The basic problem with these equations is that the test is more stringent for large a than for small a, because parameters that gave an expected 1 random grounding for a assertions would give only ¼ of a random grounding for a/4 assertions.
I spent 2 weeks devising a more-sophisticated approach, that doesn’t share this problem. The shape of the resulting curve for assertions needed as a function of dictionary size turns out to be exactly the same. The reason is that the transition between zero alternate groundings, and an astronomical number of groundings, is so sharp, that it doesn’t matter at all that the test performed for a dictionary size of 100,000 effectively allows only half as many errors as for a dictionary size of 50,000. Increasing the number of assertions by 1 decreases the number of possible alternate groundings by about a factor of 10; decreasing the number of assertions by 1 increases it by about a factor of 10. So this translates into a difference between 200,000 assertions and 200,001.
The basic problem with these equations is that the test is more stringent for large a than for small a, because parameters that gave an expected 1 random grounding for a assertions would give only ¼ of a random grounding for a/4 assertions.
Actually, not!
I spent 2 weeks devising a more-sophisticated approach, that doesn’t share this problem. The shape of the resulting curve for assertions needed as a function of dictionary size turns out to be exactly the same. The reason is that the transition between zero alternate groundings, and an astronomical number of groundings, is so sharp, that it doesn’t matter at all that the test performed for a dictionary size of 100,000 effectively allows only half as many errors as for a dictionary size of 50,000. Increasing the number of assertions by 1 decreases the number of possible alternate groundings by about a factor of 10; decreasing the number of assertions by 1 increases it by about a factor of 10. So this translates into a difference between 200,000 assertions and 200,001.