By Eric C.R. Hehner

There are a number of theories of programming. the 1st usable idea, referred to as ''Hoare's Logic'', remains 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 can be outlined in due course). one other well known and heavily comparable thought through Dijkstra makes use of the weakest precondition predicate transformer, that's a functionality from courses and postconditions to preconditions. lones's Vienna improvement technique 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 one more formalism that introduces a few targeted operators and quantifiers to explain a few features 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 concept is usually extra basic than these simply pointed out, utilising 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 functions

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**Example text**

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. With the theory, we only need to prove these six refinements, and we are confident that execution will not give us a wrong answer. 4 Program Theory 46 This solution has been constructed to make it difficult to follow the execution.

The axioms are as follows ( v is a name, A and B are bunches, b is a boolean expression, n is a number expression, and x is an element). ∀v: null· b = † ∀v: x· b = 〈v: x→b〉 x ∀v: A,B· b = (∀v: A· b) ∧ (∀v: B· b) ∃v: null· b = ƒ ∃v: x· b = 〈v: x→b〉 x ∃v: A,B· b = (∃v: A· b) ∨ (∃v: B· b) Σv: null· n = 0 Σv: x· n = 〈v: x→n〉 x (Σv: A,B· n) + (Σv: A‘B· n) = (Σv: A· n) + (Σv: B· n) Πv: null· n = 1 Πv: x· n = 〈v: x→n〉x (Πv: A,B· n) × (Πv: A‘B· n) = (Πv: A· n) × (Πv: B· n) Care is required when translating from the English words “all” and “some” to the formal notations ∀ and ∃ .

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.