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

By Eric C.R. Hehner

Knowing programming and programming languages calls for wisdom of the underlying theoretical version. This ebook explores elements of programming which are amenable to mathematical facts. the writer describes a programming idea 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 regular implication. the writer develops a pragmatic and large process for writing detailed necessities and designing courses whose executions most likely fulfill the standards. starting with preparatory fabric in good judgment, numbers, units, lists, features and kinfolk, the ebook advances extra into application concept, the guts of the publication. next chapters should be chosen or passed over in line with direction emphasis. The textual content may be beneficial to scholars in classes on programming method 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 booklet comprises various workouts and worked-out ideas for particular routines. Transparency masters and strategies for the remainder routines can be found from the writer.

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**Example text**

Suppose x is a nonlocal variable, and we reintroduce it in an inner scope. 〈y→ x y 〈x→ x y 〉 x y 〉x The Application Axiom tells us to substitute x for all occurrences of y . All three uses of y are the variable introduced by the outer scope, so all three must be replaced by the nonlocal x used as argument. But that will place a nonlocal x inside a scope that reintroduces x , making it look local. Before we substitute, we must use the Renaming Axiom for the inner scope. Choosing fresh name z , we get = 〈y→ x y 〈z→ z y 〉 x y 〉x by renaming, and then substitution gives = ( x x 〈z→ z x 〉 x x ) The Application Axiom (for element x: D ) 〈v: D→b〉 x = (substitute x for v in b ) provides us with a formal notation for substitution.

For example, the Specialization and Generalization laws at the back of the book say that if x: D , ∀v: D· b ⇒ 〈v: D→b〉 x 〈v: D→b〉 x ⇒ ∃v: D· b Together they can be written as follows: if x: Δf ∀f ⇒ f x ⇒ ∃f If f results in T for all its domain elements, then f results in T for domain element x . And if f results in T for domain element x , then there is a domain element for which f results in T . The One-Point Laws say that if x: D , and v does not appear in x , then ∀v: D· v=x ⇒ b = 〈v: D→b〉 x ∃v: D· v=x ∧ b = 〈v: D→b〉 x For instance, in the universal quantification ∀n: nat· n=3 ⇒ n<10 , we see an implication whose antecedent equates the variable to an element.

Similarly, S∨R is a specification that is satisfied by any computation that satisfies either S or R . Similarly, ¬S is a specification that is satisfied by any computation that does not satisfy S . A particularly useful operator is if b then S else R where b is a boolean expression of the initial state; it can be implemented by a computer that evaluates b , and then, depending on the value of b , behaves according to either S or R . The ∨ and if then else operators have the nice property that if their operands are implementable, so is the result; the operators ∧ and ¬ do not have that property.