Computability by monadic second-order logic

Joost Engelfriet

doi:10.1016/j.ipl.2020.106074

Abstract

A binary relation on graphs is recursively enumerable if and only if it can be computed by a formula of monadic second-order logic. The latter means that the formula defines a set of graphs, in the usual way, such that each “computation graph” in that set determines a pair consisting of an input graph and an output graph.

Highlights

There are many characterizations of computability, but the one presented here does not seem to appear explicitly in the literature.1 it is a natural and simple characterization, based on the intuitive idea that a computation of a machine, or a derivation of a grammar, can be represented by a graph satisfying a formula of monadic second-order (MSO) logic
Assuming the reader to be familiar with MSO logic on graphs, the MSO-computability of a binary relation on graphs can be given in half a page, see below
The MSO formula φ that computes the composition of R1 and R2, uses the auxiliary alphabet ∪ {ν1, ν2, d} and defines computation graphs h that are obtained as the disjoint union of a computation graph h1 of φ1 and a computation graph h2 of φ2, enriched by d-edges that establish an isomorphism between out(h1) and in(h2), and by ν-edges from in(h1) to out(h2)

Summary

Introduction

There are many characterizations of computability, but the one presented here does not seem to appear explicitly in the literature.1 it is a natural and simple characterization, based on the intuitive idea that a computation of a machine, or a derivation of a grammar, can be represented by a graph satisfying a formula of monadic second-order (MSO) logic. By definition, the set of all computation graphs h over ∪ is MSO-definable, and the sets of nodes V in(h) and V out(h) can be expressed in MSO logic. The set H consists of computation graphs h such that V h = V in(h) ∪ V out(h), in(h) and out(h) are graphs over , and the d-edges form an isomorphism from out(h) to an induced subgraph of in(h).

Objectives

Results

Conclusion