Student Z took the course 《Principles of Compliers》this semester, and he was very interested in the grammar.
A grammar can be represented by $G=(V_T, V_N, S, P)$.
$V_T$ represents the terminal symbol.
$V_N$ represents a non-terminal symbol.
$S$ represents the start symbol, which is the initial symbol of the derivation process , and can also be regarded as a special non-terminal symbol.
$P$ represents the production. The production is divided into left and right sides connected by“$->$”. It means that the left side of the production can be replaced by the right side. For example,the production which left is $S$ and right is $Aa$ is $S->Aa$ ,means that the symbol string $Aa$ can be used to replace the symbol string $S$.
A derivation is the process of replacing a non-terminal in a symbol string with the right side of the corresponding production.
The leftmost derivation result of a grammar starts from the start symbol. Each derivation will replace the leftmost non-terminal symbol of the symbol string with the symbol string on the right side of the corresponding production. After several derivations the resulting string of symbols.
For example, for the grammar "G[S]:S->aaBB B->bb":"a""b" are terminal symbol,"B" is non-terminal symbol,"S" is start symbol,"S->aaBB" and "B->bb" is production,then S->aaBB->aabbB->aabbbb is a leftmost derivation of this grammar,"aabbbb" is the result.
Student Z couldn't wait to write a grammar, but the grammar he wrote was relatively simple and met the following conditions:
1. Each terminal symbol of this grammar is a lowercase English letter, so there are only 26 terminal symbols in total.
2. There are a total of $n$ non-terminal symbols, and the $i$ non-terminal symbol is represented by "$[i]$".
3. There is only one symbol on the left of each production, and it is a non-terminal symbol.
4. A non-terminal symbol can only appear on the left of a production once, that is to say, there will be a total of n productions in the grammar.
5. The first character on the right side of each production must be a terminal symbol, that is, there will be no such situation as "$[1]->[2]a$".
6. Since Student Z didn't think about which symbol to use as the start symbol, any non-terminal symbol may become the start symbol of the grammar.
However, because Student Z does not have a good grasp of the rules of grammar, there is a recursive definition in the grammar (that is, it derives itself from a non-terminal symbol), then it may not eventually be possible to generate a symbol string composed entirely of terminal symbols. Therefore, for each leftmost derivation, he will perform at most $100^{100}$ derivations to ensure that the first $100^{100}$ characters of the final symbol string are terminal symbols.
Student z doesn't care about the recursive definition, He just wants to know what is the yth terminal symbol in the final string. And he will reselects a non-terminal symbol as the start symbol each time.