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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
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