Reducing the number of symbols in this string system

Question

This is a bit like Boolean algebra (except it isn't, it is tangle composition - but a lot of main rules agree):

1. | is legit
2. If t is legit, (t) is
3. If t and T are legit, tT is (don't worry, concatenation is associative)

(4. If this helps, ((t))=t - think of (t) as negation)

A string analyzer would be much easier to handle if I could write this only with two characters. Of course this is possible (read string as base 3, convert to binary) but then simplifying (like 4.) would become a nightmare. I could define a Sheffer NAND, but it's not associative and I still need brackets.

Any idea for writing this with only two basic symbols AND backconversion to the form above is as context-free as possible? (For example, I can't just drop closing brackets: (11) and (1)1 look the same then)