NDLI: On the Complexity of the Model Checking Problem

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Madelaine, Florent R. Martin, Barnaby D.
Copyright Year	2018
Abstract	The complexity of the model checking problem for various fragments of first-order logic (FO) has attracted much attention over the last two decades, in particular for the fragment induced by $\exists$ and $\wedge$ and that induced by $\forall, \exists$, and $\wedge$, which are better known as the constraint satisfaction problem and the quantified constraint satisfaction problem, respectively. The former was conjectured to follow a dichotomy between P and NP-complete by Feder and Vardi [SIAM J. Comput., 28 (1998), pp. 57--104]. For the latter, there are several partial trichotomy results between P, NP-complete, and Pspace-complete, and Chen [Meditations on quantified constraint satisfaction, in Logic and Program Semantics, Springer, Heidelberg, 2012, pp. 35--49] ventured a conjecture regarding Pspace-completeness vs. membership in NP in the presence of constants. We give a comprehensive account of the whole field of the complexity of model checking similar syntactic fragments of FO. The above two fragments are in fact the only ones for which there is currently no known complexity classification. Indeed, we consider all other similar syntactic fragments of FO, induced by the presence or absence of quantifiers and connectives, and fully classify the complexities of the parameterization of the model-checking problem by a finite model $\mathcal{D}$, that is, the expression complexities for certain finite $\mathcal{D}$. Perhaps surprisingly, we show that for most of these fragments, tractability is witnessed by a generic solving algorithm which uses quantifier relativization. Our classification methodology relies on tailoring suitably the algebraic approach pioneered by Jeavons, Cohen, and Gyssens [J. ACM, 44 (1997), pp. 527--548] for the constraint satisfaction problem and by Börner et al. [Inform. and Comput., 207 (2009), pp. 923--944] for the quantified constraint satisfaction problem. Most fragments under consideration can be relatively easily classified, either directly or using Schaefer's dichotomy theorems for SAT and QSAT, with the notable exception of the positive equality-free fragment induced by $\exists,\forall, \wedge$, and $\vee$. This outstanding fragment can also be classified and enjoys a tetrachotomy: according to the model, the corresponding model checking problem is either tractable, NP-complete, co-NP-complete, or Pspace-complete.
Starting Page	769
Ending Page	797
Page Count	29
File Format	PDF
ISSN	00975397
DOI	10.1137/140965715
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3 (Special Section on the Forty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2015))
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-06-12
Access Restriction	Subscribed
Subject Keyword	quantified constraints Galois connection Factorization of matrices Determinants, permanents, other special matrix functions logic in computer science Matrix inversion, generalized inverses universal algebra computational complexity
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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