Terminal coalgebras for a functor serve as semantic domains for state-based systems of various types. For example, behaviors of CCS processes, streams, infinite trees, formal languages and non-well-founded sets form terminal coalgebras. We present a uniform account of the semantics of recursive definitions in terminal coalgebras by combining two ideas: (1) abstract GSOS rules l specify additional algebraic operations on a terminal coalgebra; (2) terminal coalgebras are also initial completely iterative algebras (cias). We also show that an abstract GSOS rule leads to new extended cia structures on the terminal coalgebra. Then we formalize recursive function definitions involving given operations specified by l as recursive program schemes for l, and we prove that unique solutions exist in the extended cias. From our results it follows that the solutions of recursive (function) definitions in terminal coalgebras may be used in subsequent recursive definitions which still have unique solutions. We call this principle modularity. We illustrate our results by the five concrete terminal coalgebras mentioned above, e.\,g., a finite stream circuit defines a unique stream function.

• 18C10 - Theories (e.g., algebraic theories), structure, and semantics
• 68Q65 - Abstract data types; algebraic specification