On the Non-Derivability of Operators in CCS
If a new operator is to be added to a calculus, defined directly by axioms or operational semantics, the question arises whether it can be derived from the other operators. While it is easy to show that an operator is derived, it is usually difficult to prove non-derivability. The general technique is to find a property preserved by all the other operators of the calculus, but not by the new one. We present a simple new technique, a special case of the general one, which consists of looking for congruences preserved by all operators except the new one. The new technique is particularly applicable if we use operational semantics rather than axioms to define operators, as in our application area, CCS-like calculi for reasoning about concurrent systems. A good way to use these calculi in practice is to use new operators that reflect the structure of the problem at hand. These new operators are often defined without reference to the existing ones, and derivability is a natural question.