By Eric C.R. Hehner
Figuring out programming and programming languages calls for wisdom of the underlying theoretical version. This publication explores features of programming which are amenable to mathematical evidence. the writer describes a programming conception that's a lot less complicated 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 realistic and huge technique for writing detailed necessities and designing courses whose executions most likely fulfill the requisites. starting with preparatory fabric in good judgment, numbers, units, lists, services and relatives, the ebook advances additional into software conception, the center of the e-book. next chapters will be chosen or passed over in response to direction emphasis. The textual content may be beneficial to scholars in classes on programming method 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 verified within the e-book at any place attainable. No complicated mathematical wisdom or programming language is believed. The ebook comprises a variety of routines and worked-out recommendations for particular routines. Transparency masters and suggestions for the rest workouts can be found from the writer.
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Additional info for A Practical Theory of Programming
F (g, h) = f g, f h (f, g) h = f h, g h Operators and functions are similar; each applies to its operands to produce a result. Just as we compose functions, we can compose operators, and we can compose an operator with a function. For example, we can compose – with any function f that produces a number to obtain a new function. (–f) x = –(f x) In particular, (–suc) 3 = –(suc 3) = –4 Similarly if p is a predicate, then (¬p) x = ¬(p x) We can compose ¬ with even to obtain odd again. #L . n for some natural n , and whose body is an item, can sometimes be regarded as a kind of list.
We do not change the theory at all; the time variable is treated just like any other variable, as part of the state. The state σ = t; x; y; ... now consists of a time variable t and some memory variables x , y , ... The interpretation of t as time is justified by the way we use it. In an implementation, the time variable does not require space in the computer's memory; it simply represents the time at which execution occurs. We use t for the initial time, the time at which execution starts, and t′ for the final time, the time at which execution ends.
0 Function Inclusion and Equality optional A function f is included in a function g according to the Function Inclusion Law: f: g = Δg: Δf ∧ ∀x: Δg· fx: gx Using it both ways round, we find function equality is as follows: f = g = Δf = Δg ∧ ∀x: Δf· fx = gx We now prove suc: nat→nat . Function suc was defined earlier as suc = 〈n: nat→n+1〉 . Function nat→nat is an abbreviation of 〈n: nat→nat〉 , which has an unused variable. It is a nondeterministic function whose result, for each element of its domain nat , is the bunch nat .
A Practical Theory of Programming by Eric C.R. Hehner