Supported Sets -- A New Foundation For Nominal Sets And Automata

TL;DR

This paper introduces supported sets as a unified, finite foundational framework for various types of nominal sets and automata, enabling systematic transformations and representations.

Contribution

It establishes supported sets as a canonical base for different nominal sets, showing they are monadic over supported sets and facilitating automata transformations.

Findings

Supported sets provide a uniform foundation for various nominal sets.

All these categories are monadic over supported sets.

Enables finite representation and transformation of automata.

Abstract

The present work proposes and discusses the category of supported sets which provides a uniform foundation for nominal sets of various kinds, such as those for equality symmetry, for the order symmetry, and renaming sets. We show that all these differently flavoured categories of nominal sets are monadic over supported sets. Thus, supported sets provide a canonical finite way to represent nominal sets and the automata therein, e.g. register automata. Name binding in supported sets is modelled by a functor following the idea of de Bruijn indices. This functor lifts to the well-known abstraction functor in nominal sets. Together with the monadicity result, this gives rise to a transformation process that takes the finite representation of a register automaton in supported sets and transforms it into its configuration automaton in nominal sets.