Constructing Correct Software (Formal Approaches to by D. John Cooke

By D. John Cooke

Vital to Formal equipment is the so-called Correctness Theorem which relates a specification to its right Implementations. This theorem is the aim of conventional application checking out and, extra lately, of software verification (in which the theory needs to be proved). Proofs are tough, although in spite of using strong theorem provers. This quantity explains and illustrates another procedure, which permits the development of (necessarily right) algorithms from a specification utilizing algebraic variations and refinement suggestions which stop the advent of mistakes. in accordance with educating fabric used broadly at Loughborough college, John Cooke introduces the fundamentals, utilizing uncomplicated examples and many specified operating (which can usually be re-used). developing right software program will supply worthwhile interpreting for college students and practitioners of laptop technology and software program Engineering to whom correctness of software program is of major value.

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9 Referring to the figure and definition of R1, R 1 –{”x,y’: X Ù Y | x = y + 1 } it is clear that it is fruitless to attempt to find an answer (an answer compatible with R 1 ) starting from the value 1 as input. e. that x is in the domain, is: x>1 . This condition, often called the weakest pre-condition and denoted by wp(R 1 ), is the least restrictive condition upon values of the appropriate source type for which the specification includes ‘answers’. Notice that this condition is derived from the relation and hence is, in some sense, ‘going backwards’, from output values to inputs.

The logical notion of “existential quantification” (the symbol ∃) works in exactly the s ame way as local variables in block-structured programming languages — no more, no less. ] Also defined is the mirror-image concept of the range of the relation R. This is written R and is defined: R – {y:Y | (∃x:X)(”x,y’˜R) } where R: (X Ù Y) This gives the set of all values in Y that are to be found at the pointed end of the arrows which comprise R. 9. In this figure, the relation R1 is used as the example.

Its type is indicated thus: post-function_name: X Ù Y — These specification components are sufficient in any situation that is not required to indicate a desired change of value. , spec-function_name – {”x,y’: X Ù Y | pre-function_name  post-function_name}. A function of type X — Y with f = X is said to be a total function or mapping. When f is a mapping, pre-f(x) – True, and the pre-condition may be omitted without information loss. 9 So clearly the specification is a subset of X Ù Y and is of type (X Ù Y).

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