Matthew's Function

Here are the rules for Matthew's Function. The ↑ symbol is supposed to represent an up arrow and means the rest of the equation. This is very similar to BEAF, however it fails to reach the well-defined limit of BEAF.My function on Google Sites 
A↑+↑B=A↑↑↑↑...↑↑↑A with B arrows
A↑...↑↑B=A↑...↑(A↑...↑(A↑...↑(...(A↑...↑)...))) with B copies of A
A↑...↑↑↑B=A↑...↑↑(A↑...↑↑(A↑...↑↑(...(A↑...↑↑)...))) with B copies of A
A↑...↑+↑B=A↑...↑↑↑...↑↑↑A with B arrows
A↑++↑B=A↑+↑+↑...↑+↑+↑A with B +'s
A↑...↑++↑B=A↑...↑+↑+↑...↑+↑+↑A with B +'s
A↑+++↑B=A↑++↑++↑...↑++↑++↑A with B ++'s
A↑*↑B=A↑+++...+++↑A with B +'s
A↑...↑*↑B=A↑...↑+++...+++↑A with B+'s
A↑**↑B=A↑*↑*↑...↑*↑*↑A with B *'s
== Comparision with BEAF ==
A↑+↑B={A,A,B}
A↑+↑↑B={A,B,1,2}
A↑+↑↑↑B={A,B,2,2}
A↑+↑+↑↑B={A,B,1,3}
A↑+↑+↑↑↑B={A,B,2,3}
A↑+↑+↑+↑↑B={A,B,1,4}
A↑++↑B={A,A,1,B+1}
A↑++↑↑B={A,B,1,1,2}
A↑++↑↑↑B={A,B,2,1,2}
A↑++↑+↑↑B={A,B,1,2,2}
A↑++↑+↑+↑B={A,B,1,3,2}
A↑++↑++↑↑B={A,B,1,1,3}
A↑++↑++↑+↑B={A,B,1,2,3}
A↑++↑++↑+↑+↑B={A,B,1,3,3}
A↑++↑++↑++↑↑B={A,B,1,1,4}
A↑+++↑↑B={A,B,1,1,1,2}
A↑++++↑↑B={A,B,1,1,1,1,2}
A↑*↑B={A,B+2(1)2}
A↑*↑↑B={A,B,2(1)2}
A↑*↑↑↑B={A,B,3(1)2}
A↑*↑+↑↑B={A,B,1,2(1)2}
A↑*↑++↑↑B={A,B,1,1,2(1)2}
A↑*↑+++↑↑B={A,B,1,1,1,2(1)2}
A↑*↑*↑B={A,B(1)3}
A↑*↑*↑*↑B={A,B(1)4}
A↑**↑↑B={A,B(1)1,2}
A↑**↑*↑B={A,B(1)2,2}
A↑**↑*↑*↑↑B={A,B(1)3,2}
A↑**↑**↑↑B={A,B(1)1,3}
A↑**↑**↑*↑↑B={A,B(1)2,3}
A↑**↑**↑**↑B={A,B(1)1,4}
A↑***↑↑B={A,B(1)1,1,2}
Let o be a symbol such that A↑o↑B=A↑***...***↑B with B stars
A↑o↑↑B={A,B(1)(1)2}
A↑o↑o↑↑B={A,B(1)(1)3}
A↑o↑o↑o↑↑B={A,B(1)(1)4}
A↑oo↑↑B={A,B(1)(1)1,2}
A↑oo↑o↑B={A,B(1)(1)2,2}
A↑oo↑o↑o↑B={A,B(1)(1)3,2}
A↑oo↑oo↑B={A,B,(1)(1)1,3}
A↑oo↑oo↑oo↑B={A,B(1)(1)1,4}
A↑ooo↑B={A,B(1)(1)1,1,2}
A↑oooo↑={A,B(1)(1)1,1,1,2}
If + is the 1st symbol, * is the second symbol, and o is the third symbol, then → represents the Bth symbol.
A↑→↑B={A,B(2)2}
== Sources ==
Category:Functions
Category:Notations
Category:Incomplete