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Problem 910

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$L$-expressions II

An $L$-expression is defined as any one of the following:

• a natural number;
• the symbol $A$;
• the symbol $Z$;
• the symbol $S$;
• a pair of $L$-expressions $u,v$, which is written as $u(v)$.

An $L$-expression can be transformed according to the following rules:

• $A(x)\to x+1$ for any natural number $x$;
• $Z(u)(v)\to v$ for any $L$-expressions $u,v$;
• $S(u)(v)(w)\to v(u(v)(w))$ for any $L$-expressions $u,v,w$.

For example, after applying all possible rules, the $L$-expression $S(Z)(A)(0)$ is transformed to the number $1$:
$$S(Z)(A)(0)\to A(Z(A)(0))\to A(0)\to 1.$$
Similarly, the $L$-expression $S(S)(S(S))(S(Z))(A)(0)$ is transformed to the number $6$ after applying all possible rules.

Define the following $L$-expressions:

• $C_0=Z$;
• $C_i=S(C_{i-1})$ for $i\ge 1$;
• $D_i=C_i(S)(S)$.

For natural numbers $a,b,c,d,e$, let $F(a,b,c,d,e)$ denote the result of the $L$-expression $D_a(D_b)(D_c)(C_d)(A)(e)$ after applying all possible rules.

Find the last nine digits of $F(12,345678,9012345,678,90)$.

Note: it can be proved that the $L$-expression in question can only be transformed a finite number of times, and the final result does not depend on the order of the transformations.

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$L$-表达式（二）

一条$L$-表达式可以是以下任意一种：

• 一个自然数；
• 符号$A$；
• 符号$Z$；
• 符号$S$；
• 一对$L$-表达式$u,v$，并记作$u(v)$。

$L$-表达式可以按照以下规则进行转换：

• 对任意自然数$x$，$A(x)\to x+1$；
• 对任意$L$-表达式$u,v$，$Z(u)(v)\to v$；
• 对任意$L$-表达式$u,v,w$，$S(u)(v)(w)\to v(u(v)(w))$。

例如，在应用所有可能的规则后，$L$-表达式$S(Z)(A)(0)$可以被转换为数$1$：
$$S(Z)(A)(0)\to A(Z(A)(0))\to A(0)\to 1.$$
类似地，$L$-表达式$S(S)(S(S))(S(Z))(A)(0)$在应用所有可能的规则后被转换为数$6$。

定义如下的$L$-表达式：

• $C_0=Z$;
• $C_i=S(C_{i-1})$ for $i\ge 1$;
• $D_i=C_i(S)(S)$.

对于自然数$a,b,c,d,e$，令$F(a,b,c,d,e)$表示表达式$D_a(D_b)(D_c)(C_d)(A)(e)$在应用所有可能的规则后得到的结果。

求$F(12,345678,9012345,678,90)$的最后九位数字。

注意：可以证明，本题所涉及的$L$-表达式只能被应用有限次转换，且最终结果不依赖于转换的顺序。

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