I’m probably better at concrete examples. Consider a list of N comparable items. An optimizing process that orders them from least to greatest (sorts them) preserving relative order has optimization measure 1/N!. A less optimal process by the measure doesn’t maintain relative order and at worst has optimization measure N!/N! (completely reordering a list of N identical items) and at best (N-M)!/N! where M is the number of unique values among the items.
Sorting maintaining order is optimal under the measure. Sorting ignoring order is variable under the measure, depending entirely on the input. I think this where we were getting confused about how to talk about an oil refinery. Clearly an oil refinery is not optimal, so it must be variable. But then what is its measure? Is it the expected value over all possible initial states? Even that may not be unique; consider a list of N items that can take a value from the set {1} versus a list of N items that can take a value from the set {0,1}. Clearly the former is completely unoptimized by the order-ignoring sort, and the latter has expected optimization value (N/2)!/N!. Taking the value of the N items from the natural numbers the process would achieve 1/N! expected optimization. So taking into consideration the complexity of the problem domain won’t help; {1} is simpler than {0,1} is simpler than the natural numbers, but the optimization is inversely related to this progression.
I think this means that optimization is very context dependent. I think it means if I had to send a minimum-length message about the sorting algorithm I would be able to send a shorter message predicting the un-optimal output of sorting a list of N items of value “1” than a message predicting the optimal output of sorting a list of N natural numbers. If I had to gauge the power of the sorting mind I would get different answers depending on the input despite the fact that all sorting algorithms are O(n log n). If I had to gauge the efficiency of sorting any arbitrary output I could clearly say that the order-preserving sort was optimal, but if we simply change the problem domain so that we can no longer recognize relative order of equal values in the list then both algorithms are less optimal since the maximum measure of optimization drops to 1/(N-M)! So it appears that the same physical processes (or algorithms) can be more or less optimal depending on the preferences themselves. This means it may be impossible to infer the actual preferences from the evidence.
I’m probably better at concrete examples. Consider a list of N comparable items. An optimizing process that orders them from least to greatest (sorts them) preserving relative order has optimization measure 1/N!. A less optimal process by the measure doesn’t maintain relative order and at worst has optimization measure N!/N! (completely reordering a list of N identical items) and at best (N-M)!/N! where M is the number of unique values among the items.
Sorting maintaining order is optimal under the measure. Sorting ignoring order is variable under the measure, depending entirely on the input. I think this where we were getting confused about how to talk about an oil refinery. Clearly an oil refinery is not optimal, so it must be variable. But then what is its measure? Is it the expected value over all possible initial states? Even that may not be unique; consider a list of N items that can take a value from the set {1} versus a list of N items that can take a value from the set {0,1}. Clearly the former is completely unoptimized by the order-ignoring sort, and the latter has expected optimization value (N/2)!/N!. Taking the value of the N items from the natural numbers the process would achieve 1/N! expected optimization. So taking into consideration the complexity of the problem domain won’t help; {1} is simpler than {0,1} is simpler than the natural numbers, but the optimization is inversely related to this progression.
I think this means that optimization is very context dependent. I think it means if I had to send a minimum-length message about the sorting algorithm I would be able to send a shorter message predicting the un-optimal output of sorting a list of N items of value “1” than a message predicting the optimal output of sorting a list of N natural numbers. If I had to gauge the power of the sorting mind I would get different answers depending on the input despite the fact that all sorting algorithms are O(n log n). If I had to gauge the efficiency of sorting any arbitrary output I could clearly say that the order-preserving sort was optimal, but if we simply change the problem domain so that we can no longer recognize relative order of equal values in the list then both algorithms are less optimal since the maximum measure of optimization drops to 1/(N-M)! So it appears that the same physical processes (or algorithms) can be more or less optimal depending on the preferences themselves. This means it may be impossible to infer the actual preferences from the evidence.