Generalising automaticity to modal properties of finite structures

Authors:
Anuj Dawar;Stephan Kreutzer
Affiliations:
University of Cambridge Computer Laboratory, Cambridge CB3 0FD, UK;Institut für Informatik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany
Venue:
Theoretical Computer Science
Year:
2007

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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.