By Eric C.R. Hehner

Figuring out programming and programming languages calls for wisdom of the underlying theoretical version. This booklet explores points of programming which are amenable to mathematical facts. the writer describes a programming idea that's a lot easier and extra complete than the present theories to this point. within the theoretical version, a specification is simply a boolean expression and refinement is simply a regular implication. the writer develops a realistic and huge procedure for writing exact standards and designing courses whose executions most likely fulfill the standards. starting with preparatory fabric in common sense, numbers, units, lists, services and family, the ebook advances extra into application thought, the center of the e-book. next chapters will be chosen or passed over in response to path emphasis. The textual content may be helpful 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 e-book anyplace attainable. No complex mathematical wisdom or programming language is thought. The ebook comprises a variety of routines and worked-out recommendations for particular routines. Transparency masters and suggestions for the remainder routines can be found from the writer.

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At the other extreme is the specification ⊥ , which is not satisfied by any computer behavior. But ⊥ is not the only unimplementable specification. Here is another. x≥0 ∧ y′=0 If the initial value of x is nonnegative, the specification can be satisfied by setting variable y to 0 . But if the initial value of x is negative, there is no way to satisfy the specification. Perhaps the specifier has no intention of providing a negative input; in that case, the specifier should have written x≥0 ⇒ y′=0 For nonnegative initial x , this specification still requires variable y to be assigned 0 .

Usually when we introduce names and axioms we want them for some local purpose. The reader is supposed to understand their scope, the region where they apply, and not use them beyond it. Though the names and axioms are formal (expressions in our formalism), up to now we have introduced them informally by English sentences. But the scope of informally introduced names and axioms is not always clear. In this chapter we present a formal notation for introducing a local name and axiom. A variable is a name that is introduced for the purpose of instantiation (replacing it).

To talk about time, we just add a time variable. 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.

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