JabRef references

Mahmood Y, Meier A and Schmidt J (2023), "Parameterized Complexity of Logic-Based Argumentation in Schaefer's Framework", ACM Trans. Comput. Logic. New York, NY, USA, may, 2023. Vol. 24(3) Association for Computing Machinery.

[Abstract] [BibTeX] [DOI] [URL]

Abstract: Argumentation is a well-established formalism dealing with conflicting information by generating and comparing arguments. It has been playing a major role in AI for decades. In logic-based argumentation, we explore the internal structure of an argument. Informally, a set of formulas is the support for a given claim if it is consistent, subset-minimal, and implies the claim. In such a case, the pair of the support and the claim together is called an argument. In this article, we study the propositional variants of the following three computational tasks studied in argumentation: ARG (exists a support for a given claim with respect to a given set of formulas), ARG-Check (is a given set a support for a given claim), and ARG-Rel (similarly as ARG plus requiring an additionally given formula to be contained in the support). ARG-Check is complete for the complexity class DP, and the other two problems are known to be complete for the second level of the polynomial hierarchy (Creignou et al. 2014 and Parson et al., 2003) and, accordingly, are highly intractable. Analyzing the reason for this intractability, we perform a two-dimensional classification: First, we consider all possible propositional fragments of the problem within Schaefer's framework (STOC 1978) and then study different parameterizations for each of the fragments. We identify a list of reasonable structural parameters (size of the claim, support, knowledge base) that are connected to the aforementioned decision problems. Eventually, we thoroughly draw a fine border of parameterized intractability for each of the problems showing where the problems are fixed-parameter tractable and when this exactly stops. Surprisingly, several cases are of very high intractability (para-NP and beyond).

BibTeX:

@article{10.1145/3582499,
  author = {Mahmood, Yasir and Meier, Arne and Schmidt, Johannes},
  title = {Parameterized Complexity of Logic-Based Argumentation in Schaefer's Framework},
  journal = {ACM Trans. Comput. Logic},
  publisher = {Association for Computing Machinery},
  year = {2023},
  volume = {24},
  number = {3},
  url = {https://doi.org/10.1145/3582499},
  doi = {10.1145/3582499}
}

Haak A, Meier A, Prakash O and Rao BVR (2023), "Parameterised Counting in Logspace", Algorithmica.

[Abstract] [BibTeX] [DOI] [URL]

Abstract: Logarithmic space-bounded complexity classes such as $$backslashtextbfL $$and $$backslashtextbfNL $$play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators $$backslashtextbfpara_backslashtextbfW$$and $$backslashtextbfpara_backslashbeta $$for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators $$backslashtextbfpara_backslashtextbfW$$and $$backslashtextbfpara_backslashbeta $$by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, $$backslashtextbfpara_backslashtextbfW[1]$$and $$backslashtextbfpara_backslashbeta backslashtextbftail$$. Then, we consider counting versions of all four operators and apply them to the class $$backslashtextbfL $$. We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is $$backslash#backslashtextbfpara_backslashbeta backslashtextbftailbackslashtextbfL $$-hard and can be written as the difference of two functions in $$backslash#backslashtextbfpara_backslashbeta backslashtextbftailbackslashtextbfL $$. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of $$backslash#backslashtextbfpara_backslashbeta backslashtextbftailbackslashtextbfL $$under parameterised logspace parsimonious reductions coincides with $$backslash#backslashtextbfpara_backslashbeta backslashtextbfL $$. In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research.

BibTeX:

@article{Haak:2023aa,
  author = {Haak, Anselm and Meier, Arne and Prakash, Om and Rao, B. V. Raghavendra},
  title = {Parameterised Counting in Logspace},
  journal = {Algorithmica},
  year = {2023},
  url = {https://doi.org/10.1007/s00453-023-01114-2},
  doi = {10.1007/s00453-023-01114-2}
}

Kontinen J, Meier A and Mahmood Y (2022), "A parameterized view on the complexity of dependence and independence logic", J. Log. Comput..

[BibTeX] [DOI] [URL]

BibTeX:

@article{10.1093/logcom/exac070,
  author = {Kontinen, Juha and Meier, Arne and Mahmood, Yasir},
  title = {A parameterized view on the complexity of dependence and independence logic},
  journal = {J. Log. Comput.},
  year = {2022},
  url = {https://doi.org/10.1093/logcom/exac070},
  doi = {10.1093/logcom/exac070}
}

Haak A, Meier A, Müller F and Vollmer H (2022), "Enumerating teams in first-order team logics", Ann. Pure Appl. Log.. Vol. 173(10), pp. 103163.