Representations of Stream Processors Using Nested Fixed Points

Abstract

Abstract. 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 (finitepath) 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.

ΠΣ: Dependent Types without the Sugar

“…In particular, it facilitates the definition of mixed inductive/coinductive types such as the type of stream processors (Hancock et al 2009), a concrete representation of functions on streams:…”

Truly Modular (Co)datatypes for Isabelle/HOL

“…Stream processors were introduced by Hancock et al [13] and have rapidly become the standard example for demonstrating mixed fixpoints in proof assistants [1-3, 9, 10]. Thanks to the new (co)datatype package, Isabelle finally joins this good company.…”

Program Extraction from Nested Definitions

“…For our type-0 representation of uniformly continuous functions we adopt socalled read-write machines [3] or stream processors [11,12]. These are W-cototal R W -total ideals where…”

“…Related work Nested definitions are used by Ghani, Hancock and Pattinson [11,12] to define uniformly continuous functions. They are also studied by Bird and Meertens [6] from a purely programming perspective.…”