# A (co)algebraic theory of succinct automata

## TL;DR

This paper develops a unified algebraic framework for automata, generalizing classical constructions to various algebraic structures and automata types, revealing new ways to associate succinct automata with languages.

## Contribution

It introduces a general construction linking algebraic structures to automata, extending classical determinization to new automata classes using a coalgebraic approach.

## Key findings

- Unified algebraic framework for automata types
- Construction transforming deterministic automata into succinct forms
- Application to automata with symmetries and weights

## Abstract

The classical subset construction for non-deterministic automata can be generalized to other side-effects captured by a monad. The key insight is that both the state space of the determinized automaton and its semantics---languages over an alphabet---have a common algebraic structure: they are Eilenberg-Moore algebras for the powerset monad. In this paper we study the reverse question to determinization. We will present a construction to associate succinct automata to languages based on different algebraic structures. For instance, for classical regular languages the construction will transform a deterministic automaton into a non-deterministic one, where the states represent the join-irreducibles of the language accepted by a (potentially) larger deterministic automaton. Other examples will yield alternating automata, automata with symmetries, CABA-structured automata, and weighted automata.

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Source: https://tomesphere.com/paper/1905.05519