a. You can use portions of the text as (is) permitted by law.
- ||Using portions of the text|| ||is permitted|| by law||. [passive]
Ergo, due to structural equivalence, one can write: either "is permitted" or "permitted".
1) You can use them as ||is permitted|| by law.
2) You can use them as ||permitted|| by law.
3) You can use them as the law permits. [active]
1) and 2) mean the same thing. They just reflect different parses, both of which come from an implied passive.
b. As (is) permitted by law, you can use portions of the text.
See above. The same analysis would apply.
c. As (is) well known [by people], the country is rich in oil.
- It ||is well-known|| [by people] the country is rich in oil. [passive]
The implied passive is the same as in the other sentences. "As is" and "as" basically are the same thing. It depends on whether you choose to include the is/are portion of the implied passive.
As is well known [by everybody] = As well known [by everybody]
There is no difference in meaning. as here means "when considered in a specified form or relation" and is used before a participle or preposition. Definition from: Merriam Webster- as
As are given in the index, all the items are plural.
1) As are given in the index 2) As given in the index