Counting Complexity for Reasoning in Abstract Argumentation

Fichte, Johannes Klaus; Hecher, Markus; Meier, Arne

doi:10.1609/aaai.v33i01.33012827

Cited by 15 publications

(12 citation statements)

References 27 publications

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“…• problems relevant to abstract argumentation [Dvořák et al, 2012;Fichte et al, 2019a], where subset-minimization is also achieved by means of counter-witnesses,…”

Section: Outlook On Dynamic Programming For Other Formalismsmentioning

confidence: 99%

Advanced tools and methods for treewidth-based problem solving

Hecher¹

2023

It - Information Technology

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Computer programs, so-called solvers, for solving the well-known Boolean satisfiability problem (Sat) have been improving for decades. Among the reasons, why these solvers are so fast, is the implicit usage of the formula’s structural properties during solving. One of such structural indicators is the so-called treewidth, which tries to measure how close a formula instance is to being easy (tree-like). This work focuses on logic-based problems and treewidth-based methods and tools for solving them. Many of these problems are also relevant for knowledge representation and reasoning (KR) as well as artificial intelligence (AI) in general. We present a new type of problem reduction, which is referred to by decomposition-guided (DG). This reduction type forms the basis to solve a problem for quantified Boolean formulas (QBFs) of bounded treewidth that has been open since 2004. The solution of this problem then gives rise to a new methodology for proving precise lower bounds for a range of further formalisms in logic, KR, and AI. Despite the established lower bounds, we implement an algorithm for solving extensions of Sat efficiently, by directly using treewidth. Our implementation is based on finding abstractions of instances, which are then incrementally refined in the process. Thereby, our observations confirm that treewidth is an important measure that should be considered in the design of modern solvers.

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“…• problems relevant to abstract argumentation [Dvořák et al, 2012;Fichte et al, 2019a], where subset-minimization is also achieved by means of counter-witnesses,…”

Section: Outlook On Dynamic Programming For Other Formalismsmentioning

confidence: 99%

Advanced tools and methods for treewidth-based problem solving

Hecher¹

2023

It - Information Technology

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“…Treewidth is widely used for fine-grained complexity analyzes and to establish algorithms that provide tractability when bounding the treewidth, e.g., artificial intelligence (Gottlob and Szeider 2007), knowledge representation (Gottlob, Pichler, and Wei 2006), abduction in Datalog (Gottlob, Pichler, and Wei 2007), and databases (Grohe 2007). In the context of counting and projected counting treewidth has been considered in various areas (Fichte, Hecher, and Meier 2018;Fichte et al 2017;Fichte and Hecher 2019;Fichte, Hecher, and Schindler 2018;Fichte et al 2018a). Also competitive implementations are available (Fichte et al 2018bHecher, Thier, and Woltran 2020;Dudek, Phan, and Vardi 2020).…”

Section: Syntaxmentioning

confidence: 99%

Knowledge-Base Degrees of Inconsistency: Complexity and Counting

Fichte

Hecher

Meier

2021

AAAI

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Description logics (DLs) are knowledge representation languages that are used in the field of artificial intelligence (AI). A common technique is to query DL knowledge bases, e.g., by Boolean Datalog queries, and ask for entailment. But real world knowledge-bases are often obtained by combining data from various sources. This, inherently, might result in certain inconsistencies (with respect to a given query) and requires to estimate a degree of inconsistency before using a knowledge-base. In this paper, we provide a complexity analysis of fixed-domain non-entailment (NE) on Datalog programs for well-established families of knowledge bases (KBs). We exhibit a detailed complexity map for the decision cases, counting and projected counting, which may serve as a quantitative measure for inconsistency of a KB with respect to a query. Our results show that NE is natural for the second, third, and fourth level of the polynomial (counting) hierarchy depending on the type of the studied query (stratified, normal, disjunctive) and one level higher for the projected versions. Further, we show fixed-parameter tractability by bounding the treewidth, provide a constructive algorithm, and show its theoretical limitation in terms of conditional lower bounds.

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“…The two mentioned parameterized complexity related papers (Fichte, Hecher, and Meier 2019;Mahmood, Meier, and Schmidt 2020) both are about different formalisms that are slightly related to our setting (the first is about abstract argumentation, the second on abduction). Due to space limitations, for results marked with a , the proof can be found in the technical report of the paper (Mahmood, Meier, and Schmidt 2021).…”

Section: Contributionsmentioning

confidence: 99%

Parameterized Complexity of Logic-Based Argumentation in Schaefer's Framework

Mahmood

Meier

Schmidt

2021

AAAI

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Logic-based argumentation is a well-established formalism modeling nonmonotonic reasoning. It has been playing a major role in AI for decades, now. 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 paper, 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 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, and then study different parameterizations for each of the fragment. 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 (paraNP and beyond).

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Counting Complexity for Reasoning in Abstract Argumentation

Cited by 15 publications

References 27 publications

Advanced tools and methods for treewidth-based problem solving

Advanced tools and methods for treewidth-based problem solving

Knowledge-Base Degrees of Inconsistency: Complexity and Counting

Parameterized Complexity of Logic-Based Argumentation in Schaefer's Framework

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