I'm thinking the objective function could have constraints on the expected number of times the AI breaks the law, or the probability that it breaks the law, e.g.
- only actions with a probability of breaking any law < 0.0001 are permissible, or
- only actions for which the expected number of broken laws is < 0.001 are permissible.
There could also be separate constraints for individual laws or groups of laws, and these could depend on the severity of the penalties.
Looser constraints like this seem like they could avoid issues of lexicality and prioritizing avoidance of breaking the law over everything we want the AI to actually do, since the surest way to avoid breaking the law completely would be to never do anything (although we could also have a separate constraint for this).
Of course, the constraints should depend on breaking the law, not just being caught breaking the law, so the AI should predict whether or not it will break the law, not merely whether or not it will be caught breaking the law.
The AI could also predict whether or not it will break laws that don't exist now but will in the future (possibly even in response to its actions).
What are the challenges and problems with such an approach? Would it be too difficult to capture such constraints? Are laws too imprecise or ambiguous for this? Can we just have the AI consider multiple interpretations of the laws or try to predict how a human (or human judge) would interpret the law and apply it to its actions given the information the AI has?
How much work should the AI spend on estimating the probabilities that it will break laws?
What kinds of cases would it miss, say, given current laws?
I suspect current laws capture enough of what we care about that if an AGI followed them "properly", this wouldn't lead to worse outcomes than without AGI at all in expectation, but there could be holes to exploit and "properly" is where the challenge is, as you suggest. Many laws would have to be interpreted more broadly than before, perhaps.
Isn't interpreting statements (e.g. laws) and checking if they apply to a given action a narrower, more structured and better-defined problem than getting AI to do what we want it to do? If the AI can find an interpretation of a law according to which an action would break it with high enough probability, then that action would be ruled out. This seems like it could be a problem of reasoning and understanding language, instead of the problem of understanding and acting in line with human values.
To illustrate, "Maximize paperclips without killing anyone" is not an interpretation of "Maximize paperclips", but "Any particular person dies at least 1 day earlier with probability > p than they would have by inaction" could be an interpretation of "produce death" (although it might be better to rewrite laws in more specific numeric terms in the first place).
Defining a good search space (and search method) for interpretations of a given statement might still be a very difficult problem, though.