Topological theories and automata

TL;DR

This paper explores the deep connection between topological theories for one-manifolds with defects and automata, revealing how Boolean semiring-valued theories relate to finite automata, regular languages, and their algebraic structures.

Contribution

It introduces a topological framework for automata and regular languages using Boolean semiring-valued theories, unifying automata theory with topological and categorical concepts.

Findings

Constructs a monoidal category of Boolean semilinear combinations of cobordisms with defects.

Shows the relation between automata, regular languages, and topological theories.

Identifies conditions under which the theory forms a Boolean TQFT.

Abstract

The paper explains the connection between topological theories for one-manifolds with defects and values in the Boolean semiring and automata and their generalizations. Finite state automata are closely related to regular languages. To each pair of a regular language and a circular regular language we associate a topological theory for one-dimensional manifolds with zero-dimensional defects labelled by letters of the language. This theory takes values in the Boolean semiring. Universal construction of topological theories gives rise in this case to a monoidal category of Boolean semilinear combinations of one-dimensional cobordisms with defects modulo skein relations. The latter category can be interpreted as a semilinear rigid monoidal closure of standard structures associated to a regular language, including minimal deterministic and nondeterministic finite state automata for the language and the syntactic monoid. The circular language plays the role of a regularizer, allowing to define the rigid closure of these structures. When the state space of a single point for a regular language describes a distributive lattice, there is a unique associated circular language such that the resulting theory is a Boolean TQFT.