By Eric C.R. Hehner
ISBN-10: 0387941061
ISBN-13: 9780387941066
Realizing programming and programming languages calls for wisdom of the underlying theoretical version. This publication explores points of programming which are amenable to mathematical facts. the writer describes a programming idea that is a lot less complicated and extra accomplished than the present theories thus far. within the theoretical version, a specification is simply a boolean expression and refinement is simply a normal implication. the writer develops a pragmatic and large process for writing certain necessities and designing courses whose executions most likely fulfill the standards. starting with preparatory fabric in common sense, numbers, units, lists, services and kinfolk, the booklet advances extra into application concept, the center of the ebook. next chapters might be chosen or passed over in keeping with direction emphasis. The textual content could be precious to scholars in classes on programming technique 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 established within the publication at any place attainable. No complicated mathematical wisdom or programming language is believed. The publication includes a number of workouts and worked-out strategies for particular workouts. Transparency masters and options for the rest routines can be found from the writer.
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Extra resources for A Practical Theory of Programming
Sample text
A parameter stands for an element of the domain, and the Application Law requires the argument to be an element of the domain, but functions are not elements. Therefore we consider a higher-order function applied to an argument, as written above, to be an abbreviation for 〈f: (A→B)→ ... ~f ... 〉 {g} and the set brackets { } just make the parameter and argument into The power operator elements, as required, and the content operator ~ then removes the set structure. Here is a predicate whose parameter is a function.
It is also the bunch of all functions whose domain includes nat and whose result is included in nat . suc: nat→nat use Function Inclusion Law = nat: Δsuc ∧ ∀n: nat· suc n: nat definition of suc = nat: nat ∧ ∀n: nat· n+1: nat reflexivity, and nat construction axiom = T We can prove similar inclusions about other functions defined in the first section of this chapter. max: xrat→xrat→xrat min: xrat→xrat→xrat even: int→bool odd: int→bool divides: (nat+1)→int→bool And, more generally, f: A→B = A: Δf ∧ fA: B End of Function Inclusion and Equality We earlier defined suc by the axiom suc = 〈n: nat→n+1〉 This equation can be written instead as Δsuc = nat ∧ ∀n: nat· suc n = n+1 We could have defined suc by the weaker axiom nat: Δsuc ∧ ∀n: nat· suc n = n+1 which is almost as useful in practice, and allows suc to be extended to a larger domain later, if desired.
After a time, the output from the computation is the final state, or poststate. Although the memory contents may physically be a string of bits, we can consider it to be a string of any items; we only need to group the bits and view them through a code. 14 The indexes of the items in a state are usually called “addresses”. The bunch of possible states is called the state space. 20); char; rat If the memory is in state σ , then the items in memory are σ0 , σ1 , σ2 , and so on. Instead of using addresses, we find it much more convenient to refer to items in memory by distinct names such as i , n , c , and x .