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 elements of programming which are amenable to mathematical facts. the writer describes a programming thought that's a lot less complicated and extra accomplished than the present theories thus far. within the theoretical version, a specification is simply a boolean expression and refinement is simply a standard implication. the writer develops a pragmatic and large procedure for writing designated requirements and designing courses whose executions most likely fulfill the requirements. starting with preparatory fabric in common sense, numbers, units, lists, features and kin, the e-book advances extra into software thought, the center of the ebook. next chapters could be chosen or passed over in accordance with path emphasis. The textual content should be precious to scholars in classes on programming method 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 ebook anywhere attainable. No complex mathematical wisdom or programming language is thought. The booklet includes a variety of workouts and worked-out strategies for particular routines. Transparency masters and suggestions for the remainder routines can be found from the writer.

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**Example text**

The other assignment x:= 3 is superfluous, and our solution would be simpler without it; we have included it just to make the point that it is allowed by the specification. The next refinement makes y′=2x in two steps: first y′=2x–1 and then double y . The antecedent x>0 ensures that 2x–1 will be natural. The last two refinements again contain superfluous assignments. Without the theory of programming, we would be very worried that these superfluous assignments might in some way make the result wrong.

A function that always produces at least one result is called “total”. A function that always produces at most one result is called “deterministic”. A function that sometimes produces more than one result is called “nondeterministic”. n〉 is both partial and nondeterministic. A union of functions applied to an argument gives the union of the results: (f, g) x = fx, gx A function applied to a union of arguments gives the union of the results: f null = null f (A, B) = f A, f B f (§g) = §y: f (Δg)· ∃x: Δg· fx=y ∧ gx In other words, function application distributes over bunch union.

C∧Q ⇐ P∧C′. Q P. Q ⇐ P∧C′. C⇒Q Precondition Law: C is a sufficient precondition for P to be refined by S if and only if C⇒P is refined by S . Postcondition Law: C′ is a sufficient postcondition for P to be refined by S if and only if C′⇒P is refined by S . 4 Programs A program is a description or specification of computer behavior. A computer executes a program by behaving according to the program, by satisfying the program. People often confuse programs with computer behavior. They talk about what a program “does”; of course it just sits there on the page or screen; it is the computer that does something.

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

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