3 research outputs found
Product representation for default bilattices: an application of natural duality theory
Bilattices (that is, sets with two lattice structures) provide an algebraic
tool to model simultaneously the validity of, and knowledge about, sentences in
an appropriate language. In particular, certain bilattices have been used to
model situations in which information is prioritised and so can be viewed
hierarchically. These default bilattices are not interlaced: the lattice
operations of one lattice structure do not preserve the order of the other one.
The well-known product representation theorem for interlaced bilattices does
not extend to bilattices which fail to be interlaced and the lack of a product
representation has been a handicap to understanding the structure of default
bilattices. In this paper we study, from an algebraic perspective, a hierarchy
of varieties of default bilattices, allowing for different levels of default.
We develop natural dualities for these varieties and thereby obtain a concrete
representation for the algebras in each variety. This leads on to a form of
product representation that generalises the product representation as this
applies to distributive bilattices
Ordering Default Theories
In first-order logic, a theory T 1 is considered stronger than another theory T 2 if every formula derived from T 2 is also derived from T 1 . Such an order relation is useful to know relative value between different theories. In the context of default logic, a theory contains default information as well as definite information. To order default theories, it is necessary to assess the information content of a default theory. To this end, we introduce a multi-valued interpretation of default theories based on a nine-valued bilattice. It distinguishes definite and credulous/skeptical default information derived from a theory, and is used for ordering default theories based on their information contents. The technique is also applied to order nonmonotonic logic programs. The results of this paper provide a method for comparing different default theories and have important application to learning nonmonotonic theories
Ordering Default Theories
In first-order logic, a theory T1 is considered stronger than another theory T2 if every formula derived from T2 is also derived from T1. Such an order relation is useful to know relative value between different theories. In the context of default logic, a theory contains default information as well as definite information. To order default theories, it is necessary to assess the information content of a default theory. To this end, we introduce a multi-valued interpretation of default theories based on a nine-valued bilattice. It distinguishes definite and credulous/skeptical default information derived from a theory, and is used for ordering default theories based on their information contents. The technique is also applied to order nonmonotonic logic programs. The results of this paper provide a method for comparing different default theories and have important application to learning nonmonotonic theories.