I don’t see the point in adding so much complexity to such a simple matter. AIXI is an incomputable agent who’s proofs of optimality require a computable environment. It requires a specific configuration of the classic agent-environment-loop where the agent and the environment are independent machines. That specific configuration is only applicable to a sub-set of real-world problems in which the environment can be assumed to be much “smaller” than the agent operating upon it. Problems that don’t involve other agents and have very few degrees of freedom relative the agent operating upon them.
Marcus Hutter already proposed computable versions of AIXI like AIXI_lt. In the context of agent-environment loops, AIXI_lt is actually more general than AIXI because AIXI_lt can be applied to all configurations of the agent-environment loop including the embedded agent configuration. AIXI is a special case of AIXI_lt where the limits of “l” and “t” go to infinity.
Some of the problems you bring up seem to be concerned with the problem of reconciling logic with probability while others seem to be concerned with real-world implementation. If your goal is to define concepts like “intelligence” with mathematical formalizations (which I believe is necessary), then you need to delineate that from real-world implementation. Discussing both simultaneously is extremely confusing. In the real world, an agent only has is empirical observations. It has no “seeds” to build logical proofs upon. That’s why scientists talk about theories and evidence supporting them rather than proofs and axioms.
You can’t prove that the sun will rise tomorrow, you can only show that it’s reasonable to expect the sun to rise tomorrow based on your observations. Mathematics is the study of patterns, mathematical notation is a language we invented to describe patterns. We can prove theorems in mathematics because we are the ones who decide the fundamental axioms. When we find patterns that don’t lend themselves easily to mathematical description, we rework the tool (add concepts like zero, negative numbers, complex numbers, etc.). It happens that we live in a universe that seems to follow patterns, so we try to use mathematics to describe the patterns we see and we design experiments to investigate the extent to which those patterns actually hold.
The branch of mathematics for characterizing systems with incomplete information is probability. If you wan’t to talk about real-world implementations, most non-trivial problems fall under this domain.
I don’t see the point in adding so much complexity to such a simple matter. AIXI is an incomputable agent who’s proofs of optimality require a computable environment. It requires a specific configuration of the classic agent-environment-loop where the agent and the environment are independent machines. That specific configuration is only applicable to a sub-set of real-world problems in which the environment can be assumed to be much “smaller” than the agent operating upon it. Problems that don’t involve other agents and have very few degrees of freedom relative the agent operating upon them.
Marcus Hutter already proposed computable versions of AIXI like AIXI_lt. In the context of agent-environment loops, AIXI_lt is actually more general than AIXI because AIXI_lt can be applied to all configurations of the agent-environment loop including the embedded agent configuration. AIXI is a special case of AIXI_lt where the limits of “l” and “t” go to infinity.
Some of the problems you bring up seem to be concerned with the problem of reconciling logic with probability while others seem to be concerned with real-world implementation. If your goal is to define concepts like “intelligence” with mathematical formalizations (which I believe is necessary), then you need to delineate that from real-world implementation. Discussing both simultaneously is extremely confusing. In the real world, an agent only has is empirical observations. It has no “seeds” to build logical proofs upon. That’s why scientists talk about theories and evidence supporting them rather than proofs and axioms.
You can’t prove that the sun will rise tomorrow, you can only show that it’s reasonable to expect the sun to rise tomorrow based on your observations. Mathematics is the study of patterns, mathematical notation is a language we invented to describe patterns. We can prove theorems in mathematics because we are the ones who decide the fundamental axioms. When we find patterns that don’t lend themselves easily to mathematical description, we rework the tool (add concepts like zero, negative numbers, complex numbers, etc.). It happens that we live in a universe that seems to follow patterns, so we try to use mathematics to describe the patterns we see and we design experiments to investigate the extent to which those patterns actually hold.
The branch of mathematics for characterizing systems with incomplete information is probability. If you wan’t to talk about real-world implementations, most non-trivial problems fall under this domain.