Inigo MontoyaYou keep using that word. I do not think it means what you think it means

In their upcoming paper

*The Search for a Search: Measuring the Information Cost of Higher Level Search*W. Dembski and R.Marks introduce a

*Horizontal No Free Lunch Theorem*.

At Uncommon Descent, I tried to reach W. Dembski with the following post

**The**:

*Horizontal No free Lunch Theorem*doesn't work for a search of length m > 1 for a target T ⊆ ΩLet Ω be a finite search-space, T ⊆ Ω the target - a non-empty subset of Ω. A search is a (finite) sequence of Ω-valued random variables (φ

_{1}, φ

_{2}, ..., , φ

_{m}). A search is successful, if φ

_{n}∈ T for one n, 1 ≤ n ≤ m.

I suppose we do agree here. Now, we look at a search Φ as a Ω

^{m}-valued random variable, i.e., Φ := (φ

_{1}, φ

_{2}, ..., , φ

_{m}).

When is it successful? If we are still looking for a T ⊆ Ω we can say that we found T during our search if

Φ ∈ Ω

^{m}\ (Ω \ T)

^{m}

Let's define Θ as the subspace of Ω

^{m}which exists from the representations of targets in Ω, i.e.,

Θ := {Ω

^{m}\ (Ω \ T)

^{m}|T non-empty subset of Ω}

Obviously, Θ is much smaller than Ω

^{m}.

But this Θ is the space of feasible targets. And if you take an exhaustive partition of Θ instead of Ω

^{m}in Theorem III.1

*Horizontal No Free Lunch*, you'll find that you can indeed have positive values for the

*active entropy*as defined in the same theorem.

But that's not much of a surprise, as random sampling without repetition works better than random sampling with repetition.

But if you allow T to be any subset of Ω

^{m}, your results get somewhat trivial, as you are now looking at m independent searches of length 1 for different targets.

The searches which you state as examples in this paper and the previous one all work with a fixed target, i.e., elements of Θ. You never mention the possibility that the target changes between the steps of the search (one possible interpretation of taking arbitrary subsets of Ω

^{m}into account).

So, I'm faced with two possibilities:

- You didn't realize the switch from stationary targets to moving ones when you introduced searching for an arbitrary subset of Ω
^{m} - You realized this switch to a very different concept, but chose not to stress the point.

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