Limits to computing: Even in the age of AI, some problems are just too difficult

By Jie Wang

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Not every problem can be solved

May not need to be. Gödel’s incompleteness theorem addressed this and Asimov in his story "The Last Question".

LLMs like ChatGPT look to be leading toward Artificial General Intelligence which could be a bridge to Artificial Superior Intelligence.

“The question of whether a computer can think is no more interesting than the question of whether a submarine can swim.”

Edsger W. Dijkstra

I like this quote because it shows that ASI could be not very human-like in its type of intelligence.

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Godels's theorem makes a different point to the NP problem. The incompleteness result states that there are mathematical theorems which are correct but there can be no proof constructed to demonstrate that. In contrast, we know the Travelling Salesman problem has a solution even if we cannot compute it in an acceptable length of time. The number of different routes is finite - although very numerous - and therefore a minimum exists for the finte set.

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It’s just no problem at all if you would understand that all those computations are only theoretical models, but far away from reality. It’s important to broaden the algorithms and insert some more fuzziness instead of only binary outcomes. Two easy examples. What’s the average of dice throws? Yes, 3.5, but neither side of the dice has a 3.5 on it. Another one, tossing a coin, everyone says , the probability for head or tail is 0.5 But it’s not, it’s much lower, because in quite some reality scenarios you haven’t even an outcome of head or tails, when the coin sinks into sea, when an astronaut throws it into outer space, and if a steel worker throws it into the melting pot you won’t even have that coin anymore. And all that generalized is when the new quantum computation comes into play. Many unsolvable problems might become solvable soon, but of course again not all of them.

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AI has quite low functionality ceilings when dealing with stuff that humans regularly deal with.

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