier and extra accomplished than the present theories up to now. within the theoretical version, a specification is simply a boolean expression and refinement is simply a normal implication. the writer develops a realistic and wide strategy for writing specific requisites and designing courses whose executions most likely fulfill the requisites. starting with preparatory fabric in good judgment, numbers, units, lists, services and family members, the e-book advances additional into application concept, the center of the ebook. next chapters could be chosen or passed over in keeping with direction emphasis. The textual content can be precious to scholars in classes on programming technique or verification on the complicated 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 publication at any place attainable. No complex mathematical wisdom or programming language is thought. The e-book comprises quite a few workouts and worked-out suggestions for particular workouts. Transparency masters and suggestions for the rest routines can be found from the writer.

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Perhaps the specifier has no intention of providing a negative input; in that case, the specifier should have written x≥0 ⇒ y′=0 For nonnegative initial x , this specification still requires variable y to be assigned 0 . If we never provide a negative value for x then we don't care what would happen if we did. That's what this specification says: for negative x any result is satisfactory. It allows an implementer to provide an error indication when x is initially negative. If we want a particular error indication, we can strengthen the specification to say so.

Then x′ = x+1 ∧ y′ = y specifies the behavior of a computer that increases the value of x by 1 and leaves y unchanged. Let us check that it is implementable. We replace ∀σ by either ∀x, y or ∀y, x and we replace ∃σ′ by either ∃x′, y′ or ∃y′, x′ ; according to the Commutative Laws, the order does not matter. We find One-Point Law twice ∀x, y· ∃x′, y′· x′ = x+1 ∧ y′ = y = ∀x, y· T Identity Law twice = T The specification is implementable. It is also deterministic for each prestate. In the same state variables, here is a second specification.

If p is a predicate, then universal quantification ∀p is the boolean result of applying p to all its domain elements and conjoining all the results. Similarly, existential quantification ∃p is the boolean result of applying p to all its domain elements and disjoining all the results. If f is a function with a numeric result, then Σf is the numeric result of applying f to all its domain elements and adding up all the results; and Πf is the numeric result of applying f to all its domain elements and multiplying together all the results.

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A Practical Theory of Programming by Eric C.R. Hehner

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