I see several good answers, but most are assuming that inferential infinite loop is a thing of the past, only related to logical AI (the famous GOFAI). But it's not.
An infinite loop can happen in any program, whether it's adaptive or not. And as @SQLServerSteve pointed out, humans can also get stuck in obsessions and paradoxes.
Modern approaches are mainly using probabilistic approaches. As they are using floating numbers, it seems to people that they are not vulnerable to reasoning failures (since most are devised in binary form), but that's wrong: as long as you are reasoning, some intrinsic pitfalls can always be found that are caused by the very mechanisms of your reasoning system. Of course, probabilistic approaches are less vulnerable than monotonic logic approaches, but they are still vulnerable. If there was a single reasoning system without any paradoxes, much of philosophy would have disappeared by now.
For example, it's well known that Bayesian graphs must be acyclic, because a cycle will make the propagation algorithm fail horribly. There are inference algorithms such as Loopy Belief Propagation that may still work in these instances, but the result is not guaranteed at all and can give you very weird conclusions.
On the other hand, modern logical AI overcame the most common logical paradoxes you will see, by devising new logical paradigms such as non-monotonic logics. In fact, they are even used to investigate ethical machines, which are autonomous agents capable of solving dilemmas by themselves. Of course, they also suffer from some paradoxes, but these degenerate cases are way more complex.
The final point is that inferential infinite loop can happen in any reasoning system, whatever the technology used. But the "paradoxes", or rather the degenerate cases as they are technically called, that can trigger these infinite loops will be different for each system depending on the technology AND implementation (AND what the machine learned if it is adaptive).
OP's example may work only on old logical systems such as propositional logic. But ask this to a Bayesian network and you will also get an inferential infinite loop:
- There are two kinds of ice creams: vanilla or chocolate.
- There's more chances (0.7) I take vanilla ice cream if you take chocolate.
- There's more chances (0.7) you take vanilla ice cream if I take chocolate.
- What is the probability that you (the machine) take a vanilla ice cream?
And wait until the end of the universe to get an answer...
Disclaimer: I wrote an article about ethical machines and dilemmas (which is close but not exactly the same as paradoxes: dilemmas are problems where no solution is objectively better than any other but you can still choose, whereas paradoxes are problems that are impossible to solve for the inference system you use).
/EDIT: How to fix inferential infinite loop.
Here are some extrapolary propositions that are not sure to work at all!
- Combine multiple reasoning systems with different pitfalls, so if one fails you can use another. No reasoning system is perfect, but a combination of reasoning systems can be resilient enough. It's actually thought that the human brain is using multiple inferential technics (associative + precise bayesian/logical inference). Associative methods are HIGHLY resilient, but they can give non-sensical results in some cases, hence why the need for a more precise inference.
- Parallel programming: the human brain is highly parallel, so you never really get into a single task, there are always multiple background computations in true parallelism. A machine robust to paradoxes should foremost be able to continue other tasks even if the reasoning gets stuck on one. For example, a robust machine must always survive and face imminent dangers, whereas a weak machine would get stuck in the reasoning and "forget" to do anything else. This is different from a timeout, because the task that got stuck isn't stopped, it's just that it doesn't prevent other tasks from being led and fulfilled.
As you can see, this problem of inferential loops is still a hot topic in AI research, there will probably never be a perfect solution (no free lunch, no silver bullet, no one size fits all), but it's advancing and that's very exciting!