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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}

Sources Edit

My function on Google Sites]

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