User blog:Kite in rainbow/Kite in rainbow's bracket-comma notation,it can easily beyond ψ(I)!

to be continued...

1.we define a set BCS (bracket-comma string): (1)any nonnegitive integer is in BCS (2)if a_1 and a_2 and ... and a_n is in BCS,then the string a_1,a_2,...,a_n is in BCS (3)if a_1 and a_2 and ... and a_(n+1) and x_1 and x_2 and ... and x_n is in BCS,then the string a_1[x_1]a_2[x_2]...[x_n]a_(n+1) is in BCS for any integer n

2.we recursive define an order on BCS: for BCS x and y (1)if x has "outside" comma more than y,then x>y (2)if both have the same number of "outside" comma,let x=u,v y=s,t if u>s then x>y if u=s and v>t then x>y (3)if both don't have outside comma,write x as 1.(3),let the biggest x_i is u,the biggest x_i of y is v.if v is empty and u is not then x>y. if u>v then x>y. if u=v, write x=a[u]b , y=c[u]d , both a and c do not have the substring [u] "in outside",if a>c then x>y,if a=c and b>d then x>y. (4)if x,y both are integer and x>y then x>y.

3.we recursive define a subset LBCS of BCS (here L means legal).we call an element of LBCS legal,else in BCS is legal. (1)legal BCS mustn't have "bad zero" we call a letter 0 of a BCS x to be bad iff the letter in this 0's left is [ or nothing, or [u]0[v] is a substring of x and u>v,this 0 is that 0,or ",0[" is a substring of x. (2)