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, referred to as ''Hoare's Logic'', continues to be the most well known. 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 similar conception by means of Dijkstra makes use of the weakest precondition predicate transformer, that's a functionality from courses and postconditions to preconditions. lones's Vienna improvement strategy 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 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 usual implication. This conception can also be extra normal than these simply pointed out, making use of to either terminating and nonterminating computation, to either sequential and parallel computation, to either stand-alone and interactive computation. And it contains time bounds, either for set of rules type and for tightly restricted real-time functions

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2 Function Fine Points 3 Function Theory optional Consider a function in which the body is a bunch: each element of the domain is mapped to zero or more elements of the range. For example, 〈n: nat→n, n+1〉 maps each natural number to two natural numbers. 0 0 1 1 2 2 3 3 :: 4 :: Application works as usual: 〈n: nat→n, n+1〉 3 = 3, 4 A function that sometimes produces no result is called “partial”. 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 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. The range of function f is f (Δf) . In general, we cannot apply a function to a non-elementary bunch using the Application Law. For example, if we define double = 〈n: nat→n+n〉 we can say double (2, 3) this step is right = double 2, double 3 = 4, 6 but we cannot say double (2, 3) this step is wrong = (2, 3) + (2, 3) = 4, 5, 6 Suppose we really do want to apply a function to a collection of items, for example to report if there are too many items in the collection.

4 Limits and Reals optional Let f: nat→rat so that f0; f1; f2; ... is a sequence of rationals. The limit of the function (limit of the sequence) is expressed as LIM f . 718281828459 . We define the LIM quantifier by the following Limit Axiom: (MAX m· MIN n· f(m+n)) ≤ (LIM f) ≤ (MIN m· MAX n· f(m+n)) 33 3 Function Theory with all domains being nat . This axiom gives a lower bound and an upper bound for LIM f . When those bounds are equal, the Limit Axiom tells us LIM f exactly. For example, LIM n· 1/(n+1) = 0 For some functions, the Limit Axiom tells us a little less.

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