When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2016.13
URN: urn:nbn:de:0030-drops-65537
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6553/
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### Monadic Second Order Finite Satisfiability and Unbounded Tree-Width

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### Abstract

The finite satisfiability problem of monadic second order logic is decidable only on classes of structures of bounded tree-width by the classic result of Seese. We prove that the following problem is decidable: Input: (i) A monadic second order logic sentence alpha, and (ii) a sentence beta in the two-variable fragment of first order logic extended with counting quantifiers. The vocabularies of alpha and beta may intersect. Output: Is there a finite structure which satisfies alpha and beta such that the restriction of the structure to the vocabulary of alpha has bounded tree-width? (The tree-width of the desired structure is not bounded.) As a consequence, we prove the decidability of the satisfiability problem by a finite structure of bounded tree-width of a logic MS^{exists card} extending monadic second order logic with linear cardinality constraints of the form |X_{1}|+...+|X_{r}| < |Y_{1}|+...+|Y_{s}| on the variables X_i, Y_j of the outer-most quantifier block. We prove the decidability of a similar extension of WS1S.

### BibTeX - Entry

```@InProceedings{kotek_et_al:LIPIcs:2016:6553,
author =	{Tomer Kotek and Helmut Veith and Florian Zuleger},
title =	{{Monadic Second Order Finite Satisfiability and Unbounded Tree-Width}},
booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
pages =	{13:1--13:20},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-022-4},
ISSN =	{1868-8969},
year =	{2016},
volume =	{62},
editor =	{Jean-Marc Talbot and Laurent Regnier},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6553},
URN =		{urn:nbn:de:0030-drops-65537},
doi =		{10.4230/LIPIcs.CSL.2016.13},
annote =	{Keywords: Monadic Second Order Logic MSO, Two variable Fragment with Counting C2, Finite decidability, Unbounded Tree-width, WS1S with Cardinality Constraints}
}
```

 Keywords: Monadic Second Order Logic MSO, Two variable Fragment with Counting C2, Finite decidability, Unbounded Tree-width, WS1S with Cardinality Constraints Seminar: 25th EACSL Annual Conference on Computer Science Logic (CSL 2016) Issue Date: 2016 Date of publication: 25.08.2016

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