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Monadic Second Order Finite Satisfiability and Unbounded Tree-Width

Authors Tomer Kotek, Helmut Veith, Florian Zuleger

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Tomer Kotek
Helmut Veith
Florian Zuleger

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Tomer Kotek, Helmut Veith, and Florian Zuleger. Monadic Second Order Finite Satisfiability and Unbounded Tree-Width. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.CSL.2016.13

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.

Subject Classification

Keywords
  • Monadic Second Order Logic MSO
  • Two variable Fragment with Counting C2
  • Finite decidability
  • Unbounded Tree-width
  • WS1S with Cardinality Constraints

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