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From: Nick Tudor < >

Date: Mon, 9 Mar 2015 17:02:14 +0000

Now back in the office...for a short while.

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The System Safety Mailing List

systemsafety_at_xxxxxx Received on Mon Mar 09 2015 - 18:02:22 CET

Date: Mon, 9 Mar 2015 17:02:14 +0000

Now back in the office...for a short while.

Good point David - well put.

I would have responded: There exists a person N who knows a bit about mathematics. Person N applies some mathematics and asserts Truth. Unfortunately, because of the incorrect application of the mathematics, the claims N now makes cannot be relied upon. The maths might well be correct, but the application is wrong because - and I have to say it yet again - the application misses fails to acknowledge that it is the environment that is random rather than the software. Software essentially boils down to a string of one's and nought's. Given the same inputs (and that always comes from the chaotic environment) then the output will always be the same. It therefore makes no sense to talk about 'software reliability'.

Nick Tudor

Tudor Associates Ltd

Mobile: +44(0)7412 074654

www.tudorassoc.com

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On 9 March 2015 at 12:26, David Haworth <david.haworth_at_xxxxxx wrote:

*> 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_xxxxxx
>
> --
> David Haworth B.Sc.(Hons.), OS Kernel Developer
> david.haworth_at_xxxxxx
> Tel: +49 9131 7701-6154 Fax: -6333 Keys:
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> Elektrobit Automotive GmbH Am Wolfsmantel 46, 91058 Erlangen,
> Germany
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> 4886
>
> Disclaimer: my opinion, not necessarily that of my employer.
>
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>
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The System Safety Mailing List

systemsafety_at_xxxxxx Received on Mon Mar 09 2015 - 18:02:22 CET

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