Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

On the KLM properties of a fuzzy DL with Typicality

The paper investigates the properties of a fuzzy logic of typicality. The extension of fuzzy logic with a typicality operator was proposed in recent work to define a fuzzy multipreference semantics for Multilayer Perceptrons, by regarding the deep neural network as a conditional knowledge base. In this paper, we study its properties. First, a monotonic extension of a fuzzy ALC with typicality is considered (called ALC^FT) and a reformulation the KLM properties of a preferential consequence relation for this logic is devised. Most of the properties are satisfied, depending on the reformulation and on the fuzzy combination functions considered. We then strengthen ALC^FT with a closure construction by introducing a notion of faithful model of a weighted knowledge base, which generalizes the notion of coherent model of a conditional knowledge base previously introduced, and we study its properties.Comment: 15 page

One-variable fragments of intermediate logics over linear frames

A correspondence is established between one-variable fragments of (first-order) intermediate logics defined over a fixed countable linear frame and Gödel modal logics defined over many-valued equivalence relations with values in a closed subset of the real unit interval. It is also shown that each of these logics can be interpreted in the one-variable fragment of the corresponding constant domain intermediate logic, which is equivalent to a Gödel modal logic defined over (crisp) equivalence relations. Although the latter modal logics in general lack the finite model property with respect to their frame semantics, an alternative semantics is defined that has this property and used to establish co-NP-completeness results for the one-variable fragments of the corresponding intermediate logics both with and without constant domains

Adding Threshold Concepts to the Description Logic EL

We introduce an extension of the lightweight Description Logic EL that allows us to de_ne concepts in an approximate way. For this purpose, we use a graded membership function, which for each individual and concept yields a number in the interval [0, 1] expressing the degree to which the individual belongs to the concept. Threshold concepts C~t for ~ then collect all the individuals that belong to C with degree ~ t. We generalize a well-known characterization of membership in EL concepts to construct a specific graded membership function deg, and investigate the complexity of reasoning in the Description Logic τEL(deg), which extends EL by threshold concepts defined using deg. We also compare the instance problem for threshold concepts of the form C>t in τEL(deg) with the relaxed instance queries of Ecke et al

Probabilistic description logics for subjective uncertainty

We propose a family of probabilistic description logics (DLs) that are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to temporal DLs and are well-suited for representing subjective probabilities. We carry out a detailed study of reasoning in the new family of logics, concentrating on probabilistic extensions of the DLs ALC and EL, and showing that the complexity ranges from PTime via ExpTime and 2ExpTime to undecidable

Planeamento de redes de distribuição de energia

O planeamento de redes de distribuição tem como objetivo assegurar a existência de capacidade nas redes para a fornecimento de energia elétrica com bons níveis de qualidade de serviço tendo em conta os fatores económicos associados. No âmbito do trabalho apresentado na presente dissertação, foi elaborado um modelo de planeamento que determina a configuração de rede resultante da minimização de custos associados a: 1) perdas por efeito de joule; 2) investimento em novos componentes; 3) energia não entregue. A incerteza associada ao valor do consumo de cada carga é modelada através de lógica difusa. O problema de otimização definido é resolvido pelo método de decomposição de benders que contempla dois trânsitos de potências ótimos (modelo DC e modelo AC) no problema mestre e escravo respectivamente para validação de restrições. Foram também definidos critérios de paragem do método de decomposição de benders. O modelo proposto classifica-se como programação não linear inteira mista e foi implementado na ferramenta de otimização General Algebraic Modeling System (GAMS). O modelo desenvolvido tem em conta todos componentes das redes para a otimização do planeamento, conforme podemos analisar nos casos de estudo implementados. Cada caso de estudo é definido pela variação da importância que cada uma das variáveis do problema toma, tendo em vista cobrir de alguma todos os cenários de operação expetáveis. Através destes casos de estudo verifica-se as várias configurações que a rede pode tomar, tendo em conta as importâncias atribuídas a cada uma das variáveis, bem como os respetivos custos associados a cada solução. Este trabalho oferece um considerável contributo no âmbito do planeamento de redes de distribuição, pois comporta diferentes variáveis para a execução do mesmo. É também um modelo bastante robusto não perdendo o ‘norte’ no encontro de solução para redes de grande dimensão, com maior número de componentes.The distribution network planning aims to ensure that there is capacity in the networks for the supply of electricity with good levels of service quality taking into account the economic factors. The work presented in this thesis has produced a planning model that determines the resulting network configuration minimizing costs associated with: 1) the effect of joule losses; 2) investment in new components; 3) energy undeliverable. The uncertainty in the consumption of each load is modeled using fuzzy logic. The defined optimization problem is solved by benders decomposition method that comprises an optimal power flow (DC model) in the slave problem for validation restrictions. It were also defined stopping criteria of benders decomposition method. The proposed model is classified as mixed integer non-linear programming. It was implemented in the General Algebraic Modeling System (GAMS) tool. The model takes into account all components of networks to optimize the planning, as we analyze the cases of study implemented. Each case is defined by the study variation of the importance of each variable taking the problem in view of a cover all expected operating scenarios. Through these case studies, it results that various network configurations can be taken, in view of the importance attributed to each of these variables, as well as the costs associated with each solution. This work offers a considerable contribution in planning distribution networks, because it holds different variables for execution. It is also a fairly robust model not losing the 'north' in meeting solution for large networks with a larger number of components

Adding Threshold Concepts to the Description Logic EL

We introduce a family of logics extending the lightweight Description Logic EL, that allows us to define concepts in an approximate way. The main idea is to use a graded membership function m, which for each individual and concept yields a number in the interval [0,1] expressing the degree to which the individual belongs to the concept. Threshold concepts C~t for ~ in {,>=} then collect all the individuals that belong to C with degree ~t. We further study this framework in two particular directions. First, we define a specific graded membership function deg and investigate the complexity of reasoning in the resulting Description Logic tEL(deg) w.r.t. both the empty terminology and acyclic TBoxes. Second, we show how to turn concept similarity measures into membership degree functions. It turns out that under certain conditions such functions are well-defined, and therefore induce a wide range of threshold logics. Last, we present preliminary results on the computational complexity landscape of reasoning in such a big family of threshold logics