The continuity of monadic stream functions

Abstract

Abstract-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 Escardó proved that it is also inconsistent in type theory. We propose a reformulation of the continuity principle that may be more faithful to the original meaning by Brouwer. It applies to 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).

Computability Beyond Church-Turing via Choice Sequences

“…For example, CP was proven in Nuprl using exceptions [39]. In [13] the authors formalized choice sequences as monadic streams and internally proved CP for natural monadic stream functions. It has also been shown that one can use references to obtain CP [32].…”

“…We derive that from A≡ lib A ′ and B≡ lib B ′ , and from the monotonicity of BITT. 13 We use these characterization lemmas to validate introduction and elimination rules for BITT's types, such as the following introduction rule for union types, which states that if a is a member of A (and B is a type) then inl(a) is a member of A+B: 14 H ⊢ A ⌊ext a⌋ H ⊢ B ∈ U i H ⊢ A+B ⌊ext inl(a)⌋ In addition to proving the validity of such rules, we have also proved that BITT is weakly consistent w.r.t. Coq's consistency, in the sense that the proposition False is not derivable.…”

“…See https://github.com/vrahli/NuprlInCoq/tree/beth/per/nuprl_props.v 13. See https://github.com/vrahli/NuprlInCoq/tree/beth/per/per_props_union.v.…”

Almost Sure Productivity

“…In this paper we consider productivity, which informally ensures that one can compute arbitrarily precise approximations of infinite objects in finite time. Productivity has been studied extensively for standard, non-probabilistic coinductive languages [20,16,1,13,8], but the probabilistic setting introduces new subtleties and challenges.…”