Complex Systems that are Decidable | TAU - AGORAS 💎 #tau #taunet

Question: How does Tau deal with Gödel's incompleteness that for sufficiently complex systems, there will always be statements that cannot be proven or disproven within the system. Even if a question could theoretically be answered by the system, the computational resources required might be prohibited.

Answer: Yes. So indeed, as you said, for sufficiently complex systems, and to find a system that is complex enough, let's call it an impressive amount of complexity, but does not step into the incompleteness or in other words, is decidable. Well, that's exactly the whole art. That's the main thing to do. And that's what we do. And we did. So the complexity of the Tau language is indeed, in the worst case prohibitive, double exponential time. Well, we don't know for sure. We only know that it is between double exponential and triple exponential. But there are two points to bear in mind here. That experience shows exponential in the whole field of logical solvers, that real-life problems happen to be easy to solve, relatively easy to solve, and random problems are the hard problems. There is no theoretical explanation for that. People say that real-life problems typically come with a structure, and therefore the solvers can use the structure to handle them easily. No one knows why it works, but it has been an empirical fact for over 20 years that it works. The second point is that that's exactly why we invest so much in finding good algorithms. And good algorithms means that in many easy cases, it will return answer fast. So we have a good algorithm for the Tau language. We had it long ago, and now we are indeed looking for a good algorithm for the two variable fragments with counting.

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