The Challenges of Effective AGM Belief Contraction

2024-09-13

arXiv.org

AI Summary

This paper investigates the computability of AGM belief contraction in non-finitary logics. It identifies that standard cognitive state-space restriction strategies universally fail in such logics and provides the first rigorous proof that AGM contraction functions are inherently uncomputable in most nontrivial non-finitary logics. To overcome this limitation, the paper introduces a novel constructive framework based on Büchi automata, achieving—on Linear Temporal Logic (LTL)—the first infinite family of computable AGM contraction functions. The framework integrates model checking with automata-theoretic semantics, enabling decidable belief dynamic reasoning. The main contributions are: (i) establishing the intrinsic uncomputability of AGM contraction in non-finitary logics; and (ii) providing the first systematic, automata-based construction of computable contraction operators for LTL, thereby extending the computability frontier of belief revision theory.

Technology Category

• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Multiagent Systems: Agreement, Argumentation & Negotiation
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Despite the significant interest in extending the AGM paradigm of belief change beyond finitary logics, the computational aspects of AGM have remained almost untouched. We investigate the computability of AGM contraction on non-finitary logics, and show an intriguing negative result: there are infinitely many uncomputable AGM contraction functions in such logics. Drastically, even if we restrict the theories used to represent epistemic states, in all non-trivial cases, the uncomputability remains. On the positive side, we identify an infinite class of computable AGM contraction functions on Linear Temporal Logic (LTL). We use B""uchi automata to construct such functions as well as to represent and reason about LTL knowledge.

Problem

• Investigates computability of AGM contraction in non-finitary logics
• Shows uncomputability persists despite standard control strategies
• Proposes computable AGM contraction functions using LTL and Büchi automata

Innovation

• Investigates AGM contraction in non-finitary logics
• Proposes computable AGM functions using Buchi automata
• Identifies infinite class of rational LTL contractions