Traces, Feedback, and the Geometry of Computation (Abstract)

Abstract

The notion of feedback and information flow is a fundamental concept arising in many classical areas of computing. In the late 1980s and early 1990s, an algebraic structure dealing with cyclic operations emerged from various fields, including flowchart schemes, dataflow with feedback, iteration theories, action calculi in concurrency theory, proof theory (Linear Logic and Geometry of Interaction), and network algebra, as well as in pure mathematics. This structure, now known as a “traced monoidal category”, was formally introduced in an influential paper of Joyal, Street and Verity (1996) from current work in topology and knot theory. However these authors were also keenly aware of its potential applicability. Since then, such structures – with variations – have been pursued in several areas of mathematics, logic and theoretical computer science, including game semantics, quantum programming languages and quantum protocols, and computational biology. We shall survey applications in logic and theoretical computer science and discuss progress towards an abstract geometry of algorithms.