By Eric C.R. Hehner
Realizing programming and programming languages calls for wisdom of the underlying theoretical version. This publication explores elements of programming which are amenable to mathematical evidence. the writer describes a programming concept that is a lot easier and extra accomplished than the present theories so far. within the theoretical version, a specification is simply a boolean expression and refinement is simply a typical implication. the writer develops a realistic and huge technique for writing designated necessities and designing courses whose executions most likely fulfill the requisites. starting with preparatory fabric in good judgment, numbers, units, lists, capabilities and family members, the ebook advances extra into software idea, the guts of the e-book. next chapters should be chosen or passed over in accordance with direction emphasis. The textual content should be valuable 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 confirmed within the publication anywhere attainable. No complex mathematical wisdom or programming language is believed. The publication includes a variety of workouts and worked-out suggestions for particular routines. Transparency masters and ideas for the remainder workouts can be found from the writer.
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Extra info for A Practical Theory of Programming
Y:= 2 Although x does not appear in y:= 2 , the answer is not y:= 2 . We must remember that y:= 2 is defined by an axiom, and it depends on x . = x:= 1. x′=x ∧ y′=2 = x′=1 ∧ y′=2 (It is questionable whether x′=1 ∧ y′=2 is a “simplification” of x:= 1. ) (h) x:= 1. P where P = y:= 2 This one just combines the points of parts (f) and (g). = x′=1 ∧ y′=2 (i) x:= 1. y:= 2. x:= x+y In part (g) we saw that x:= 1. y:= 2 = x′=1 ∧ y′=2 . If we use that, we are then faced with a dependent composition x′=1 ∧ y′=2.
In practice, to prove that P is refined by S , we work within the universal quantifications and prove P ⇐ S . In this context, we can pronounce P ⇐ S as “ P is refined by S ”. 4 Program Theory 40 Here are some examples of refinement. x′>x ⇐ x′=x+1 ∧ y′=y x′=x+1 ∧ y′=y ⇐ x:= x+1 x′≤x ⇐ if x=0 then x′=x else x′
Here is an example in which the gaps represent uninteresting parts. 〈x→ x 〈x→ x 〉 x 〉3 Variable x is introduced twice: it is reintroduced in the inner scope even though it was already introduced in the outer scope. Inside the inner scope, the x is the one introduced in the inner scope. The outer scope is a function, which is being applied to argument 3 . Assuming 3 is in its domain, the Application Axiom says that this expression is equal to one obtained by substituting 3 for x . The intention is to substitute 3 for the x introduced by this function, the outer scope, not the one introduced in the inner scope.
A Practical Theory of Programming by Eric C.R. Hehner