observational equivalence

This work addresses the problem of characterizing behavioral equivalence for time-varying stateful processes based solely on finite input-output observations, without relying on inaccessible internal states. To this end, it introduces a novel structure called the “discard double category,” which provides a unified framework for modeling partial, nondeterministic, probabilistic, and quantum processes, and constructs a functorial semantics into the free feedback category. By leveraging preordered enriched monoidal categories and closed relations over compact Hausdorff spaces, the paper establishes a categorical compactness theorem. This framework not only subsumes Willems’ behavioral theory of linear time-invariant systems as a special case but also offers a unified, internal-state-free foundation for behavioral semantics across a broad spectrum of process types.

Quantifying behavioral similarity for continuous-time systems—including purely continuous, purely jump, and hybrid dynamical systems—remains challenging due to the lack of appropriate behavioral metrics. Method: This paper introduces the first extension of discrete-time bisimulation metrics to continuous time, establishing a unified behavioral pseudometric framework. The metric is defined equivalently via a fixed-point equation and real-valued modal logic, integrating Lipschitz functional analysis with semantics from continuous-time Markov processes. Contribution/Results: Theoretical analysis confirms its applicability to Brownian motion, Poisson jump processes, and hybrid diffusion-jump systems. As the first general-purpose continuous-time behavioral metric supporting quantitative behavioral comparison and approximate verification, it overcomes limitations of prior discrete-time or single-dynamics approaches. This work lays a foundational basis for formal verification and robustness analysis of stochastic systems.

This paper addresses the quantification of behavioral similarity between states in continuous-time Markov processes—particularly diffusion processes. To this end, it introduces, for the first time, a second class of behavioral pseudometrics based on trajectory-level semantics, complementing the existing first class grounded in time-indexed Markov kernels. The two classes are rigorously constructed via functional iteration and real-valued logical distance, respectively, and unified within a fixed-point theoretical framework. Theoretical analysis establishes that both pseudometrics are functionally equivalent and logically expressible, and that each converges to behavioral equivalence. This work not only extends the theoretical foundations of behavioral metrics for continuous-time stochastic systems but also provides the first trajectory-driven characterization of behavioral similarity for diffusion processes, thereby bridging the long-standing theoretical gap between kernel-based and path-based approaches.

This paper addresses the challenges of fragility analysis and poor constructivity in observational equivalence proofs. We propose a stepwise reasoning method based on hypergraph rewriting. Our key contributions are threefold: (1) We formally characterize robustness as a critical sufficient condition for observational equivalence—a novel formulation; (2) We establish a neighborhood-based local reasoning framework that supports generalized observational equivalence definitions and verification under syntactically restricted contexts and quantitative step bounds; (3) Leveraging a hypergraph rewriting abstract machine inspired by geometric interaction, we structurally model function abstraction and application in higher-order stateful lambda calculus. Experimental evaluation on call-by-value lambda calculus demonstrates that our approach significantly improves modularity, constructivity, and analyzability of fragility in observational equivalence proofs.

Formalizing open stochastic systems under combined uncertainty from incomplete information and probabilistic perturbations remains challenging within categorical frameworks. Method: We introduce *copartiality* to model uncertainty propagation, categorify Willems-style open stochastic systems for the first time, define *extended Gaussian distributions*—a unification of Gaussian probability measures and linear relations—and construct the *category of extended Gaussian maps* to support compositional modeling and semantic consistency verification. Contribution: This work bridges categorical probability theory and control theory (e.g., signal-flow graphs), rigorously formalizes physical noise constraints and Bayesian noninformative priors, and establishes a novel structural paradigm for modeling and analyzing open linear stochastic systems. It provides a unified categorical foundation wherein stochasticity, partiality, and linearity are coherently integrated, enabling principled compositionality, abstraction, and verification in uncertain dynamical systems.

Program analyses often lack robustness in the face of code changes. This work introduces, for the first time, a unified framework grounded in category theory that formalizes programs and their properties as categorical objects, capturing various forms of robustness—such as variable renaming and semantic refinement—via structure-preserving functors. Two implementation pathways are proposed: one lifts constructions from restricted computational models to general-purpose programs, while the other ensures stability in the composition of robust operators within algebraic program analyses. The framework not only uncovers common principles underlying loop summarization and termination analysis but also provides a theoretical foundation and predictability guarantees for developing program analyses that are more resilient to program transformations.

This study investigates the descriptive complexity of bisimilarity for nondeterministic labelled Markov processes with finite-support measure assignments over uncountable standard Borel spaces. Employing tools from descriptive set theory—specifically the theories of Borel and analytic sets together with reduction techniques—it establishes, for the first time, that bisimilarity for well-founded processes is Borel definable. Moreover, by reducing the tail equivalence relation $E_0$ on binary sequences to this bisimulation problem, the work provides a sharp lower bound on its complexity. The results demonstrate that bisimilarity in the well-founded case resides precisely within the Borel hierarchy and further show that the countable fragment of basic modal logic cannot fully characterize bisimilarity even for low-rank processes, thereby strictly limiting the expressive power of modal logic for capturing such behavioral equivalences.

This work aims to unify the characterization of behavioral equivalences across the linear-time–branching-time spectrum in labeled transition systems. By leveraging topos theory, it interprets behavioral equivalence as localization and, for the first time, establishes a semantic foundation for the process algebraic equivalence spectrum within geometric logic by integrating Grothendieck topologies with an energy-game framework. The main contributions include a geometric closure theorem revealing that the equivalence spectrum forms a bi-Heyting algebra, and the construction of a 30-element closure lattice \( L_{30} \) encompassing 13 classical and 17 novel hybrid equivalences. All results are constructively proved and formally verified in Lean 4/Mathlib.

This work addresses the challenge of uniformly characterizing both qualitative behaviors and quantitative discrepancies—such as accumulated rewards—in system equivalence analysis. It proposes a unified framework grounded in fibrational coalgebra, which for the first time incorporates reward sensitivity into coalgebraic bisimulation. By leveraging categorical gluing techniques, the approach seamlessly integrates graded (metric) and ungraded (relational) bisimulations. The framework demonstrates broad applicability across diverse system models, subsuming relational bisimulation for reward-augmented automata and metric bisimulation for labeled Markov processes, thereby validating its generality and expressiveness.

This work addresses a critical limitation in existing methods for verifying quantum protocols, which erroneously distinguish physically equivalent processes due to the introduction of probabilistic nondeterminism inconsistent with physical observations. To resolve this, the paper proposes lqCCS, a quantum concurrent process calculus that eliminates unphysical nondeterminism while preserving the expressiveness required to model real-world protocols. This is achieved through a semantics based on quantum generalized probability distributions, constraints imposed by physically admissible schedulers—a novel integration into quantum process calculi—and a saturation-based labeled bisimulation. The framework establishes behavioral equivalence as a congruence with respect to parallel composition and proves the adequacy of the bisimulation under mixtures of indistinguishable quantum states. It enables compositional reasoning and successfully supports equivalence checking for a broad class of processes and formal verification of multiple realistic quantum protocols.