The complexity of first-order and monadic second-order logic revisited

Frick, Markus; Grohe, Martin

doi:10.1016/j.apal.2004.01.007

Cited by 175 publications

(166 citation statements)

References 15 publications

Supporting

Mentioning

162

Contrasting

Unclassified

Order By: Relevance

“…Since we start with a satisfying assignment the same can be said for every clause, thus S is also true at the root state. Now, we need to show that the produced modal formula has very small depth and the hardness result will follow in a way very similar to the results of [8].…”

Section: Lemma 1 φ P Is Satisfiable If and Only If φ M Is Satisfiablementioning

confidence: 60%

“…We will prove this under the assumption that P = NP, by reducing the problem of propositional satisfiability to our problem. Our proof follows ideas similar to those found in [8].…”

Section: Theorem 1 [10] Modal Satisfiability For the Logic K Is Fpt Wmentioning

confidence: 82%

“…Our hardness proof follows an approach of encoding a propositional formula into a modal formula with very small modality depth. This draws a nice connection with previously known lower bound results of this form which also use a similar idea to prove the hardness of some model checking problems for first and second-order logic ( [8] and the relevant chapter in [7]). …”

Section: Introductionmentioning

confidence: 83%

“…Although the complexity of satisfiability for modal logic has been studied extensively in the past, to the best of our knowledge this is the first time this has been done from an explicitly parameterized perspective. Moreover, the parameterized complexity of logic problems has been a fruitful field of research and we hope to extend this success to modal logic (some examples are the celebrated theorem of Courcelle [3] or the results of [8]; for an excellent survey on the interplay between logic, graph problems and parameterized complexity see [9]). …”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Parameterized Modal Satisfiability

2011

View full text Add to dashboard Cite

We investigate the parameterized computational complexity of the satisfiability problem for modal logic and attempt to pinpoint relevant structural parameters which cause the problem's combinatorial explosion, beyond the number of propositional variables v. To this end we study the modality depth, a natural measure which has appeared in the literature, and show that, even though modal satisfiability parameterized by v and the modality depth is FPT, the running time's dependence on the parameters is a tower of exponentials (unless P = NP). To overcome this limitation we propose possible alternative parameters, namely diamond dimension and modal width. We show fixed-parameter tractability results using these measures where the exponential dependence on the parameters is much milder (doubly and singly exponential respectively) than in the case of modality depth thus leading to FPT algorithms for modal satisfiability with much more reasonable running times. We also give lower bound arguments which prove that our algorithms cannot be improved significantly unless the Exponential Time Hypothesis fails.Keywords Modal logic · Parameterized complexity · Satisfiability problems A preliminary version of this paper has appeared in ICALP 2010.A. Achilleos · M. Lampis ( ) · V. Mitsou

show abstract

Section: Lemma 1 φ P Is Satisfiable If and Only If φ M Is Satisfiablementioning

confidence: 60%

“…We will prove this under the assumption that P = NP, by reducing the problem of propositional satisfiability to our problem. Our proof follows ideas similar to those found in [8].…”

Section: Theorem 1 [10] Modal Satisfiability For the Logic K Is Fpt Wmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Parameterized Modal Satisfiability

2011

View full text Add to dashboard Cite

show abstract

“…It has also been investigated whether it's possible to obtain similar results for larger graph classes (see [7] for a metatheorem for bounded cliquewidth graphs, [15,16] for corresponding hardness results and [23] for hardness results for graphs of small but unbounded treewidth). Finally, lower bound results have been shown proving that the running times predicted by Courcelle's theorem can not be improved significantly in general [18].…”

mentioning

confidence: 96%

Algorithmic Meta-theorems for Restrictions of Treewidth

Lampis

2011

Algorithmica

142

View full text Add to dashboard Cite

Possibly the most famous algorithmic meta-theorem is Courcelle's theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time's dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees.We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded max-leaf number. We prove stronger algorithmic meta-theorems for these more restricted classes of graphs. More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.

show abstract

Incremental Verification and Coverage Analysis of Strongly Distributed Systems

Ravve¹,

Volkovich²

2018

Analytic Methods in Systems and Software Testing

View full text Add to dashboard Cite

The complexity of first-order and monadic second-order logic revisited

Cited by 175 publications

References 15 publications

Parameterized Modal Satisfiability

Parameterized Modal Satisfiability

Algorithmic Meta-theorems for Restrictions of Treewidth

Incremental Verification and Coverage Analysis of Strongly Distributed Systems

Contact Info

Product

Resources

About