How exactly would one use the Profession (Cook) skill?

For one of the campaigns I’m going to be starting soon, most of our time will be spent out of ‘civilization,’ so our GM ruled that we would have to either acquire our own food from the wilderness (through uses of the Survival skill), or we would have to buy enough food to sustain us as we travel.

After hearing this, I ask our GM if it would be helpful for me to take ranks in Profession (Cook) to prepare food while we were out (because it might be possible that we could buy some ingredients and then make a lower DC Survival check to find enough food to supplement the ingredients we had already bought). He said that this would be a great idea, I just had to find reasonable rules for buying ingredients.

So I pored over Ultimate Equipment trying to find ingredients, and I found these “ingredients:” Bread, Caviar, Cheese, Chocolate, Fortune cookie, Honey, Ice cream, Maple syrup, Meat, Travel cake mix, and Yogurt, within the “food and drink” section; and Allspice, Basil, Beans, Cardamom, Chicken, Chilies, Cinnamon, Citrus, Cloves, Coffee beans, Cumin, Dill, Fennel, Flour, Garlic, Ginger, Mint, Mustard, Nutmeg, Nuts, Oregano, Pepper, Potatoes, Rosemary, Saffron, Salt, Sugar, Tobacco, Turnips, Vanilla, Wheat in the “trade goods” section.

This is a decent amount of ingredients yes, but there arises a different question, how much of what is needed to make a given meal? Then, how could it be edited to fit the survivalists helping supplement?

Does Cook and Ruzzo’s result also hold for logspace-uniform AC0?

In Cook’s famous paper on $ \mathsf{NC}$ , he cites the following result:

PROPOSITION 4.7 (Cook and Ruzzo, 1983). $ \mathsf{AC}^k$ consists of those problems solvable by uniform unbounded fan-in circuit families in $ O(\log^k n)$ depth and $ n^{O(1)}$ size.

where $ \mathsf{AC}^k$ here is defined for $ k \ge 1$ as the class of problems solvable by an ATM (with direct access) in $ O(\log n)$ space and $ O(\log^k n)$ alternation depth. (Of course, the definition of $ \mathsf{AC}^k$ usually goes the other way around, but given the equivalence it is all just a matter of presentation.)

Note the above is the logspace-uniform version of $ \mathsf{AC}$ . It was later acknowledged that for $ \mathsf{AC}^0$ the more restrictive dlogtime-uniformity is preferable. There is also an interesting result regarding the (in)equality of dlogtime- and logspace-uniformity of $ \mathsf{AC}^0$ . We also know that dlogtime-uniform $ \mathsf{AC}^0$ is characterizable as the class of problems solvable by ATMs in $ O(\log n)$ time (instead of space above) and $ O(1)$ alternation depth.

Given that dlogtime- and logspace-uniformity most likely does make a difference for $ \mathsf{AC}^0$ I am interested whether the Cook and Ruzzo result can be extended to the case of $ k = 0$ and logspace-uniform $ \mathsf{AC}^0$ . Unfortunately, in Cook’s paper, “Cook and Ruzzo, 1983” is only listed as “unpublished theorem,” so I was unable to check whether the proof also works for $ k=0$ (and Cook and Ruzzo had simply neglected it for some reason).

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