Skeptical Reasoning with Preferred Semantics in Abstract Argumentation without Computing Preferred Extensions

We address the problem of deciding skeptical acceptance wrt. preferred semantics of an argument in abstract argumentation frameworks, i.e., the problem of deciding whether an argument is contained in all maximally admissible sets, a.k.a. preferred extensions. State-of-the-art algorithms solve this problem with iterative calls to an external SAT-solver to determine preferred extensions. We provide a new characterisation of skeptical acceptance wrt. preferred semantics that does not involve the notion of a preferred extension. We then develop a new algorithm that also relies on iterative calls to an external SAT-solver but avoids the costly part of maximising admissible sets. We present the results of an experimental evaluation that shows that this new approach significantly outperforms the state of the art. We also apply similar ideas to develop a new algorithm for computing the ideal extension.

Fuzzy Extension of Coupled Γ-semiring: Some Properties

The concept of coupled Γ-semiring first appeared in [1]. In the present paper assuming M is a Γ-semiring, we introduce concepts of anti-fuzzy prime ideal, anti-fuzzy semi prime ideal, and anti-fuzzy ideal extension, respectively, of Some properties associated with these new concepts are obtained. The work in this paper takes inspiration from [2].

An Efficient Algorithm for Skeptical Preferred Acceptance in Dynamic Argumentation Frameworks

Though there has been an extensive body of work on efficiently solving computational problems for static Dung's argumentation frameworks (AFs), little work has been done for handling dynamic AFs and in particular for deciding the skeptical acceptance of a given argument. In this paper we devise an efficient algorithm for computing the skeptical preferred acceptance in dynamic AFs. More specifically, we investigate how the skeptical acceptance of an argument (goal) evolves when the given AF is updated and propose an efficient algorithm for solving this problem. Our algorithm, called SPA, relies on two main ideas: i) computing a small portion of the input AF, called "context-based" AF, which is sufficient to determine the status of the goal in the updated AF, and ii) incrementally computing the ideal extension to further restrict the context-based AF. We experimentally show that SPA significantly outperforms the computation from scratch, and that the overhead of incrementally maintaining the ideal extension pays off as it speeds up the computation.

Skew left braces with non-trivial annihilator

We describe the class of all skew left braces with non-trivial annihilator through ideal extension of a skew left brace. The ideal extension of skew left braces is a generalization to the non-abelian case of the extension of left braces provided by Bachiller in [D. Bachiller, Extensions, matched products, and simple braces, J. Pure Appl. Algebra 222 (2018) 1670–1691].

USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS

In this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.