Parametrised Complexity of Satisfiability in Temporal Logic

Lück, Martin; Meier, Arne; Schindler, Irena

doi:10.1145/3001835

Cited by 3 publications

(4 citation statements)

References 31 publications

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“…This substitution does not increase the treewidth. Then the result follows by Lück et al [38,Cor. 4.7].…”

Section: Corollary 21supporting

confidence: 60%

“…However, in this setting usually CNF-formulas are considered. Coping with this restriction, Lück et al [38] defined syntax circuits for temporal logic formulas that also allow arbitrary formulas. In our setting, we continue in this direction and define the syntax (or formula) structure with respect to a PDL-formula.…”

Section: Representation Of Inputs As Graphsmentioning

confidence: 99%

“…However, we are unable to immediately utilise their result because Samer and Szeider study CNF-formulas and we have arbitrary formulas instead. Yet, Lück et al [38,Cor. 4.7] studying temporal logics under the parameterised approach, classified, as a byproduct, the propositional satisfiability problem with respect to arbitrary formulas to be fixed-parameter tractable.…”

Section: Corollary 21mentioning

confidence: 99%

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Parameterised complexity of model checking and satisfiability in propositional dependence logic

Mahmood

Meier

2021

Ann Math Artif Intell

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Dependence Logic was introduced by Jouko Väänänen in 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parameterisations with respect to central decision problems. The model checking problem (MC) of PDL is NP-complete (Ebbing and Lohmann, SOFSEM 2012). The subject of this research is to identify a list of parameterisations (formula-size, formula-depth, treewidth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) which classically is known to be NP-complete as well (Lohmann and Vollmer, Studia Logica 2013). There we are presenting a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is FPT. Finally, we introduce a variant of the satisfiability problem, asking for a team of a given size, and show for this problem an almost complete picture.

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“…This substitution does not increase the treewidth. Then the result follows by Lück et al [38,Cor. 4.7].…”

Section: Corollary 21supporting

confidence: 60%

Section: Representation Of Inputs As Graphsmentioning

confidence: 99%

Section: Corollary 21mentioning

confidence: 99%

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Parameterised complexity of model checking and satisfiability in propositional dependence logic

Mahmood

Meier

2021

Ann Math Artif Intell

Self Cite

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“…The technique of Courcelle's theorem [4] (see Prop. 6) has been used in different contexts: temporal logic [23], knowledge representation [13], and nonmonotonic logic [25]. Elberfeld et al [10] enriched Courcelle's theorem to also yield results for the complexity class XL.…”

Section: Related Workmentioning

confidence: 99%

Parameterised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

Mahmood

Meier

2020

Lecture Notes in Computer Science

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In this paper, we initiate a systematic study of the parametrised complexity in the field of Dependence Logics which finds its origin in the Dependence Logic of Väänänen from 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parametrisations with respect to the central decision problems. The model checking problem (MC) of PDL is NP-complete. The subject of this research is to identify a list of parametrisations (formula-size, treewidth, treedepth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) showing a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is in FPT. Finally, we introduce a variant of the satisfiability problem, asking for teams of a given size, and show for this problem an almost complete picture. ACM Subject Classification Theory of computation → Complexity theory and logic; Theory of computation → Parameterized complexity and exact algorithmsContributions In Table 1, we give an overview of our results. We study eight different parametrisations for each of the model checking problem, the satisfiability problem, as well as a variant of satisfiability (this problem asks for a team of a given size). Thereby, we prove dichotomies for MC and SAT: depending on the parameter the problem is either fixed-parameter tractable or paraNP-complete. Only the satisfiability variant and the parameters treewidth and dep-arity resist a complete classification and are left for further research. Related workThe technique of Courcelle's theorem [4] (see Prop. 6) has been used in different contexts: temporal logic [23], knowledge representation [13], and nonmonotonic logic [25]. Elberfeld et al. [10] enriched Courcelle's theorem to also yield results for the complexity class XL. This improvement applies to our results utilising this theorem as well and affects Theorem 12, 16, and 25.Organisation of the article At first, we introduce some required notions and definitions in (parametrised) complexity theory, dependence logic, and first-order logic. Then we study the parametrised complexity of the model checking problem. Proceed with the satisfiability problem and study a variant of it. Finally, we conclude and discuss open questions. Proofs of results marked with a ( ) can be found in the appendix.

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Parameterized Complexity of Some Prefix-Vocabulary Fragments of First-Order Logic

Bustamante

Martins

2018

Logic, Language, Information, and Computation

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Parametrised Complexity of Satisfiability in Temporal Logic

Cited by 3 publications

References 31 publications

Parameterised complexity of model checking and satisfiability in propositional dependence logic

Parameterised complexity of model checking and satisfiability in propositional dependence logic

Parameterised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

Parameterized Complexity of Some Prefix-Vocabulary Fragments of First-Order Logic

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