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Clock systems for stochastic and non-deterministic categorical systems theories

Owen Lynch, David Jaz Myers, Eigil Fjeldgren Rischel, Sam Staton

Abstract

One of the characteristic features of categorical systems theory is that the behavior of systems can be characterized by certain morphisms into them. In other words, behaviors form a representable covariant functor to Set. And more generally, in the compositional setting, behaviors form a representable double functor to Span. Clock systems are convenient because behavior functors represented by clock systems are automatically well-behaved. It was previously not known whether stochastic and non-deterministic systems theories have clock systems. In this paper, we show that indeed they do have clock systems. Moreover, the clock systems for non-deterministic systems point to generalized notions of behavior for non-linear time.

Clock systems for stochastic and non-deterministic categorical systems theories

Abstract

One of the characteristic features of categorical systems theory is that the behavior of systems can be characterized by certain morphisms into them. In other words, behaviors form a representable covariant functor to Set. And more generally, in the compositional setting, behaviors form a representable double functor to Span. Clock systems are convenient because behavior functors represented by clock systems are automatically well-behaved. It was previously not known whether stochastic and non-deterministic systems theories have clock systems. In this paper, we show that indeed they do have clock systems. Moreover, the clock systems for non-deterministic systems point to generalized notions of behavior for non-linear time.

Paper Structure

This paper contains 15 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Moore machines for a monad, morphisms of Moore machines, and clocks
  4. Organization of the rest of the paper
  5. Systems theories via tangencies
  6. Representable behaviors for discrete-time deterministic Moore machines
  7. Representable behaviors for discrete-time stochastic Moore machines
  8. Representable behaviors for discrete-time nondeterministic Moore machines
  9. Profiting from representable behaviors
  10. Compositionality of behavior
  11. Sheaves of behaviors
  12. Future work
  13. Representable behaviors for imprecise probability Moore machines
  14. Representable behaviors for continuous-time stochastic Moore machines
  15. Supplementary lemmas for probability sheaves

Key Result

Proposition 1

The behavior functor $B \colon \mathsf {Sys} \to \mathsf {Set}$ as constructed in rt-000B is represented by maps out of the clock system in rt-000I.

Theorems & Definitions (31)

  • Example 1: Bundles of arrows
  • Definition 1: Tangency
  • Example 2: The tangency for discrete-time deterministic Moore machines
  • Example 3: The tangency for discrete-time stochastic Moore machines
  • Example 4: The tangency for discrete-time nondeterministic Moore machines
  • Definition 2: Behaviors of discrete-time deterministic Moore machines
  • Definition 3: The clock for discrete-time deterministic Moore machines
  • Proposition 1: Behaviors as maps out of the clock
  • Definition 4: Filtration
  • Definition 5: Process adapted to a filtration
  • ...and 21 more