By Eric C.R. Hehner
There are a number of theories of programming. the 1st usable conception, referred to as ''Hoare's Logic'', continues to be essentially the most well known. In it, a specification is a couple of predicates: a precondition and postcondition (these and all technical phrases could be outlined in due course). one other renowned and heavily similar 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 certain operators and quantifiers to explain a few facets of computation. the idea during this ebook is easier than any of these simply pointed out. In it, a specification is simply a boolean expression. Refinement is simply usual implication. This thought is usually extra common 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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We also need a variable to serve as index in the list, saying how many items have been considered; let us take natural variable n for that. We must begin by assigning 0 to both s and n to indicate that we have summed zero items so far. We complete the task by adding the remaining items (which means all of them) to the sum. s′ = ΣL ⇐ s:= 0. n:= 0. ) This theorem is easily proven by two applications of the Substitution Law. #L] . When we refine this new problem, we must ignore the context in which it arose; in particular, we ignore that s=0 ∧ n=0 .
The exact postcondition for P to be refined by S is ∀σ· P⇐S . For example, although x′>5 is not refined by x:= x+1 , we can calculate (in one integer variable) (the exact precondition for x′>5 to be refined by x:= x+1 ) = ∀x′· x′>5 ⇐ (x:= x+1) = ∀x′· x′>5 ⇐ x′=x+1 One-Point Law = x+1 > 5 = x>4 This means that a computation satisfying x:= x+1 will also satisfy x′>5 if and only if it starts with x>4 . If we are interested only in prestates such that x>4 , then we should weaken our problem with that antecedent, obtaining the refinement x>4 ⇒ x′>5 ⇐ x:= x+1 There is a similar story for postconditions.
Here are two examples. max = 〈x: xrat→〈y: xrat→if x≥y then x else y〉〉 min = 〈x: xrat→〈y: xrat→if x≤y then x else y〉〉 If we apply max to an argument we obtain a function of one variable, max 3 = 〈y: xrat→if 3≥y then 3 else y〉 which can be applied to an argument to obtain a number. max 3 5 = 5 A predicate is a function whose body is a boolean expression. Two examples are even = 〈i: int→i/2: int〉 odd = 〈i: int→¬ i/2: int〉 A relation is a function whose body is a predicate. Here is an example: divides = 〈n: nat+1→〈i: int→i/n: int〉〉 divides 2 = even divides 2 3 = ƒ One more operation on functions is selective union.
A Practical Theory of Programming by Eric C.R. Hehner
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