By Eric C.R. Hehner

ISBN-10: 0387941061

ISBN-13: 9780387941066

Figuring out programming and programming languages calls for wisdom of the underlying theoretical version. This publication explores elements of programming which are amenable to mathematical facts. the writer describes a programming concept that's a lot easier and extra entire than the present theories up to now. within the theoretical version, a specification is simply a boolean expression and refinement is simply a standard implication. the writer develops a pragmatic and extensive process for writing distinctive requirements and designing courses whose executions most likely fulfill the requirements. starting with preparatory fabric in good judgment, numbers, units, lists, services and family members, the ebook advances extra into software thought, the center of the e-book. next chapters could be chosen or passed over in response to path emphasis. The textual content should be valuable to scholars in classes on programming technique or verification on the complicated 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 e-book anywhere attainable. No complicated mathematical wisdom or programming language is believed. The publication includes quite a few routines and worked-out strategies for particular routines. Transparency masters and ideas for the rest routines can be found from the writer.

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**Example text**

If A ⇐ B and C ⇐ D are theorems, then A∧C ⇐ B∧D is a theorem. When we add to our repertoire of programming operators in later chapters, the new operators must obey similar Refinement by Steps and Refinement by Parts laws. Our final refinement law is Refinement by Cases P ⇐ if b then Q else R is a theorem if and only if P ⇐ b ∧ Q and P ⇐ ¬b ∧ R are theorems. 1 List Summation As an example of program development, let us do Exercise 142: write a program to find the sum of a list of numbers. Let L be the list of numbers, and let s be a number variable whose final value will be the sum of the items in L .

Suppose there are two state variables x and y each with domain int . Then x′ = x+1 ∧ y′ = y specifies the behavior of a computer that increases the value of x by 1 and leaves y unchanged. Let us check that it is implementable. We replace ∀σ by either ∀x, y or ∀y, x and we replace ∃σ′ by either ∃x′, y′ or ∃y′, x′ ; according to the Commutative Laws, the order does not matter. We find One-Point Law twice ∀x, y· ∃x′, y′· x′ = x+1 ∧ y′ = y = ∀x, y· T Identity Law twice = T The specification is implementable.

Y:= 7 You can reduce the number of refinements by applying the Stepwise Refinement Law. A ⇐ if x=0 then (y:= 1. x:= 3) else (x:= x–1. y:= 7. A. y:= 2×y. x:= 5) You can translate this into a programming language that is available on a computer near you. For example, in C it becomes int x, y; void A (void) {if (x==0) {y = 1; x = 3;} else {x = x–1; y = 7; A ( ); y = 2*y; x = 5;}} You can then test it on a variety of x values. For example, execution of x = 5; A ( ); printf ("%i", y); will print 32 .

### A Practical Theory of Programming by Eric C.R. Hehner

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