# Substitution semiring?

Let G be a [ CF ] grammar, and let elements of semiring be sets of rules.

Define multiplication as:

$$x\otimes y = \{ t| \exists r \in x \exists s \in y (t=subst(r,s))\}$$

where $subst(r,s)$ is rewriting of grammar rule $r$ by formal substitution of one symbol in the rule body matching $s$ head with $s$ body. For example, the rule

expr : expr oper expr


matches itself and rewrites into

expr : expr oper expr oper expr


Next, define addition $\oplus$ as set union.

Is it known semiring?

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add apostrophes before dollar signs to fix that IIRC. –  Will Sawin Nov 19 '11 at 20:16