The continuity of monadic stream functions

Brouwer's continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently Martín Escardό proved that it is also inconsistent in type theory. This shows that we cannot internalize the meta-theoretical observation that every definable function is continuous. However, we can adapt Brouwer's ideas to an important class of functions and propose a reformulation of the continuity principle that is internally provable. We note that Brouwer talked about functions on choice sequences, which are described as free progressions of values not necessarily generated by a rule. The functions must produce their results independently of how the sequences are generated. We formalize them as monadic streams, potentially unending sequences of values produced by steps triggered by a monadic action, possibly involving side effects. We consider functions on them that are uniform, in the sense that they operate in the same way independently of the particular monad that provides the specific side effects. Formally this is done by requiring a form of naturality in the monad. Functions on monadic streams have not only a foundational importance, but have also practical applications in signal processing and reactive programming. We give algorithms to determine the modulus of continuity of monadic stream functions and to generate dialogue trees for them (trees whose nodes and branches describe the interaction of the process with the environment).