Complexity of logic-based argumentation in Post's framework

Creignou, Nadia; Schmidt, Johannes; Thomas, Michael E.; Woltran, Stefan

doi:10.1080/19462166.2011.629736

Cited by 8 publications

(7 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us recall that in the full framework of propositional logic these three problems, Arg, Arg-Check and Arg-Rel, are respectively Σ p 2 -complete [PWA03], DP-complete and Σ p 2complete (see e.g. [CSTW11]).…”

Section: Argumentation Problemsmentioning

confidence: 99%

“…A first step towards an extensive study of the complexity of argumentation in fragments of propositional logic was taken in [CSTW11] in Post's framework, where the authors considered formulae built upon a restricted set of connectives. They obtained a full classification of various argumentation problems depending on the set of allowed connectives.…”

Section: Introductionmentioning

confidence: 99%

“…A wide range of algorithmic problems have been studied in this context (for a survey see [CV08]), and in particular the abduction problem [CZ06,NZ08]. Preliminary results concerning argumentation have been obtained in [CES12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Complexity Classifications for Logic-Based Argumentation

Creignou

Egly

Schmidt

2014

ACM Trans. Comput. Logic

Self Cite

View full text Add to dashboard Cite

We consider logic-based argumentation in which an argument is a pair ( , α), where the support is a minimal consistent set of formulae taken from a given knowledge base (usually denoted by ) that entails the claim α (a formula). We study the complexity of three central problems in argumentation: the existence of a support ⊆ , the verification of a support, and the relevance problem (given ψ, is there a support such that ψ ∈ ?). When arguments are given in the full language of propositional logic, these problems are computationally costly tasks: the verification problem is DP-complete; the others are p 2 -complete. We study these problems in Schaefer's famous framework where the considered propositional formulae are in generalized conjunctive normal form. This means that formulae are conjunctions of constraints built upon a fixed finite set of Boolean relations (the constraint language). We show that according to the properties of this language , deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete, or p 2 -complete. We present a dichotomous classification, P or DPcomplete, for the verification problem and a trichotomous classification for the relevance problem into either polynomial, NP-complete, or p 2 -complete. These last two classifications are obtained by means of algebraic tools.

show abstract

Section: Argumentation Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Complexity Classifications for Logic-Based Argumentation

Creignou

Egly

Schmidt

2014

ACM Trans. Comput. Logic

Self Cite

View full text Add to dashboard Cite

show abstract

“…The most prominent result under this approach is the dichotomy theorem of Lewis [18] which classifies propositional satisfiability into polynomialtime solvable cases and intractable ones depending merely on the existence of specific Boolean operators. This approach has been followed many times in a wealth of different contexts [1,2,6,10,19,20,29] as well as in the context of abduction itself [22,9]. Interestingly, in the scope of constraint satisfaction problems, the investigation of co-clones allows one to proceed a similar kind of classification.…”

Section: Introductionmentioning

confidence: 99%

Parameterised Complexity of Abduction in Schaefer’s Framework

Mahmood

Meier

Schmidt

2019

Logical Foundations of Computer Science

Self Cite

View full text Add to dashboard Cite

Abductive reasoning is a non-monotonic formalism stemming from the work of Peirce. It describes the process of deriving the most plausible explanations of known facts. Considering the positive version asking for sets of variables as explanations, we study, besides asking for existence of the set of explanations, two explanation size limited variants of this reasoning problem (less than or equal to, and equal to). In this paper, we present a thorough classification regarding the parameterised complexity of these problems under a wealth of different parameterisations. Furthermore, we analyse all possible Boolean fragments of these problems in the constraint satisfaction approach with co-clones. Thereby, we complete the parameterised picture started by Fellows et al. (AAAI 2012), partially building on results of Nordh and Zanuttini (Artif. Intell. 2008). In this process, we outline a fine-grained analysis of the inherent intractability of these problems and pinpoint their tractable parts. ACM Subject ClassificationTheory of computation → Parameterized complexity and exact algorithms; Computing methodologies → Knowledge representation and reasoning

show abstract

“…The difficult nature of argumentation has been underlined by studies concerning the complexity of finding individual arguments Parsons, Wooldridge, & Amgoud (2003), the complexity of some decision problems concerning the instantiation of argument graphs with classical logic arguments and the direct undercut attack relation Wooldridge, Dunne, & Parsons (2006), and the complexity of finding argument trees Hirsch & Gorogiannis (2009). Encodation of these tasks as quantified Boolean formulae also indicates that development of algorithms is a difficult challenge (Besnard, Hunter, & Woltran 2009), and Post's framework has been used to give a breakdown of where complexity lies in logic-based argumentation (Creignou, Schmidt, Thomas, & Woltran 2011).…”

Section: Automated Reasoning For Deductive Argumentationmentioning

confidence: 99%

Constructing argument graphs with deductive arguments: a tutorial

Besnard¹,

Hunter²

2014

Argument & Computation

View full text Add to dashboard Cite

A deductive argument is a pair where the first item is a set of premises, the second item is a claim, and the premises entail the claim. This can be formalised by assuming a logical language for the premises and the claim, and logical entailment (or consequence relation) for showing that the claim follows from the premises. Examples of logics that can be used include classical logic, modal logic, description logic, temporal logic, and conditional logic. A counterargument for an argument A is an argument B where the claim of B contradicts the premises of A. Different choices of logic, and different choices for the precise definitions of argument and counterargument, give us a range of possibilities for formalising deductive argumentation. Further options are available to us for choosing the arguments and counterarguments we put into an argument graph. If we are to construct an argument graph based on the arguments that can be constructed from a knowledgebase, then we can be exhaustive in including all arguments and counterarguments that can be constructed from the knowledgebase. But there are other options available to us. We consider some of the possibilities in this review.

show abstract

Complexity of logic-based argumentation in Post's framework

Cited by 8 publications

References 34 publications

Complexity Classifications for Logic-Based Argumentation

Complexity Classifications for Logic-Based Argumentation

Parameterised Complexity of Abduction in Schaefer’s Framework

Constructing argument graphs with deductive arguments: a tutorial

Contact Info

Product

Resources

About