By Eric C.R. Hehner
Knowing programming and programming languages calls for wisdom of the underlying theoretical version. This booklet explores points of programming which are amenable to mathematical evidence. the writer describes a programming idea that is a lot easier and extra accomplished than the present theories up to now. within the theoretical version, a specification is simply a boolean expression and refinement is simply a typical implication. the writer develops a pragmatic and extensive strategy for writing targeted requirements and designing courses whose executions most likely fulfill the requirements. starting with preparatory fabric in common sense, numbers, units, lists, capabilities and kin, the ebook advances additional into software idea, the guts of the ebook. next chapters will be chosen or passed over in line with direction emphasis. The textual content should be worthwhile to scholars in classes on programming method or verification on the complex undergraduate or starting graduate point, in addition to for software program engineers within the box. All technical phrases are defined after which proven within the booklet anywhere attainable. No complicated mathematical wisdom or programming language is believed. The publication comprises quite a few routines and worked-out ideas for particular workouts. Transparency masters and suggestions for the rest workouts can be found from the writer.
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When there is no danger of confusion, we may write fx without a space between, but when we use multicharacter names, we must put a space between the function and the argument. As an example of application, if suc = 〈n: nat→n+1〉 , then suc 3 = 〈n: nat→n+1〉 3 = 3+1 = 4 Here is the Application Axiom. If element x: D , then 〈v: D→b〉 x = (substitute x for v in b ) Operators and functions are similar; just as we apply operator – to operand x to get –x , we apply function f to argument x to get f x . A function of more than one variable is a function whose body is a function.
They talk about what a program “does”; of course it just sits there on the page or screen; it is the computer that does something. They ask whether a program “terminates”; of course it does; it is the behavior that may not terminate. A program is not behavior, but a specification of behavior. Furthermore, a computer may not behave as specified by a program for a variety of reasons: a disk head may crash, a compiler may have a bug, or a resource may become exhausted (stack overflow, number overflow), to mention a few.
4 Limits and Reals 3 Function Theory 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)) 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.
A Practical Theory of Programming by Eric C.R. Hehner