It looks like the UK Post Office’s Horizon IT system is going to have a significant impact on the prosecution of cases that revolve around the reliability of software systems, at least in the UK. I have discussed the evidence illustrating the fallacy of the belief that “most computer error is either immediately detectable or results from error in the data entered into the machine.” This post discusses what can be learned about the reliability of a program after a fault experience has occurred, or alleged to have occurred in the Horizon legal proceedings.

Sub-postmasters used the Horizon IT system to handle their accounts with the Post Office. In some cases money that sub-postmasters claimed to have transferred did not appear in the Post Office account. The sub-postmasters claimed this was caused by incorrect behavior of the Horizon system, the Post Office claimed it was due to false accounting and prosecuted or fired people and sometimes sued for the ‘missing’ money (which could be in the tens of thousands of pounds); some sub-postmasters received jail time. In 2019 a class action brought by 550 sub-postmasters was settled by the Post Office, and the presiding judge has passed a file to the Director of Public Prosecutions; the Post Office may be charged with instituting and pursuing malicious prosecutions. The courts are working their way through reviewing the cases of the sub-postmasters charged.

How did the Post Office lawyers calculate the likelihood that the missing money was the result of a ‘software bug’?

Horizon trial transcript, day 1, Mr De Garr Robinson acting for the Post Office: “Over the period 2000 to 2018 the Post Office has had on average 13,650 branches. That means that over that period it has had more than 3 million sets of monthly branch accounts. It is nearly 3.1 million but let’s call it 3 million and let’s ignore the fact for the first few years branch accounts were weekly. That doesn’t matter for the purposes of this analysis. Against that background let’s take a substantial bug like the Suspense Account bug which affected 16 branches and had a mean financial impact per branch of £1,000. The chances of that bug affecting any branch is tiny. It is 16 in 3 million, or 1 in 190,000-odd.”

That 3.1 million comes from the calculation: 19-year period times 12 months per year times 13,650 branches.

If we are told that 16 events occurred, and that there are 13,650 branches and 3.1 million transactions, then the likelihood of a particular transaction being involved in one of these events is 1 in 194,512.5. If all branches have the same number of transactions, the likelihood of a particular branch being involved in one of these 16 events is 1 in 853 (13650/16 -> 853); the branch likelihood will be proportional to the number of transactions it performs (ignoring correlation between transactions).

This analysis does not tell us anything about the likelihood that 16 events will occur, and it does not tell us anything about whether these events are the result of a coding mistake or fraud.

We don’t know how many of the known 16 events are due to mistakes in the code and how many are due to fraud. Let’s ask the question: What is the likelihood of one fault experience occurring in a software system that processes a total of 3.1 million transactions (the number of branches is not really relevant)?

The reply to this question is that it is not possible to calculate an answer, because all the required information is not specified.

A software system is likely to contain some number of coding mistakes, and given the appropriate input any of these mistakes may produce a fault experience. The information needed to calculate the likelihood of one fault experience occurring is:

- the number of coding mistakes present in the software system,
- for each coding mistake, the probability that an input drawn from the distribution of input values produced by users of the software will produce a fault experience.

Outside of research projects, I don’t know of any anyone who has obtained the information needed to perform this calculation.

The Technical Appendix to Judgment (No.6) “Horizon Issues” states that there were 112 potential occurrences of the Dalmellington issue (paragraph 169), but does not list the number of transactions processed between these ‘issues’ (which would enable a likelihood to be estimated for that one coding mistake).

The analysis of the Post Office expert, Dr Worden, is incorrect in a complicated way (paragraphs 631 through 635). To ‘prove’ that the missing money was very unlikely to be the result of a ‘software bug’, Dr Worden makes a calculation that he claims is the likelihood of a particular branch experiencing a ‘bug’ (he makes the mistake of using the number of known events, not the number of unknown possible events). He overlooks the fact that while the likelihood of a particular branch experiencing an event may be small, the likelihood of any one of the branches experiencing an event is 13,630 times higher. Dr Worden’s creates complication by calculating the number of ‘bugs’ that would have to exist for there to be a 1 in 10 chance of a particular branch experiencing an event (his answer is 50,000), and then points out that 50,000 is such a large number it could not be true.

As an analogy, let’s consider the UK National Lottery, where the chance of winning the Thunderball jackpot is roughly 1 in 8-million per ticket purchased. Let’s say that I bought a ticket and won this week’s jackpot. Using Dr Worden’s argument, the lottery could claim that my chance of winning was so low (1 in 8-million) that I must have created a counterfeit ticket; they could even say that because I did not buy 0.8 million tickets, I did not have a reasonable chance of winning, i.e., a 1 in 10 chance. My chance of winning from one ticket is the same as everybody else who buys one ticket, i.e., 1 in 8-million. If millions of tickets are bought, it is very likely that one of them will win each week. If only, say, 13,650 tickets are bought each week, the likelihood of anybody winning in any week is very low, but eventually somebody will win (perhaps after many years).

The difference between the likelihood of winning the Thunderball jackpot and the likelihood of a Horizon fault experience is that we have enough information to calculate one, but not the other.

The analysis by the defence team produced different numbers, i.e., did not conclude that there was not enough information to perform the calculation.

Is there any way that the information needed to calculate the likelihood of a fault experience occurring?

In theory fuzz testing could be used. In practice this is probably completely impractical. Horizon is a data driven system, and so a copy of the database would need to be used, along with a copy of all the Horizon software. Where is the computer needed to run this software+database? Yes, use of the Post Office computer system would be needed, along with all the necessary passwords.

Perhaps if we wait long enough, a judge will require that one party make all the software+database+computer+passwords available to the other party.