# Generalising Automaticity to Modal Properties of Finite Structures

## Abstract

We introduce a complexity measure of modal properties of finite structures which generalises the automaticity of languages. It is based on graph-automata like devices called labelling systems. We define a measure of the size of a structure that we call *rank*, and show that any modal property of structures can be approximated up to any fixed rank *n* by a labelling system. The function that takes *n* to the size of the smallest labelling system doing this is called the *labelling index* of the property. We demonstrate that this is a useful and fine-grained measure of complexity and show that it is especially well suited to characterise the expressive power of modal fixed-point logics. From this we derive several separation results of modal and non-modal fixed-point logics, some of which are already known whereas others are new.

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