By Eric C.R. Hehner

Figuring out programming and programming languages calls for wisdom of the underlying theoretical version. This ebook explores facets of programming which are amenable to mathematical evidence. the writer describes a programming idea that is a lot less complicated and extra accomplished than the present theories to this point. within the theoretical version, a specification is simply a boolean expression and refinement is simply a normal implication. the writer develops a realistic and vast strategy for writing distinctive requirements and designing courses whose executions most likely fulfill the requirements. starting with preparatory fabric in good judgment, numbers, units, lists, features and family members, the e-book advances extra into software conception, the center of the publication. next chapters will be chosen or passed over in line with path emphasis. The textual content can be helpful 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 thought. The e-book includes various routines and worked-out recommendations for particular routines. Transparency masters and recommendations for the rest workouts can be found from the writer.

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Refinement by Steps allows us to introduce one programming construct at a time into our ultimate solution. The next law allows us to break the problem into parts in a different way. Refinement by Parts (monotonicity, conflation) If A ⇐ if b then C else D and E ⇐ if b then F else G are theorems, then A∧E ⇐ if b then C∧F else D∧G is a theorem. F are theorems, then A∧D ⇐ B∧E. C∧F is a theorem. If A ⇐ B and C ⇐ D are theorems, then A∧C ⇐ B∧D is a theorem. When we add to our repertoire of programming operators in later chapters, the new operators must obey similar Refinement by Steps and Refinement by Parts laws.

X′=x ∧ y′=y = x′=1 ∧ y′=y = x′=x ∧ y′=y , so it depends on x . 39 4 Program Theory (g) x:= 1. 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.

If f is a function with a numeric result, then Σf is the numeric result of applying f to all its domain elements and adding up all the results; and Πf is the numeric result of applying f to all its domain elements and multiplying together all the results. Here are four examples. ” Π〈n: nat+1→(4×n2)/(4×n2–1)〉 For the sake of tradition and convenience, we allow two abbreviated quantifier notations. First, we allow the scope brackets 〈 〉 following a quantifier to be omitted; now we have to change the arrow to a raised dot to avoid ambiguity.