Complexity Classifications for logic-based Argumentation

We consider logic-based argumentation in which an argument is a pair (Fi,al), where the support Fi is a minimal consistent set of formulae taken from a given knowledge base (usually denoted by De) that entails the claim al (a formula). We study the complexity of three central problems in argumentation: the existence of a support Fi ss De, the validity of a support and the relevance problem (given psi is there a support Fi such that psi ss Fi?). When arguments are given in the full language of propositional logic these problems are computationally costly tasks, the validity problem is DP-complete, the others are SigP2-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 build upon a fixed finite set of Boolean relations Ga (the constraint language). We show that according to the properties of this language Ga, deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete or SigP2-complete. We present a dichotomous classification, P or DP-complete, for the verification problem and a trichotomous classification for the relevance problem into either polynomial, NP-complete, or SigP2-complete. These last two classifications are obtained by means of algebraic tools

When win-argument pedagogy is a loss for the composition classroom

Despite the effort educators put into developing in students the critical writing and thinking skills needed to compose effective arguments, undergraduate college students are often accused of churning out essays lacking in creative and critical thought, arguments too obviously formulated and with sides too sharply drawn. Theories abound as to why these deficiencies are rampant. Some blame students’ immature cognitive and emotional development for these lacks. Others put the blame of lackadaisical output on the assigning of shopworn writing subjects, assigned topics such as on American laws and attitudes about capital punishment and abortion. Although these factors might contribute to faulty written output in some cases, the prevailing hindrance is our very pedagogy, a system in which students are rewarded for composing the very type of argument we wish to avoid — the eristic, in which the goal is not truth seeking, but successfully disputing another’s argument. Certainly the eristic argument is the intended solution in cases when a clear‑cut outcome is needed, such as in legal battles and political campaigns when there can only be one winner. However, teaching mainly or exclusively the eristic, as is done in most composition classrooms today, halts the advancement of these higher‑order inquiry skills we try developing in our students

The Complexity of Satisfiability for Sub-Boolean Fragments of ALC

The standard reasoning problem, concept satisfiability, in the basic description logic ALC is PSPACE-complete, and it is EXPTIME-complete in the presence of unrestricted axioms. Several fragments of ALC, notably logics in the FL, EL, and DL-Lite family, have an easier satisfiability problem; sometimes it is even tractable. All these fragments restrict the use of Boolean operators in one way or another. We look at systematic and more general restrictions of the Boolean operators and establish the complexity of the concept satisfiability problem in the presence of axioms. We separate tractable from intractable cases.Comment: 17 pages, accepted (in short version) to Description Logic Workshop 201

A Plausibility Semantics for Abstract Argumentation Frameworks

We propose and investigate a simple ranking-measure-based extension semantics for abstract argumentation frameworks based on their generic instantiation by default knowledge bases and the ranking construction semantics for default reasoning. In this context, we consider the path from structured to logical to shallow semantic instantiations. The resulting well-justified JZ-extension semantics diverges from more traditional approaches.Comment: Proceedings of the 15th International Workshop on Non-Monotonic Reasoning (NMR 2014). This is an improved and extended version of the author's ECSQARU 2013 pape

Commentary on Constructing a Periodic Table of Arguments

This is the Commentary on Wagemans\u27 paper Constructing a Periodic Table of Arguments

Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment

Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number I(a) is assigned to a then a natural number I(b) can be assigned to b such that I(a) is greater than I(b). The construction of I is based on Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the lambda formulation of T and its fragments. Compared with Weiermann's 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda terms instead of combinatory logic terms. It is expected that the methods developed here can be applied to other higher order rewrite systems resulting in new powerful termination orderings since T is a paradigm for such systems