Re: [SystemSafety] Software reliability (or whatever you would prefer to call it)

From: Peter Bernard Ladkin < >
Date: Mon, 09 Mar 2015 11:48:57 +0100


Consider a mathematical function, f with domain D and range R. Given input i \in D, the output is f(i).

Consider another function, g, let us say for simplicity with the same input domain D and range R.

Define a Boolean function on D, Corr-f-g(i):

Corr-f-g(i) = 0 if and only if f(i)=g(i); Corr-f-g(i) = 1 if and only if f(i) NOT-EQUAL g(i)

If X is a random variable taking values in D, then f(X), g(X) are random variables taking values in R, and Corr-f-g(X) is a random variable taking values in {0,1}.

If S is a sequence of values of X, then let Corr-f-g(S) be the sequence of values of Corr-f-g corresponding to the sequence S of X-values.

Define Min-1(S) to be the least place in Corr-f-g(S) containing a 1; and to be 0 if there is no such place.

Suppose I construct a collection of sequences S.i, each of length 1,000,000,000, by repeated sampling from Distr(X). Suppose there are 100,000,000 sequences I construct.

I can now construct the average of Min-1(S) over all the 1,000,000,000sequences S.i.

All these things are mathematically well-defined.

Now, suppose I have deterministic software, S. Let f(i) be the output of S on input i. Let g(i) be what the specification of S says should be output by S on input i. Corr-f-g is the correctness function of S, and Mean(Min-1(S)) will likely be very close to the mean time/number-of-demands to failure of S if you believe the Laws of Large Numbers.

I have no idea why you want to suggest that all this is nonsensical and/or wrong. It is obviously quite legitimate well-defined mathematics.

PBL Prof. Peter Bernard Ladkin, Faculty of Technology, University of Bielefeld, 33594 Bielefeld, Germany Je suis Charlie
Tel+msg +49 (0)521 880 7319

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