2 research outputs found
Axiomatic Aspects of Default Inference
This paper studies axioms for nonmonotonic consequences from a
semantics-based point of view, focusing on a class of mathematical structures
for reasoning about partial information without a predefined syntax/logic. This
structure is called a default structure. We study axioms for the nonmonotonic
consequence relation derived from extensions as in Reiter's default logic,
using skeptical reasoning, but extensions are now used for the construction of
possible worlds in a default information structure.
In previous work we showed that skeptical reasoning arising from
default-extensions obeys a well-behaved set of axioms including the axiom of
cautious cut. We show here that, remarkably, the converse is also true: any
consequence relation obeying this set of axioms can be represented as one
constructed from skeptical reasoning. We provide representation theorems to
relate axioms for nonmonotonic consequence relation and properties about
extensions, and provide one-to-one correspondence between nonmonotonic systems
which satisfies the law of cautious monotony and default structures with unique
extensions. Our results give a theoretical justification for a set of basic
rules governing the update of nonmonotonic knowledge bases, demonstrating the
derivation of them from the more concrete and primitive construction of
extensions. It is also striking to note that proofs of the representation
theorems show that only shallow extensions are necessary, in the sense that the
number of iterations needed to achieve an extension is at most three. All of
these developments are made possible by taking a more liberal view of
consistency: consistency is a user defined predicate, satisfying some basic
properties.Comment: 16 pages. Originally published in proc. PCL 2002, a FLoC workshop;
eds. Hendrik Decker, Dina Goldin, Jorgen Villadsen, Toshiharu Waragai
(http://floc02.diku.dk/PCL/
Axiomatic aspects of default inference
as ⊢) have been well studied and well-understood, either with or without the presence of logical connectives. There is, however, less uniform agreement on laws for the nonmonotonic consequence relation. This paper studies axioms for nonmonotonic consequences from a semantics-based point of view, focusing on a class of mathematical structures for reasoning about partial information without a predefined syntax/logic. This structure is called a default structure. We study axioms for the nonmonotonic consequence relation derived from extensions as in Reiter’s default logic, using skeptical reasoning, but extensions are now used for the construction of possible worlds in a default information structure. In previous work we showed that skeptical reasoning arising from defaultextensions obeys a well-behaved set of axioms including the axiom of cautious cut. We show here that, remarkably, the converse is also true: any consequence relation obeying this set of axioms can be represented as one constructed from skeptical reasoning. We provide representation theorems to relate axioms for nonmonotonic consequence relation and properties about extensions, and provide a one-to-one correspondence between nonmonotonic systems which satisfies the law of cautious monotony and default structures with unique extensions. Our results give a theoretical justification for a set of basic rules governing the update of nonmonotonic knowledge bases, demonstrating the derivation of them from the more concrete and primitive construction of extensions. It is also striking to note that proofs of the representation theorems show that only shallow extensions are necessary, in the sense that the number of iterations needed to achieve an extension is at most three. All of these developments are made possible by taking a more liberal view of consistency: consistency is a user defined predicate, satisfying some basic properties