By Eric C.R. Hehner
Figuring out programming and programming languages calls for wisdom of the underlying theoretical version. This e-book explores points of programming which are amenable to mathematical evidence. the writer describes a programming concept that's a lot less complicated and extra finished than the present theories up to now. within the theoretical version, a specification is simply a boolean expression and refinement is simply a normal implication. the writer develops a pragmatic and vast procedure for writing distinct requirements and designing courses whose executions most likely fulfill the necessities. starting with preparatory fabric in common sense, numbers, units, lists, capabilities and kin, the booklet 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 might be priceless 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 established within the publication anywhere attainable. No complicated mathematical wisdom or programming language is believed. The e-book comprises quite a few routines and worked-out options for particular workouts. Transparency masters and ideas for the rest routines can be found from the writer.
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Extra resources for A Practical Theory of Programming
For example, ∃n: nat· x = 2×n says that x is an even natural. The local variable n , which could just as well have been m or any other name, is used to help say that x is an even natural. The expression is talking about x , not about n . End of Function Theory 34 4 Program Theory We begin with a very simple model of computation. A computer has a memory, and we can observe its contents, or state. Our input to a computation is to provide an initial state, or prestate. After a time, the output from the computation is the final state, or poststate.
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 function that sometimes produces more than one result is called “nondeterministic”. n〉 is both partial and nondeterministic. 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.
S) Distributive Law (if b then P else Q)∧R = if b then P∧R else Q∧R Functional-Imperative Law x:= if b then e else f = if b then x:= e else x:= f In the second Distributive Law, we can replace ∧ with any other boolean operator. We can even replace it with dependent composition with a restriction: If b is a boolean expression of the prestate (in unprimed variables), Distributive Law (if b then P else Q). R = if b then (P. R) else (Q. R) And finally, if e is any expression of the prestate (in unprimed variables), Substitution Law x:= e.
A Practical Theory of Programming by Eric C.R. Hehner