Thank you for the reply. You wrote:

Consider an algorithm that finds the WEASEL target with the following logic: It randomly selects points in the search space until it finds a point whose fitness plus the number of the query is even. In other words, if it’s the 3rd query and the fitness is 127, then the condition is satisfied. After finding such a point, it immediately goes to “METHINKS IT IS LIKE A WEASEL”.

This strategy would no longer be using a standard evolutionary strategy, which could find different targets simply by using different fitness functions, but would constitute a new search strategy/algorithm. We could also say “What about an algorithm that simply tries one query, no matter what the fitness function, then goes to the target?” or any other variation of that. But these are different search strategies, so the fitness function method I outlined isn’t directly applicable, since they aren’t really evolutionary strategies in the normal sense of the word.

However, your example wouldn’t escape the LCI.

Going with your new set-up, we can see that there exists a similar set-up for every target in your lower level search space: for example, it could go to “Meblinks it is like a weasel” after satisfying the condition. So why did we choose the one algorithm that goes to our target rather than to “Meblinks…”, “Rethinks…”, “hstjdins…” or any other of the 10^40 permutation choices we have?

More importantly, what is the minimum informational cost incurred by going from the set of all such algorithms (bounded by our original search space) to the set that chooses “Methinks…” with the same efficiency as the algorithm you constructed?

As I mentioned, the “goto” target of your algorithm could have been any of the roughly 10^40 permutations in the original search space, so we have at least 10^40 algorithms to choose from. The search for your particular algorithm (or one that performs equivalently well) is as hard as, and likely much harder, than our original search.

The LCI still holds.

Atom

Okay, I think I’ve finally got it. Sorry it took so long to sink in. I think your idea for defining a higher-order baseline is a good one, but I don’t believe it works with Marks and Dembski’s framework.

First of all, back in [168] where I agreed with your point about all algorithms performing equally over the whole set of fitness functions, I was wrong. Marks and Dembski’s model is not, in general, NFL-compatible. The problem is that Wolpert and Macready define the goodness of a search in terms of the codomain of the fitness function, but Marks and Dembski define the target independent of the fitness function, as Tom English pointed out above.

Consider an algorithm that finds the WEASEL target with the following logic: It randomly selects points in the search space until it finds a point whose fitness plus the number of the query is even. In other words, if it’s the 3rd query and the fitness is 127, then the condition is satisfied. After finding such a point, it immediately goes to “METHINKS IT IS LIKE A WEASEL”.

No matter what fitness function we use, this algorithm will likely find the target within a few queries. So how do we apply your condition that the higher-order space of fitness functions must have the same average performance as the null search?

I think that coming up with generally applicable constraints on the higher-order space definition is harder than meets the eye. As it says in the paper, the ways to search and to metasearch are endlessly varied, and the higher-order space definition can include or exclude any aspect of any conceivable search.

Sorry, I type too fast sometimes.

Atom

1. Information cost [of the higher order reduction] depends on the definition of the higher-order search space.

Correct and agreed.

2. We can define the higher-order search space to contain only good searches, thus making the information cost zero and falsifying the LCI.

No we can’t, since doing so would result in a search performance on the lower level search. If a reduction leads to search performance on the lower level search, then we cannot ignore that cost. If it leads to no search improvement (and no hinderance, since we can contribute negative active information), then we can ignore it.

3. In response to the objection that this higher-order search space must incur an information cost from an even higher-order search space, we can point out that this is true for all search spaces that have a non-zero probability of yielding a good search. If the LCI requires us to regress probabilities all the way up, then we’re stuck with an infinite information cost in every case.

Either that, or a source that can generate information without relying on search spaces. But this is a side issue.

Atom

One counterintuitive aspect of Marks and Dembski’s framework is that information cost is not based on the average performance of elements in the higher-order search space. Rather, it’s based on the fraction of those elements that perform at a level of at least q. Information cost does not tell us whether the average performance of the higher-order space is better or worse than the null search. It only tells us what the odds are of randomly selecting a search that performs at least as well as the given alternate search.

R0b,

You’ve almost got it. I didn’t say that the average performance of the higher level search was used to calculate the incurred cost, only that it can be used as an objective basis for deciding which informational costs are relevant, and hence, must be accounted for. It also provides a handy method for setting an objective baseline for for the higher level informational cost measure.

My reply has been consistent and I fail to see any issue with using the method I outlined to define the higher order space in a non-ad hoc way.

Atom

Hopefully we’ve gotten past the confusion about average performance vs. information cost. I can’t remember my train of thought from a few days ago, so I’ll just reiterate the point that you’re disputing:

1. Information cost depends on the definition of the higher-order search space.

2. We can define the higher-order search space to contain only good searches, thus making the information cost zero and falsifying the LCI.

3. In response to the objection that this higher-order search space must incur an information cost from an even higher-order search space, we can point out that this is true for all search spaces that have a non-zero probability of yielding a good search. If the LCI requires us to regress probabilities all the way up, then we’re stuck with an infinite information cost in every case.

You dropped my qualifier in “physical probability.” See more on this in the new thread Bill started.

I’m guessing that you, like me, are more engineer than philosopher. I accuse myself of a serious error in neglecting computational complexity in my investigation of NFL. Dembski and Marks are making the same error in focusing entirely on information costs. There are huge distinctions in search programs when time and memory are limited. I don’t have to go with Seth Lloyd in saying that the universe literally is a computer to say that there are analogous distinctions in nature.

This discussion has turned interesting at just the wrong time. I really need to put on the blinders and deal with the end-of-semester drudge work.

I have been thinking about my response and wanted to make a distinction. When I say we could possibly deal with the physical constraints (which I referred to as a form of “necessity”), what I meant was physical necessity, given the number of particles in the universe. I don’t want this confused with logical necessity, which wouldn’t make sense to treat as contingent (obviously, by definition).

I just wanted to make sure I was clear on that point. Given that there is no logically reason we’re aware of that the universe has this number of particles, which causes a reduction to take place, then measuring a cost on that reduction could be meaningful (via the tri-level search outlined above.) If however there is a logical necessity to that number of particles, the reduction requires no explanation, as necessary entities are their own explanation.

Atom

I think a major discrepancy in our thinking is your association of information cost with performance averaged over all of the functions in the higher-order space. For instance:

So if Reduction A results in a subset that still only performs as well as blind search, either a) the reduction didn’t improve search performance, and so incurs no informational cost…

[Emphasis mine]

One counterintuitive aspect of Marks and Dembski’s framework is that information cost is not based on the average performance of elements in the higher-order search space. Rather, it’s based on the fraction of those elements that perform at a level of at least q. Information cost does not tell us whether the average performance of the higher-order space is better or worse than the null search. It only tells us what the odds are of randomly selecting a search that performs at least as well as the given alternate search.

Consider that the set of functions that indicate proximity to a target performs no better on average than the larger set mentioned in endnote 49, i.e. they both perform on average the same as the null search. Yet Marks and Dembski say that the reduction from the latter to the former entails a heavy information cost.

More later, probably after Mothers’ Day.