Peter Hancock | Searcharxiv

arXiv2009

Representations of Stream Processors Using Nested Fixed Points

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.

cs.DS↗

arXiv2009

Programming interfaces and basic topology

A pattern of interaction that arises again and again in programming is a "handshake", in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent--the client or Angel--and concluded by the other--the server or Demon. We present a category in which the objects--called interaction structures in the paper--serve as descriptions of services provided across such handshaken interfaces. The morphisms--called (general) simulations--model components that provide one such service, relying on another. The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve (in principle) as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain. This category is then shown to coincide with the subcategory of "generated" basic topologies in Sambin's terminology, where a basic topology is given by a closure operator whose induced sup-lattice structure need not be distributive; and moreover, this operator is inductively generated from a basic cover relation. This coincidence provides topologists with a natural source of examples for non-distributive formal topology. It raises a number of questions of interest both for formal topology and programming. The extra structure needed to make such a basic topology into a real formal topology is then interpreted in the context of interaction structures.

cs.LO↗