The model checking problem for prefix classes of second-order logic: a survey

Authors:
Thomas Eiter;Georg Gottlob;Thomas Schwentick
Affiliations:
Institute of Information Systems, Vienna University of Technology, Austria;Computing Laboratory, Oxford University, United Kingdom;Fakultät für Informatik, Technische Universität Dortmund, Germany
Venue:
Fields of logic and computation
Year:
2010

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Abstract

In this paper, we survey results related to the model checking problem for second-order logic over classes of finite structures, including word structures (strings), graphs, and trees, with a focus on prefix classes, that is, where all quantifiers (both first- and second-order ones) are at the beginning of formulas. A complete picture of the prefix classes defining regular and non-regular languages over strings is known, which nearly completely coincides with the tractability frontier; some complexity issues remain to be settled, though. Over graphs and arbitrary relational structures, the tractability frontier is completely delineated for the existential second-order fragment, while it is less explored for trees. Besides surveying some of the results, we mention some open issues for research.