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Where First-Order and Monadic Second-Order Logic Coincide

Where First-Order and Monadic Second-Order Logic Coincide Where First-Order and Monadic Second-Order Logic Coincide MICHAEL ELBERFELD and MARTIN GROHE, RWTH Aachen University ¨ ¨ TILL TANTAU, Universitat zu Lubeck We study on which classes of graphs first-order logic (FO) and monadic second-order logic (MSO) have the same expressive power. We show that for all classes C of graphs that are closed under taking subgraphs, FO and MSO have the same expressive power on C if and only if, C has bounded tree depth. Tree depth is a graph invariant that measures the similarity of a graph to a star in a similar way that tree width measures the similarity of a graph to a tree. For classes just closed under taking induced subgraphs, we show an analogous result for guarded second-order logic (GSO), the variant of MSO that not only allows quantification over vertex sets but also over edge sets. A key tool in our proof is a Feferman­Vaught-type theorem that works for infinite collections of structures despite being constructive. CCS Concepts: r Theory of computation Finite Model Theory; Additional Key Words and Phrases: First-order logic, graph classes, guarded second-order logic, monadic second-order logic, tree depth ACM Reference Format: Michael Elberfeld, Martin Grohe, and Till http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Where First-Order and Monadic Second-Order Logic Coincide

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References (26)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2016 by ACM Inc.
ISSN
1529-3785
DOI
10.1145/2946799
Publisher site
See Article on Publisher Site

Abstract

Where First-Order and Monadic Second-Order Logic Coincide MICHAEL ELBERFELD and MARTIN GROHE, RWTH Aachen University ¨ ¨ TILL TANTAU, Universitat zu Lubeck We study on which classes of graphs first-order logic (FO) and monadic second-order logic (MSO) have the same expressive power. We show that for all classes C of graphs that are closed under taking subgraphs, FO and MSO have the same expressive power on C if and only if, C has bounded tree depth. Tree depth is a graph invariant that measures the similarity of a graph to a star in a similar way that tree width measures the similarity of a graph to a tree. For classes just closed under taking induced subgraphs, we show an analogous result for guarded second-order logic (GSO), the variant of MSO that not only allows quantification over vertex sets but also over edge sets. A key tool in our proof is a Feferman­Vaught-type theorem that works for infinite collections of structures despite being constructive. CCS Concepts: r Theory of computation Finite Model Theory; Additional Key Words and Phrases: First-order logic, graph classes, guarded second-order logic, monadic second-order logic, tree depth ACM Reference Format: Michael Elberfeld, Martin Grohe, and Till

Journal

ACM Transactions on Computational Logic (TOCL) – Association for Computing Machinery

Published: Sep 10, 2016

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