Brouwerian stream processors

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 of 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 nonwellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some interest. In neither case are the representatives unique. There may be many distinct representatives of extensionally the same function. Nevertheless, the distinctions between them capture genuine differences in computational behaviour. These representations (the data structures and their decoding functions) have a simple (if imperfect) expression in a functional programming language such as Haskell. The ideas extend to functions on final coalgebras for a broad class of functors, though in the general case to program their representation requires a language with dependent types. In some particular cases, only polymorphic recursion is needed. We hope to publish these results in another paper.