By Eric C.R. Hehner

There are a number of theories of programming. the 1st usable conception, referred to as ''Hoare's Logic'', remains to be the most widely recognized. In it, a specification is a couple of predicates: a precondition and postcondition (these and all technical phrases should be outlined in due course). one other renowned and heavily comparable concept 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 common sense is yet one more formalism that introduces a few certain operators and quantifiers to explain a few elements of computation. the speculation during this publication is less complicated than any of these simply pointed out. In it, a specification is simply a boolean expression. Refinement is simply traditional implication. This conception can be extra common than these simply pointed out, using 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 class and for tightly limited real-time purposes

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As in this example, the shortcut usually works, but beware: it can sometimes lead to inconsistencies. ) Like application, composition distributes over bunch union. 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.

The Renaming Axiom says that if v and w are names, and neither v nor w appears in D , and w does not appear in b , then 〈v: D→b〉 = 〈w: D→(substitute w for v in b )〉 The substitution must replace every occurrence of v with w . If f is a function and x is an element of its domain, then fx “ f applied to x ” or “ f of x ” is the corresponding element of the range. This is function application, and x is the argument. Of course, parentheses may be used around any expression, so we may write f(x) if we want.

For a specification to be implementable, there must be at least one satisfactory output state for each input state. 35 4 Program Theory Here are four definitions based on the number of satisfactory outputs for each input. ¢(§σ′· S) < 1 Specification S is unsatisfiable for prestate σ : ¢(§σ′· S) ≥ 1 Specification S is satisfiable for prestate σ : ¢(§σ′· S) ≤ 1 Specification S is deterministic for prestate σ : ¢(§σ′· S) > 1 Specification S is nondeterministic for prestate σ : We can rewrite the definition of satisfiable as follows: Specification S is satisfiable for prestate σ : ∃σ′· S And finally, ∀σ· ∃σ′· S Specification S is implementable: For convenience, we prefer to write specifications in the initial values x , y , ...