[Paper Review] A Unified Framework for Nonmonotonic Reasoning with Vagueness and Uncertainty

Sandip Paul, Kumar S. Ray+1 more|arXiv (Cornell University)|Oct 1, 2019

Logic, Reasoning, and Knowledge55 references4 citations

TL;DR

This paper proposes a unified framework for nonmonotonic reasoning with vagueness and uncertainty using interval-valued fuzzy sets as truth values, ordered via a preorder-based triangle to enhance knowledge comparability. It introduces an iterative answer set computation method with convergence guarantees through difference equations, enabling intuitive reasoning with prioritized and exception-handling rules under both fuzziness and uncertainty.

ABSTRACT

An interval-valued fuzzy answer set programming paradigm is proposed for nonmonotonic reasoning with vague and uncertain information. The set of sub-intervals of $[0,1]$ is considered as truth-space. The intervals are ordered using preorder-based truth and knowledge ordering. The preorder based ordering is an enhanced version of bilattice-based ordering. The system can represent and reason with prioritized rules, rules with exceptions. An iterative method for answer set computation is proposed. The sufficient conditions for termination of iterations are identified for a class of logic programs using the notion of difference equations.

Motivation & Objective

• To address the limitations of existing fuzzy, possibilistic, and probabilistic logic programming frameworks in handling both vagueness and uncertainty simultaneously.
• To overcome the shortcomings of bilattice-based knowledge ordering in nonmonotonic reasoning, especially regarding incomparable truth values under incomplete information.
• To develop a unified semantics for logic programs that supports classical negation, negation-as-failure, and weighted rules with interval-valued truth degrees.
• To provide a mathematically grounded iterative method for answer set computation with sufficient conditions for convergence.
• To enable more intuitive and robust reasoning in applications like medical decision support systems, where both imprecision and uncertainty are inherent.

Proposed method

• The truth space is defined as the set of all sub-intervals of [0,1], representing both truth degree and certainty level.
• A preorder-based triangle replaces the traditional bilattice-based ordering to allow meaningful comparison of intervals even when one is not a subset of the other.
• Weighted rules are introduced, where rule weights are intervals indicating degrees of uncertainty, enabling modeling of exceptions and prioritized reasoning.
• A knowledge aggregation operator is used to combine positive and negative evidence, improving the intuition of nonmonotonic inference.
• The answer set computation is performed iteratively, inspired by classical ASP three-stage computation, but adapted for real-valued truth degrees.
• The convergence of iterations is analyzed using difference equations derived from value-propagation-graph representations, with sufficient conditions for termination established via the Contraction Mapping Theorem.

Experimental results

Research questions

• RQ1 How can a unified logic programming framework effectively model both vagueness and uncertainty in nonmonotonic reasoning?
• RQ2 What is a more intuitive and robust ordering of interval-valued truth degrees that supports knowledge comparison in incomplete information scenarios?
• RQ3 How can iterative answer set computation be guaranteed to terminate for a broad class of logic programs with real-valued truth degrees?
• RQ4 In what way does the proposed preorder-based triangle improve upon the bilattice-based triangle in handling nonmonotonicity and belief revision?
• RQ5 How can weighted rules with interval-valued weights support reasoning with exceptions and prioritized knowledge in a nonmonotonic setting?

Key findings

• The proposed preorder-based triangle enables meaningful comparison of interval-valued truth values even when they are not subsets of each other, resolving a key limitation of bilattice-based ordering.
• The framework supports both classical negation and negation-as-failure, enabling richer nonmonotonic reasoning than prior FASP or PFASP approaches.
• The iterative computation method for answer sets is mathematically analyzed using difference equations derived from value-propagation-graphs, providing a novel convergence analysis method.
• Sufficient conditions for convergence of the iterative process are formally identified, ensuring termination for a broad class of programs under the Contraction Mapping Theorem.
• The framework allows intuitive modeling of uncertain and vague medical knowledge, such as linguistic rules and probabilistic diagnoses, making it suitable for clinical decision support systems.
• The integration of a knowledge aggregation operator enhances the handling of conflicting evidence, improving the realism and robustness of nonmonotonic inference.