A B\"uchi-Elgot-Trakhtenbrot theorem for automata with MSO graph storage

TL;DR

This paper extends automata theory by introducing MSO graph storage types, establishing a B"uchi-Elgot-Trakhtenbrot theorem for these automata, and demonstrating that various storage transformations are MSO-expressible, including iterated pushdown types.

Contribution

It defines MSO graph storage types, proves a B"uchi-Elgot-Trakhtenbrot theorem for them, and shows that MSO-expressibility is preserved under certain storage transformations.

Findings

MSO graph storage types are isomorphic to MSO-definable graph sets.

A B"uchi-Elgot-Trakhtenbrot theorem holds for automata with MSO graph storage.

Iterated pushdown storage types are MSO-expressible.

Abstract

We introduce MSO graph storage types, and call a storage type MSO-expressible if it is isomorphic to some MSO graph storage type. An MSO graph storage type has MSO-definable sets of graphs as storage configurations and as storage transformations. We consider sequential automata with MSO graph storage and associate with each such automaton a string language (in the usual way) and a graph language; a graph is accepted by the automaton if it represents a correct sequence of storage configurations for a given input string. For each MSO graph storage type, we define an MSO logic which is a subset of the usual MSO logic on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string case and the graph case. Moreover, we prove that (i) each MSO graph transduction can be used as storage transformation in an MSO graph storage type, (ii) every automatic storage type is MSO-expressible, and (iii) the pushdown operator on storage types preserves the property of MSO-expressibility. Thus, the iterated pushdown storage types are MSO-expressible.