Axiomatic Aspects of Default Inference

W.C. Rounds and G.-Q. Zhang (2001) have proposed to study a form of disjunctive logic programming generalized to algebraic domains. This system allows reasoning with information which is hierarchically structured and forms a (suitable) domain. We extend this framework to include reasoning with default negation, giving rise to a new nonmonotonic reasoning framework on hierarchical knowledge which encompasses answer set programming with extended disjunctive logic programs. We a...

A. Tarski proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of intere...

A fact apparently not observed earlier in the literature of nonmonotonic reasoning is that Reiter, in his default logic paper, did not directly formalize informal defaults. Instead, he translated a default into a certain natural language proposition and provided a formalization of the latter. A few years later, Moore noted that propositions like the one used by Reiter are fundamentally different than defaults and exhibit a certain autoepistemic nature. Thus, Reiter had develo...

A default theory can be characterized by its sets of plausible conclusions, called its extensions. But, due to the theoretical complexity of Default Logic (Sigma_2p-complete), the problem of finding such an extension is very difficult if one wants to deal with non trivial knowledge bases. Based on the principle of natural selection, Genetic Algorithms have been quite successfully applied to combinatorial problems and seem useful for problems with huge search spaces and when n...

We show how to transform any set of prioritized propositional defaults into an equivalent set of parallel (i.e., unprioritized) defaults, in circumscription. We give an algorithm to implement the transform. We show how to use the transform algorithm as a generator of a whole family of inferencing algorithms for circumscription. The method is to employ the transform algorithm as a front end to any inferencing algorithm, e.g., one of the previously available, that handles the p...

Recently, it has been shown that probabilistic entailment under coherence is weaker than model-theoretic probabilistic entailment. Moreover, probabilistic entailment under coherence is a generalization of default entailment in System P. In this paper, we continue this line of research by presenting probabilistic generalizations of more sophisticated notions of classical default entailment that lie between model-theoretic probabilistic entailment and probabilistic entailment u...

A conditional knowledge base R is a set of conditionals of the form "If A, the usually B". Using structural information derived from the conditionals in R, we introduce the preferred structure relation on worlds. The preferred structure relation is the core ingredient of a new inference relation called system W inference that inductively completes the knowledge given explicitly in R. We show that system W exhibits desirable inference properties like satisfying system P and av...

When we work with information from multiple sources, the formalism each employs to handle uncertainty may not be uniform. In order to be able to combine these knowledge bases of different formats, we need to first establish a common basis for characterizing and evaluating the different formalisms, and provide a semantics for the combined mechanism. A common framework can provide an infrastructure for building an integrated system, and is essential if we are to understand its ...

Recent success of Bayesian methods in neuroscience and artificial intelligence gives rise to the hypothesis that the brain is a Bayesian machine. Since logic, as the laws of thought, is a product and practice of the human brain, it leads to another hypothesis that there is a Bayesian algorithm and data-structure for logical reasoning. In this paper, we give a Bayesian account of entailment and characterize its abstract inferential properties. The Bayesian entailment is shown ...

In this paper we present a transformation of finite propositional default theories into so-called propositional argumentation systems. This transformation allows to characterize all notions of Reiter's default logic in the framework of argumentation systems. As a consequence, computing extensions, or determining wether a given formula belongs to one extension or all extensions can be answered without leaving the field of classical propositional logic. The transformation propo...