Mayuko Kori

Mayuko Kori, Kazuki Watanabe

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.

Mayuko Kori

Non-wellfounded proof systems impose a global condition called the global trace condition (GTC) on a derivation tree to ensure soundness. Providing a categorical characterisation of the GTC that guarantees soundness remains challenging due to the global, non-compositional nature of these conditions and the infinitary structure of non-wellfounded proofs. We develop a coalgebraic framework for non-wellfounded proof systems where derivation trees are modelled as coalgebras of generalised polynomial functors on presheaves. Since the GTC is a constraint on infinite paths in derivation graphs, we employ graphs of coalgebras and formulate the GTC coalgebraically as a condition on these graphs. Soundness is then formulated as the existence of a unique coalgebra-to-algebra morphism from a coalgebra representing a derivation graph to an algebra specifying semantics. Within this framework, we characterise the GTC via recursive coalgebras: a coalgebra satisfies the GTC if and only if its image under a suitable adjoint is recursive. Under an appropriate assumption on the given semantic algebra, this yields soundness, that is, every proof admits a unique coalgebra-to-algebra morphism. We demonstrate our framework through a non-wellfounded proof system for the modal mu-calculus, one for higher-order fixed-point logics, and a non-wellfounded variant of Santocanale's circular proof system in mu-bicomplete categories.

Pedro H. Azevedo de Amorim, Mayuko Kori, Koko Muroya

In this paper we present a framework for modelling \emph{reward-sensitive bisimulations}, that is, bisimulations that account for quantitative differences such as accumulated rewards. To capture both qualitative and quantitative aspects uniformly, we consider two interacting notions of bisimulation: a graded variant that tracks bounded reward differences, and an ungraded one that abstracts from them. Our characterization of these notions is done in the fibrational and coalgebraic approach to (bi)simulation initiated by Hermida and Jacobs. To formally relate the graded and ungraded notions, we deploy categorical gluing, a standard technique in categorical logic. Furthermore, we show that this construction interacts well with standard coalgebra concepts, such as final coalgebras, and that it yields a unified characterization in terms of combined notions of bisimulations under mild assumptions. In order to demonstrate the versatility of our approach, we show how it encompasses various bisimulation notions for different kinds of systems, including relation-based bisimulations for automata with rewards and metric-based notions of bisimulations for labelled Markov processes.

Mayuko Kori, Ichiro Hasuo, Shin-ya Katsumata

The coincidence between initial algebras (IAs) and final coalgebras (FCs) is a phenomenon that underpins various important results in theoretical computer science. In this paper, we identify a general fibrational condition for the IA-FC coincidence, namely in the fiber over an initial algebra in the base category. Identifying (co)algebras in a fiber as (co)inductive predicates, our fibrational IA-FC coincidence allows one to use coinductive witnesses (such as invariants) for verifying inductive properties (such as liveness). Our general fibrational theory features the technical condition of stability of chain colimits; we extend the framework to the presence of a monadic effect, too, restricting to fibrations of complete lattice-valued predicates. Practical benefits of our categorical theory are exemplified by new "upside-down" witness notions for three verification problems: probabilistic liveness, and acceptance and model-checking with respect to bottom-up tree automata.

Mayuko Kori, Takeshi Tsukada, Naoki Kobayashi

A cyclic proof system allows us to perform inductive reasoning without explicit inductions. We propose a cyclic proof system for HFLN, which is a higher-order predicate logic with natural numbers and alternating fixed-points. Ours is the first cyclic proof system for a higher-order logic, to our knowledge. Due to the presence of higher-order predicates and alternating fixed-points, our cyclic proof system requires a more delicate global condition on cyclic proofs than the original system of Brotherston and Simpson. We prove the decidability of checking the global condition and soundness of this system, and also prove a restricted form of standard completeness for an infinitary variant of our cyclic proof system. A potential application of our cyclic proof system is semi-automated verification of higher-order programs, based on Kobayashi et al.'s recent work on reductions from program verification to HFLN validity checking.