Sigh...that rabbit hole.
a) There is insufficient data unless some omniscient being is commanding the fleets. There always is in stochastic systems.
b) FPS AI is easy. Maybe it's a bad example because people will latch onto the 'aimbot' concept, but aside from motor skills the actual strategy element of of FPS AI gameplay is bad. The bots are dumb no matter how good their aim is. They can't adapt to different strategies well because their AI works on limited data and is therefore highly deterministic (enemy detected here, go here, shoot enemy -- maybe with a bit of flair here or there).
c) Non-sentient AIs are completely incapable of math. Computers do computations well. (As my first analysis professor said on the first day of class 'Forget what you learned in high school. Mathematics is not compuations.') They are wholly incapable of actual mathematics because they lack sentience. If an AI were able to perform actual mathematics (i.e. proofs and such) then there'd be the so-called singularity at which point all knowledge could quickly be gained through technological means.
These things, especially (a) and (c) lie at the core of information theory and AI theory. They are irrefutable in those current theories.
Ok I see that you are latching on my simplified use of language and generalizations I used to keep it relatively short. This does not mean I don't know what I'm talking about.
To answer a)
Insufficient data is not a problem in space combat, there is no way in Stellaris to hide a ship, and there is also no way in reality to hide a ship, unless you park it behind a planet or star. But you would still see it approaching from light years away, either by some kind of FTL emissions from their method of FTL travel or if it doesn't have FTL then because it smacks into hydrogen atoms initiating fusion, if it approaches anywhere close to the speed of light, or from its infrared emissions and because it blocks out the light of the stars behind it. So no stealth in space barring the invention of a cloaking device.
This leads automatically to the next consequence. If you can't hide, fleeing or engaging are your only options. If you decide to engage, you want all your ships to have two abilities, room to maneuver and a free line of sight, especially if your fleet is more powerful. You could either use a standard formation like a wall or have a computer generated optimal distribution given that each of your ships is a known quantity. This does not require it to understand math, it just needs to do calculations to do that. And the most likely solution for attacking over large distances is by using probability to discern where the other ships are going to be, the more shots you can fire at once, the more likely you are to hit. Once you get to shorter ranges, accuracy will improve dramatically, so that's something a smaller fleet should avoid.
To be b)
I didn't claim they were intelligent, I said their lack of intelligence is compensated or more precisely overcompensated by the ability to be more accurate and with faster reaction times than you are. So much, that in order for you to have a chance they need to dial this ability down.
To c)
This is incorrect, plain and simple. Automated theorem proving is one such example. It doesn't even require AI. Formulating a theorem is currently beyond them, but proving it correct or incorrect isn't. But they are by now also capable of inductive reasoning, meaning they can make decisions based on incomplete information, just as humans can. Some years back there was an episode of jeopardy featuring the Watson AI by IBM, who was capable of that. One of the questions was to name "A long tiresome speech delivered by a frothy pie topping" and it came up with the correct answer, "What is meringue harangue". This answer wasn't anywhere in its database. A similar question asked for a rhyming "Boxing term for a hit below the belt" and it came up with the incorrect answer "A Wang bang" this was a term it invented because it didn't know the correct answer "A low blow".
Sure those examples aren't math, but they are of a similar concept. Math is just another language after all. So yes, an AI is capable of learning math, and it can actually be creative as well. Which is why the field of AI safety is so important, before we create an AI that has the potential to become a Technological singularity, we need to be able to control it, because once it reaches that stage, we will no longer be able to understand how it works.
