We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore how probabilistic reasoning under coherence is related to model- theoretic probabilistic reasoning and to default reasoning in System . In particular, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Moreover, we show that probabilistic reasoning under coherence is a generalization of default reasoning in System (...) . That is, we provide a new probabilistic semantics for System , which neither uses infinitesimal probabilities nor atomic bound (or big-stepped) probabilities. These results also provide new algorithms for probabilistic reasoning under coherence and for default reasoning in System , and they give new insight into default reasoning with conditional objects. (shrink)

The ability to conduct logical reasoning is a fundamental aspect of intelligent behavior, and thus an important problem along the way to human-level artificial intelligence. Traditionally, symbolic methods from the field of knowledge representation and reasoning have been used to equip agents with capabilities that resemble human reasoning qualities. More recently, however, there has been an increasing interest in applying alternative approaches based on machine learning rather than logic-based formalisms to tackle this kind of tasks. Here, we make use of (...) state-of-the-art methods for training deep neural networks to devise a novel model that is closely coupled to symbolic reasoning methods, and thus able to learn how to effectively perform basic ontology reasoning. This term describes an important and at the same time very natural kind of problem settings where the rules for conducting reasoning are specified alongside with the actual information. Many problems in practice may be viewed as such reasoning tasks, which is why the presented approach is applicable to a plethora of important real-world problems. To demonstrate the effectiveness of the suggested method, we present the outcomes of several experiments that have been conducted on both toy datasets as well as real-world data, which show that our model learned to perform precise reasoning on a number of diverse inference tasks that require comprehensive deductive proficiencies. Furthermore, it turned out that the suggested model suffers much less from different obstacles that prohibit symbolic reasoning. (shrink)

The ability to conduct logical reasoning is a fundamental aspect of intelligent behavior, and thus an important problem along the way to human-level artificial intelligence. Traditionally, symbolic methods from the field of knowledge representation and reasoning have been used to equip agents with capabilities that resemble human reasoning qualities. More recently, however, there has been an increasing interest in applying alternative approaches based on machine learning rather than logic-based formalisms to tackle this kind of tasks. Here, we make use of (...) state-of-the-art methods for training deep neural networks to devise a novel model that is closely coupled to symbolic reasoning methods, and thus able to learn how to effectively perform basic ontology reasoning. This term describes an important and at the same time very natural kind of problem settings where the rules for conducting reasoning are specified alongside with the actual information. Many problems in practice may be viewed as such reasoning tasks, which is why the presented approach is applicable to a plethora of important real-world problems. To demonstrate the effectiveness of the suggested method, we present the outcomes of several experiments that have been conducted on both toy datasets as well as real-world data, which show that our model learned to perform precise reasoning on a number of diverse inference tasks that require comprehensive deductive proficiencies. Furthermore, it turned out that the suggested model suffers much less from different obstacles that prohibit symbolic reasoning. (shrink)

We present new probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment, called Zλ- and lexλ-entailment, which are parameterized through a value λ ∈ [0,1] that describes the strength of the inheritance of purely probabilistic knowledge. In the special cases of λ = 0 and λ = 1, the notions of Zλ- and lexλ-entailment coincide with probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment that have been recently introduced by the author. We show (...) that the notions of Zλ- and lexλ-entailment have similar properties as their classical counterparts. In particular, they both satisfy the rationality postulates of System P and the property of Rational Monotonicity. Moreover, Zλ-entailment is weaker than lexλ-entailment, and both Zλ- and lexλ-entailment are proper generalizations of their classical counterparts. (shrink)

We propose a combination of probabilistic reasoning from conditional constraints with approaches to default reasoning from conditional knowledge bases. In detail, we generalize the notions of Pearl's entailment in system Z, Lehmann's lexicographic entailment, and Geffner's conditional entailment to conditional constraints. We give some examples that show that the new notions of z-, lexicographic, and conditional entailment have similar properties like their classical counterparts. Moreover, we show that the new notions of z-, lexicographic, and conditional entailment are proper generalizations of (...) both their classical counterparts and the classical notion of logical entailment for conditional constraints. (shrink)