On an Algebraic Characterization of Default Reasoning
Date: July 17th (Wednesday)
Time: ??.??
Abstract
This talk deals with non-monotonic logical consequence operators
defined by default reasoning.
The aim is to consider and investigate such operators within the
general framework of the theory of closure operators, in particular,
topological and algebraic closure operators used in topology, algebra,
logic and metamathematics.The main result of the paper is an
appropriate generalization of the well-known theorem of G. Birkhoff,
P. Hall and J. Schmidt, which gives an algebraic
characterization of
compact closure operators by universal algebras and also by classical,
"monotonic", deductive systems.
For this p[urpose we introduce a new type of algebraic strusture
generalizing the classical notion of universal algebra (introduced by
G. Birkhoff) which we shall call regulated algebra. Furthermore,
generalizing the concept of a classical monotonic deductive system
which is closely asociated with the theorem of G. Birkhoff,
P. Hall and
J. Schmidt, we introduce Regulated Deductive Systems
which include,
for instance Reiter's default theories as special cases.
As we shall see in the sequel, many of the classical results in
non-monotonic reasoning can be derived on the basis of this general
axiomatic approach. Furthermore, this approach gives a deeper
mathematical understanding of what default reasoning is and provides
the starting point for further inverstigations of regulated algebras
and regulated deductive systems.
