There are a number of connections I can think of. There is logic, as you have already mentioned. Another connection is cryptography. Cryptography uses some fairly high-powered number theory, which in turn uses abstract algebra.
Another connection is grammars, context-free grammars, etc.
I am by no means an expert here. You might ask for emakarov's opinion as well. I bet he could weigh in on this question profitably.