# A wager against Solomonoff induction

The universal prior assigns zero probability to non-computable universes—for instance, universes that could only be represented by Turing machines in which uncountably many locations need to be updated, or universes in which the halting problem is solved in physics. While such universes might very likely not exist, one cannot justify assigning literally zero credence to their existence. I argue that it is of overwhelming importance to make a potential AGI assign a non-zero credence to incomputable universes—in particular, universes with uncountably many “value locations”.

Here, I assume a model of universes as sets of value locations. Given a specific goal function, each element in such a set could specify an area in the universe with some finite value. If a structure contains a sub-structure, and both the structure and the sub-structure are valuable in their own regard, there could either be one or two elements representing this structure in the universe’s set of value locations. If a structure is made up of infinitely many sub-structures, all of which the goal function assigns some positive but finite value to, then this structure could (if the sum of values does not converge) possibly only be represented by infinitely many elements in the set. If the set of value locations representing a universe is countable, then the value of said universe could be the sum over the values of all elements in the set (granted that some ordering of the elements is specified). I write that a universe is “countable” if it can be represented by a finite or countably infinite set, and a universe is “uncountable” if it can only be represented by an uncountably infinite set.

A countable universe, for example, could be a regular cellular automaton. If the automaton has infinitely many cells, then, given a goal function such as total utilitarianism, the automaton could be represented by a countably infinite set of value locations. An uncountable universe, on the other hand, could be a cellular automaton in which there is a cell for each real number, and interactions between cells over time are specified by a mathematical function. Given some utility functions over such a universe, one might be able to represent the universe only by an uncountably infinite set of value locations. Importantly, even though the universe could be described in logic, it would be incomputable.

Depending on one’s approach to infinite ethics, an uncountable universe could matter much more than a countable universe. Agents in uncountable universes might—with comparatively small resource investments—be able to create (or prevent), for instance, amounts of happiness or suffering that could not be created in an entire countable universe. For instance, each cell in the abovementioned cellular automaton might consist of some (possibly valuable) structure in of itself, and the cells’ structures might influence each other. Moreover, some (uncountable) set of cells might be regarded as an agent. The agent might then be able to create a positive amount of happiness in uncountably many cells, which—at least given some definitions of value and approaches to infinite ethics—would have created more value than could ever be created in a countable universe.

Therefore, there is a wager in favor of the hypothesis that humans actually live in an uncountable universe, even if it appears unlikely given current scientific evidence. But there is also a different wager, which applies if there is a chance that such a universe exists, regardless of whether humans live in that universe. It is unclear which of the two wagers dominates.

The second wager is based on acausal trade: there might be agents in an uncountable universe that do not benefit from the greater possibilities of their universe—e.g., because they do not care about the number of individual copies of some structure, but instead care about an integral over the structures’ values relative to some measure over structures. While agents in a countable universe might be able to benefit those agents equally well, they might be much worse at satisfying the values of agents with goals sensitive to the greater possibilities in uncountable universes. Thus, due to different comparative advantages, there could be great gains from trade between agents in countable and uncountable universes.

The above example might sound outlandish, and it might be flawed in that one could not actually come up with interaction rules that would lead to anything interesting happening in the cellular automaton. But this is irrelevant. It suffices that there is only the faintest possibility that an AGI could have an acausal impact in an incomputable universe which, according to one’s goal function, would outweigh all impact in all computable universes. There probably exists a possible universe like that for most goal functions. Therefore, one could be missing out on virtually all impact if the AGI employs Solomonoff induction.

There might not only be incomputable universes represented by a set that has the cardinality of the continuum, but there might be incomputable universes represented by sets of any cardinality. In the same way that there is a wager for the former, there is an even stronger wager for universes with even higher cardinalities. If there is a universe of highest cardinality, it appears to make sense to optimize only for acausal trade with that universe. Of course, there could be infinitely many different cardinalities, so one might hope that there is some convergence as to the values of the agents in universes of ever higher cardinalities (which might make it easier to trade with these agents).

In conclusion, there is a wager in favor of considering the possibility of incomputable universes: even a small acausal impact (relative to the total resources available) in an incomputable universe could counterbalance everything humans could do in a computable universe. Crucially, an AGI employing Solomonoff induction will not consider this possibility, hence potentially missing out on unimaginable amounts of value.

## Acknowledgements

Caspar Oesterheld and I came up with the idea for this post in a conversation. I am grateful to Caspar Oesterheld and Max Daniel for helpful feedback on earlier drafts of this post.