On the Decomposition of Abstract Dialectical Frameworks and the Complexity of Naive-based Semantics

dialectical frameworks (ADFs) are a recently introduced powerful generalization of Dung’s popular abstract argumentation frameworks (AFs). Inspired by similar work for AFs, we introduce a decomposition scheme for ADFs, which proceeds along the ADF’s strongly connected components. We find that, for several semantics, the decompositionbased version coincides with the original semantics, whereas for others, it gives rise to a new semantics. These new semantics allow us to deal with pertinent problems such as odd-length negative cycles in a more general setting, that for instance also encompasses logic programs. We perform an exhaustive analysis of the computational complexity of these new, so-called naive-based semantics. The results are quite interesting, for some of them involve little-known classes of the so-called Boolean hierarchy (another hierarchy in between classes of the polynomial hierarchy). Furthermore, in credulous and sceptical entailment, the complexity can be different depending on whether we check for truth or falsity of a specific statement.

ENHANCING ADEQUACY OF GRADING STUDY PROJECTS ON THE BASE OF PARAMETRIC RELAXATION OF PAIRWISE COMPARISONS

A problem of automated assessing of students’ study projects is regarded. A heuristic algorithm based on fuzzy estimating of projects and on pairwise comparisons among them is proposed. For improving adequacy and naturalness of grades, an approach based on introducing a parameter named relaxation parameter was suggested in the paper. This enables to reduce the spread between maximum and minimum values of projects in comparison with the one in the standard scale suggested by T. Saati. Reasonable values of this parameter were selected experimentally. For estimating the best alternative, a center of mass of a fuzzy max-min composition should be calculated. An estimation algorithm for a case of non-transitive preferences based on getting strongly connected components and on pairwise comparisons between them is also suggested. In this case, relaxation parameters should be chosen separately for each subtask. So the combined technique of evaluating alternatives proposed in the paper depends of the following parameters: relaxation parameters for pairwise comparisons matrices within each strongly connected components; relaxation parameter for pairwise comparisons matrices among strongly connected components; membership function for describing the best alternative.

Symbolic Coloured SCC Decomposition

Problems arising in many scientific disciplines are often modelled using edge-coloured directed graphs. These can be enormous in the number of both vertices and colours. Given such a graph, the original problem frequently translates to the detection of the graph’s strongly connected components, which is challenging at this scale.We propose a new, symbolic algorithm that computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $$O(p\cdot n\cdot \log n)$$ O ( p · n · log n ) symbolic steps, where p is the number of colours and n the number of vertices. We evaluate the algorithm using an experimental implementation based on Binary Decision Diagrams (BDDs) and large (up to $$2^{48}$$ 2 48 ) coloured graphs produced by models appearing in systems biology.

Проверка изоморфности двух графов и поиск изоморфных подграфов: подход, основанный на анализе максимально протяженных неразветвляющихся путей / Maximal non-branching paths approach to solve (sub)graph isomorphism problem

Предложен подход к решению проблемы проверки изоморфности двух графов исходя из анализа их максимально протяженных неразветвляющихся путей. На его основе предлагается подход и алгоритм решения частного случая задачи поиска в некотором графе A всех подграфов, изоморфных заданному графу B (а именно, поиск только «вписанных» подграфов), а также определяется само понятие «вписанного» подграфа. «Вписанным» подграфом графа A здесь называется такой его подграф, который может быть «приклеен» к другим частям графа A только за счет ребер, инцидентных лишь граничным вершинам его (подграфа) неразветвляющихся путей максимальной длины (при этом граф A может содержать и иные компоненты связности). Решение частного случая задачи поиска «вписанных» подграфов обобщается для поиска в графе A всех подграфов, изоморфных данному графу-образцу B. Для этого вместо рассмотрения их максимально протяженных неразветвляющихся путей необходимо рассматривать все их ребра. Предложенные подход и алгоритм применимы: (1) как для ориентированных, так и для неориентированных графов, (2) для графов, содержащих более одной компоненты связности/ сильной связности, (3) для графов, содержащих кратные (множественные) ребра и петли. ----------- An approach based on maximal non-branching paths analysis to solve graph isomorphism problem is introduced. An algorithm to solve the particular case of the problem of finding in a some graph A all subgraphs that are isomorphic to given graph B is proposed (only “inscribed” subgraphs can be found this way, not all of them). Here we shall name a subgraph of some given graph A as "inscribed" if (1) this subgraph is "glued" to other parts of A only by edges that connected to those vertices of this subgraph that are begin/ end ones of any max-length non-branching path of it, or (2) this subgraph is a separate connected component of the graph A. The proposed algorithm of finding “inscribed” subgraphs is expanded to solve a whole problem of finding all (not only “inscribed”) subgraphs of a graph A that are isomorphic to given template graph B. To do so one should consider all edges of these graphs instead of their max-length non-branching paths. These proposed approach and algorithm may be implemented to: (1) directed or undirected graphs, (2) graphs that have more than one connected components/ strongly connected components, (3) graphs that contain multiple edges and multiple loops.

Geometrical digraphs as models of heterogeneous computer networks

New mathematical models of networks are considered — geometrical oriented graphs. Such models adequately reflect the structure of heterogeneous wireless computer, sensor and other networks. The strongly connected components of geometric digraphs, their dependence on the number of network nodes in a given area, on the distributions of the radii of the zones of reception and transmission of signals are investigated. The probabilistic characteristics of random geometric digraphs, the features of the dependences of the probability of strong connectivity on the number of nodes in the digraphs for various combinations of radii are analyzed.