Reasoning about probabilistic and higher-order beliefs

RWTH Publications (RWTH Aachen)

Abstract

In Knowledge Representation and Reasoning, the notion of belief is used to model an agent's mental state about the world and other agents. Probabilistic beliefs extend this notion by capturing degrees of belief under uncertainty, while higher-order beliefs represent beliefs about the beliefs of others. These concepts play an important role in reasoning under uncertainty, multi-agent interaction, and decision-making. Developing logical frameworks that can faithfully represent such beliefs and support effective reasoning remains a fundamental challenge in the field. The first contribution of this thesis is a measure-based semantics for probabilistic beliefs. By defining probability measures over spaces of possible worlds, the proposed logics support reasoning about arbitrary belief distributions, including continuous and mixed distributions, and allow reasoning about beliefs after actions via integration over such spaces. This provides a unification of logic and probability that goes beyond discrete probabilistic models commonly studied in the literature. The second contribution concerns reasoning about higher-order beliefs in multi-agent settings, in both static and dynamic cases. For static reasoning, a translation-based inference method is developed for inference in modal logics with beliefs. For dynamic scenarios, the thesis proposes a syntactic and decidable approach to epistemic planning in the presence of higher-order beliefs. The final contribution integrates probabilistic and higher-order beliefs into a unified logical account. By extending the notion of only-knowing to probabilistic and multi-agent settings, the resulting framework supports precise reasoning about agents' knowledge bases as well as their beliefs about others' beliefs.

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