(https://en.wikipedia.org/wiki/Reversible_computing)

It is both obvious and profound, the more information you already have, the more information you already have.

So we all know addition swap. One generalization that comes to mind is doing some other in-place transform on the two input variables. Lets keep it simple and suppose that its a linear transform. Thus the problem is to apply some matrix [[a,b],[c,d]] to two input variables [x,y] using entirely in-place operations.

We can do this by realizing that our basic operands can be expressed as matrices. x += ky is the same as the matrix [1, k] [0, 1]

likewise y += kx is equiv to the lower triangular matrix [1, 0] [k, 1]

and lastly, the = operator is equiv to a matrix with an element on the diag. x = k [k ,0] [0, 1]

y *= k [1, 0] [0, k]

From this point on it becomes a challenge of if you can construct any desired matrix into some combination of these available ones (spoiler, yes you can).

The next generalization one could contemplates is doing operations in place on more than 2 variables. Well, if one has already solved arbitrary 2x2 matrix operations, then that can be rigged to implement larger matrices one submatrix at a time.

The final generalization that comes to mind is what can we do with non-arithmetic operators? We've already seen an example of this with using xor-swap rather than addition-swap. But is there anything out there vaguely like xor-2x2-matrix-multiply?

I legit don't know. I have some thought, but I won't meander out loud if its not going to lead anywhere.

>It is clear that the risks of severe punishment for corrupt behavior are slight. A dishonest policeman knows that, even if he is caught and convicted, he will probably receive a court reprimand or, at most, a fairly short jail sentence. Considering the vast sums to be made in some plainclothes squads or in narcotics enforcement, the gains from corruption seem far to outweigh the risks. Both William Phillips and Edward Droge said that they assessed the risk of meaningful punishment and determined that they had little to fear.

I've read enough reports of this nature and experienced enough regulators to be really convinced that where you see apparent stupidity you are also likely to see a table like the one on page 250 showing virtually no prosecutions prior to the Commission.

Neither Drodge nor Phillips were wrong in their calculation. I hate to say, your estimate of a 10% chance of getting caught was way too optimistic. Going by the data, it was much much lower than that.

(My axe to grind is not with the police in particular, reports on police corruption just happen to be the most available and easily understandable).

This broadly goes for non-criminal acts too. Sometimes powerful people do seemingly dumb things because they are only dumb in the context of the incentive structures if one of us tried to do it. In their context, it would be stupid not to egregiously take advantage.

[1] https://www.youtube.com/watch?v=s93KC4AGKnY

(this is an unsolved problem that exists in many domains from long before AI)