Effective AGM Belief Contraction: A Journey beyond the Finitary Realm (Technical Report)

TL;DR

This paper explores the computational limits of AGM belief contraction in non-finitary logics, revealing inherent uncomputability issues and proposing new methods to achieve computability, exemplified through Linear Temporal Logic.

Contribution

It demonstrates the uncomputability of AGM contraction in non-finitary logics and introduces automata-based approaches to ensure computability in LTL.

Findings

Uncomputability of AGM contraction functions in non-finitary logics.

Standard restrictions on epistemic states do not eliminate uncomputability.

Automata-based methods can produce computable contraction functions in LTL.

Abstract

Despite significant efforts towards 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, we also show that the current de facto standard strategies to control computability, which rely on restricting the space of epistemic states, fail: uncomputability remains in all non-finitary cases. Motivated by this disruptive result, we propose new approaches to controlling computability beyond the finitary realm. Using Linear Temporal Logic (LTL) as a case study, we identify an infinite class of fully-rational AGM contraction functions that are computable by design. We use B\"uchi automata to construct such functions, and to represent and reason about LTL beliefs.