From Coalgebraic Determinization to Belief Construction for Partial Observability

2026-04-28

AI Summary

This work investigates how to transform partially observable systems into fully observable ones while preserving semantic equivalence. To this end, the authors develop a coalgebraic framework that lifts monads to slice categories and introduces a belief decomposition mechanism, thereby unifying belief-state construction and determinization within coalgebra theory for the first time. The approach is extended to weighted transition systems equipped with the multiset monad. The resulting belief coalgebra is shown to be semantically equivalent to the original system and, under certain conditions, coincides with a fully observable belief system. This not only recovers the classical equivalence between POMDPs and belief MDPs but also establishes novel semantic equivalences for weighted transition systems.

Abstract

The belief construction is a fundamental technique for transforming partially observable systems to fully observable ones while preserving the relevant semantics. It plays a central role in the analysis of partially observable systems, in particular partially observable Markov decision processes (POMDPs), which is a central model in artificial intelligence and formal verification. In this paper, we develop a coalgebraic framework for the belief construction. To handle observations categorically, we lift a monad to slice categories and introduce a belief decomposition that reorganizes states according to their observations. This allows us to introduce a coalgebraic generalization of the belief construction, obtained by combining the belief decomposition with the coalgebraic determinization of Silva, Bonchi, Bonsangue, and Rutten. In this framework, we show that the semantics of a partially observable system coincides with that of the corresponding belief coalgebra. We then study when the latter further agrees with the semantics of its fully observable counterpart, and use this to identify conditions under which the semantics of a partially observable system coincides with that of the corresponding fully observable belief system. As consequences, we recover the standard equivalence between POMDPs and belief MDPs, and obtain a new equivalence result for weighted transition systems with the multiset monad.

Problem

• partial observability
• belief construction
• coalgebra
• POMDP
• semantic equivalence

Innovation

• coalgebraic belief construction
• partial observability
• belief decomposition
• monad lifting
• semantic equivalence