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

[David Haworth]

Peter,

there's nothing wrong with the mathematics, but I've got
one little nit-pick about its application in the real world.

The mathematics you describe gives two functions f and g,
one of which is the model, the other is the implementation.

In practice, your implementation runs on a computer and so the
domain and range are not "the continuum". If your model is mathematical
(or even runs on a different computer), the output of one will
necessarily be different from the output of the other. That
may not be a problem in the discrete sense - you simply specify a
tolerance t > 0 in the form of:

Corr-f-g(i) = 0 if and only if |f(i)-g(i)| < t

etc.

The problem becomes much larger in the real world of control
systems where the output influences the next input of the
sequence. The implementation and the model will tend to drift
apart. In the worst case what might be nice and stable in the
model might exhibit unstable behaviour in the implementation.

You're then in the subject of mathematical chaos, where a
perfectly deterministic system exhibits unstable and unpredictable
behaviour. However, this email is too small to describe it. :-)

Cheers,
Dave

On 2015-03-09 11:48:57 +0100, Peter Bernard Ladkin wrote:
> Nick,
> 
> 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  www.rvs.uni-bielefeld.de
> 
> 
> 
> 
> _______________________________________________
> The System Safety Mailing List
> systemsafety at TechFak.Uni-Bielefeld.DE

-- 
David Haworth B.Sc.(Hons.), OS Kernel Developer    david.haworth at elektrobit.com
Tel: +49 9131 7701-6154     Fax: -6333                  Keys: keyserver.pgp.com
Elektrobit Automotive GmbH           Am Wolfsmantel 46, 91058 Erlangen, Germany
Geschäftsführer: Alexander Kocher, Gregor Zink       Amtsgericht Fürth HRB 4886

Disclaimer: my opinion, not necessarily that of my employer.
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