## FANDOM

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Matthew's Function
TypeBasic (BEAF-Related)
Based onArrow Notation and BEAF
Growth rate$$f_{\omega^{\omega^2}}(n)$$

Here is 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 much related to BEAF, however fails to be as strong as BEAF.

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 Edit

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's define o as a symbol A↑o↑B=A↑***...***↑B

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, → means the Bth symbol

A↑→↑B={A,B(2)2}