Broadening the Applicability of Conditional Syntax Splitting for Reasoning from Conditional Belief Bases

TL;DR

This paper introduces a generalized form of conditional syntax splitting that enhances reasoning capabilities from belief bases by allowing shared atoms and nontrivial conditionals, overcoming previous limitations.

Contribution

It proposes a new notion of syntax splitting that broadens applicability and supports more complex belief base decompositions for inductive reasoning.

Findings

The new splitting notion supports shared atoms and nontrivial conditionals.

It overcomes limitations of previous splitting concepts.

Some inference operators satisfy the new postulates but not the old ones.

Abstract

In nonmonotonic reasoning from conditional belief bases, an inference operator satisfying syntax splitting postulates allows for taking only the relevant parts of a belief base into account, provided that the belief base splits into subbases based on disjoint signatures. Because such disjointness is rare in practice, safe conditional syntax splitting has been proposed as a generalization of syntax splitting, allowing the conditionals in the subbases to share some atoms. Recently this overlap of conditionals has been shown to be limited to trivial, self-fulfilling conditionals. In this article, we propose a generalization of safe conditional syntax splittings that broadens the applicability of splitting postulates. In contrast to safe conditional syntax splitting, our generalized notion supports syntax splittings of a belief base {\Delta} where the subbases of {\Delta} may share atoms and nontrivial conditionals. We illustrate how this new notion overcomes limitations of previous splitting concepts, and we identify genuine splittings, separating them from simple splittings that do not provide benefits for inductive inference from {\Delta}. We introduce adjusted inference postulates based on our generalization of conditional syntax splitting, and we evaluate several popular inductive inference operators with respect to these postulates. Furthermore, we show that, while every inductive inference operator satisfying generalized conditional syntax splitting also satisfies conditional syntax splitting, the reverse does not hold.