I would like to know the name of the syntax used for the Cloud Firestore security rules as described at https://firebase.google.com/docs/firestore/security/get-started?authuser=
Firebase Security Rules is a custom DSL.
Condition expressions are JS like, and should work with one of those. The path matching framework is less common, but we're working on providing the grammar + additional tooling (syntax highlighting, parsing, type checking, evaluators) in the future.
For those interested in the history, the semantics are closely related to XACML, we just wanted to provide better (non-XML) syntax.