Download e-book for iPad: A Practical Theory of Programming by Eric C.R. Hehner

By Eric C.R. Hehner

ISBN-10: 0387941061

ISBN-13: 9780387941066

ISBN-10: 3540941061

ISBN-13: 9783540941064

There are numerous theories of programming. the 1st usable concept, known as ''Hoare's Logic'', continues to be the most widely recognized. In it, a specification is a couple of predicates: a precondition and postcondition (these and all technical phrases might be outlined in due course). one other renowned and heavily comparable idea by means of Dijkstra makes use of the weakest precondition predicate transformer, that is a functionality from courses and postconditions to preconditions. lones's Vienna improvement process has been used to virtue in a few industries; in it, a specification is a couple of predicates (as in Hoare's Logic), however the moment predicate is a relation. Temporal good judgment is one more formalism that introduces a few precise operators and quantifiers to explain a few points of computation. the speculation during this e-book is less complicated than any of these simply pointed out. In it, a specification is simply a boolean expression. Refinement is simply usual implication. This conception is additionally extra basic than these simply pointed out, utilizing to either terminating and nonterminating computation, to either sequential and parallel computation, to either stand-alone and interactive computation. And it comprises time bounds, either for set of rules category and for tightly restricted real-time functions

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A particularly useful operator is if b then S else R where b is a boolean expression of the initial state; it can be implemented by a computer that evaluates b , and then, depending on the value of b , behaves according to either S or R . The ∨ and if then else operators have the nice property that if their operands are implementable, so is the result; the operators ∧ and ¬ do not have that property. Specifications can also be combined by dependent composition, which describes sequential execution.

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. We can describe the acceptable inputs as x≥0 , but not the computer behavior. We can describe the acceptable inputs and the computer behavior together as x≥0 ∧ (x≥0 ⇒ y′=0) , which can be simplified to x≥0 ∧ y′=0 . But x≥0 ∧ y′=0 cannot be implemented as computer behavior because a computer cannot control its inputs.

F (g, h) = f g, f h (f, g) h = f h, g h 3 Function Theory 32 Operators and functions are similar; each applies to its operands to produce a result. Just as we compose functions, we can compose operators, and we can compose an operator with a function. For example, we can compose – with any function f that produces a number to obtain a new function. (–f) x = –(f x) In particular, (–suc) 3 = –(suc 3) = –4 Similarly if p is a predicate, then (¬p) x = ¬(p x) We can compose ¬ with even to obtain odd again.

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