Total correctness

Test Nnmber correct Number incorrect Total % Correct No. correct in excess of expectancy in cr values Significant difference... 

The total correction will vary from 0.6 for a poorly designed exchanger with large clearances to 0.9 for a well-designed exchanger. 

In Fig. 3 the total corrected emission spectrum of MBC in ethylcellulose filmsisshown(curve A)... 

The program below is totally correct for z - 2x and input x a nonnegative integer. 

DEFINITION Program (P,I) is totally correct with respect to input criterion A(X) and output criterion B(X,Z) if for all a in D11, whenever A(a) = TRUE, ... 

If we omit the input criterion in discussing partial or total correctness it is understood that we take as input criterion the function which is TRUE on all of Dn - i.e., all possible input vectors are regarded as legitimate input. 

There are several points to notice. We have not said how functions A and B are to be described, although for all practical considerations this is extremely important. Next, notice that B(X) = Z B(X,Z) - TRUE need not correspond to a function, i.e., B(a) could have no members or more than one member for input a. Thus there could be more than one function computed by programs totally correct for B if B(a,b) = B(a,b ) = TRUE for b b. This situation might occur if, for example, one wants the program to retrieve certain data but does not care about the order in which they are presented, or the program is to find a person or object meeting certain specifications and if more than one "correct" object exists, no preference is expressed as to which should be selected. 

Statements of either partial or of total correctness can be rendered meaningless by choice of A or B. For example, any program that always loops is partially correct for any A and B while any program is partially correct for B if B is defined as TRUE everywhere, and any program is totally correct with respect to A and B if A is defined as FALSE everywhere. 

Appendix A contains a brief summary of sane relevant ideas of satisfiability and validity of well-formed formulas in the predicate calculus. Using these ideas it gives a definition of partial and total correctness of a scheme with respect to a well-formed formula as output criterion. The treatment is cursory and nonrigorous. Readers who have not seen these ideas before should examine this appendix before we return to the treatment of correctness and program verification in Chapter V, and finally conclude this treatment in Chapjter VII. 

COROLLARY 5.4 If P is an always halting program scheme we can construct a quantifier-free well-formed formula p(P) such that P is totally correct with respect to TRUE and p(P). ... 

Of course we are usually interested in verifying not just partial correctness but also total correctness. We can do so using a variant of the previous methods, using now a formula 3 X V f [A(X) a W(P,A,N0T B,D]. ... 

We can now give the corresponding result for total correctness. First, suppose (P,I) is not totally correct for A and B. Either... 

P,I) cannot be totally correct for input criterion A and any output criterion. 

EXAMPLE A-l TOTAL CORRECTNESS FOR AN ALWAYS HALTING SCHEME... 

This scheme is totally correct with respect to TRUE and the wff ... 

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