User blog comment:MilkyWay90/Generalized Factorial Function/@comment-35392788-20180517135333/@comment-78.243.14.108-20180523142845

The function A(n,!) grows basically depending on the value of n, since it's the first parameters which determines the level of the function (addition, multiplication, exponentiation, tetration,...). Since every level of operation is describes by one step in the FGH, we can say that A(n, !) has growth rate of about f_ω. The fact that the factorial only affects the second argument side makes it almost negligible.

A(!,n) however is another level. Here you're diagonalizing over the parameter that already diagonalizes over finite FGH. This therefore is somewhere between f_ω and f_ω+1 (which is on par with the Graham function)

BB(n,!) is going to be uncomputable no matter what, and BB(n,n) already grows uncomputably fast. BB(n,!) and BB(!,n) are definitely faster than BB(n,n), but I think their growth rate remains the same at that level.

Now, your array generalization is a good step forward, but it has its flaws :

- How do you evaluate something like F(1,2,!), or F(2,3,!,1,2) ? Hell, even if there are no 1s, the fact that you decrease a whole batch of parameters at once makes us run into a problem as soon as all the parameters aren't the same. Not only that, but it makes it significantly weaker.

- Wouldn't it be cool to have something like F(!,!,3) ?

I'll lay some clues here :

Don't decrease a batch of parameters at once, it just makes your function much weaker.

A suggestion is have a base rule (similar to the one of your 2-argument notation), a "getting rid" rule, like removing trailing ones, and a "process", which goes like : start at the rightmost !, and look right: if it is > 1 do that, otherwise... blah blah blah.

If you ever have to collapse 1s, then don't just make them disappear, try to do something powerful, like copying another entry, or replacing something with a recursive call, things like that.

Now, here is something that I can compute in BEAF :

{3,!,3} = {3,{2,!,2},3} = {3,{2,{1,!,1},2},3} = {3,{2,1,2},3} = {3,2,3} = 3^3^3

{3,3,!,3} = {3,3,{2,2,!,2},3} = {3,3,{2,2,{1,1,!,1},2},3} = {3,3,{2,2,1,2},3}

= {3,3,4,3} = HUGE

I still think there's more potential.

P.S : it's pentation, not penetration, unless you're referring to that comic strip.