We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finite-path) trees of a certain kind. The underlying idea can be traced back to Brouwer's justification of bar-induction, or to Kreisel and Troelstra's elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity.
- continuous functions
- initial algebras
- final coalgebras
Hancock, P., Pattinson, D., & Ghani, N. (2009). Representations of stream processors using nested fixed points. Logical Methods in Computer Science, 5(3), 1-17. . https://doi.org/10.2168/LMCS-5(3:9)2009