Halting Problem:
The idea of General Intelligence applicable to all problems has given rise to AGI, Artificial General Intelligence, computer simulation of human thought. Computers have bested humans in limited areas such as factory work, and games like chess,GO and Jeopardy.
It isn’t clear that a AI will replace total or General Human Intelligence. AGI reminds me of the problem of old explored by Alan Turing, the Universal Turing Machine. The Universal Turing Machine could replace particular Turing machines, solving a wide variety of computations. However, Turing found his Universal machine isn’t capable all computations. Some problems were “undecidable”. One of these was called The Halting Problem, when a program had finished calculating and could stop.
I wondered, by analogy, that just as Turing found a theoretical limit to computation, undecidable problems in his Universal Machine, might not modern computer scientists hit a similar wall with AGI. Is it mathematically possible to prove a limit of computation? In fact, there comes a time for all to understand, prove and measure our own limitations. For example, the fact the physicists can’t apply known laws to the earliest microseconds of our universe’s existence means that we can’t know how it everything came to be. That’s just one example.
I asked one of the new language models, Perplexity, the following three questions:
1. Isn’t AGI merely a recent upgrade of the Universal Turing Machine concept, that is solver of all problems?
2. What is the Halting Problem and why is it important?
3. Have some theoretical limits of AGI been determined in the manner of indeterminacy of the Halting problem?
In the case of the first two questions I was informed that the concepts of Universal Turing Machine and AGI are not the same. I meant that they were somewhat analogous such that the theoretical limits of AGI might be mathematically provable. The response of the Perplexity language model to my third inquiry is worth quoting in full:
“Theoretical limits of AGI have been compared to the indeterminacy of the Halting problem in some theories. The “Inherent Limitations of AGI” theory posits that there are fundamental limitations to an AGI’s capabilities, incorporating concepts like Gödel’s Incompleteness Theorems, Heisenberg’s Uncertainty Principle, and the Halting Problem. However, to the best of current knowledge, no mathematical proof or rigorous argumentation has been published demonstrating that the AGI control problem may be solvable, even in principle, much less in practice.
The Halting problem is a fundamental issue in computability theory that asks whether a given computer program, when provided with a specific input, will terminate or continue to run forever. It has been proven to be undecidable, meaning that there is no general algorithm that can solve it for all possible program-input pairs
The comparison between the theoretical limits of AGI and the Halting problem highlights the potential fundamental constraints in the understanding and prediction of various phenomena by AGI systems.”
According to which I maintain, there is some theoretical limit to computer and human understanding. AGI being much “hyped” and in the news these days, I see as worth your consideration.