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The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in discrete probability distributions and in relational databases, using a probabilistic DIBI model and a similarly-constructed relational model. Despite the similarity of the two models, there lacks a uniform account. As a result, the laborious case-by-case verification of the frame conditions required for constructing new models hinders them from generalising the results to CI in other useful models such that continuous distribution. In this paper, we develop an abstract framework for systematically constructing DIBI models, using category theory as the unifying mathematical language. We show that DIBI models arise from arbitrary symmetric monoidal categories with copy-discard structure. In particular, we use string diagrams - a graphical presentation of monoidal categories - to give a uniform definition of the parallel composition and subkernel relation in DIBI models. Our approach not only generalises known models, but also yields new models of interest and reduces properties of DIBI models to structures in the underlying categories. Furthermore, our categorical framework enables a comparison between string diagrammatic approaches to CI in the literature and a logical notion of CI, defined in terms of the satisfaction of specific DIBI formulas. We show that the logical notion is an extension of string diagrammatic CI under reasonable conditions. 

ProbNV is a new framework for probabilistic network control plane verification that strikes a balance between generality and scalability. ProbNV is general enough to encode a wide range of features from the most common protocols (eBGP and OSPF) and yet scalable enough to handle challenging properties, such as probabilistic all-failures analysis of medium-sized networks with 100-200 devices. When there are a small, bounded number of failures, networks with up to 500 devices may be verified in seconds. ProbNV operates by translating raw CISCO configurations into a probabilistic and functional programming language designed for network verification. This language comes equipped with a novel type system that characterizes the sort of representation to be used for each data structure: concrete for the usual representation of values; symbolic for a BDD-based representation of sets of values; and multi-value for an MTBDD-based representation of values that depend upon symbolics. Careful use of these varying representations speeds execution of symbolic simulation of network models. The MTBDD-based representations are also used to calculate probabilistic properties of network models once symbolic simulation is complete. We implement the language and evaluate its performance on benchmarks constructed from real network topologies and synthesized routing policies. 

This paper presents McNetKAT, a scalable tool for verifying probabilistic network programs. McNetKAT is based on a new semantics for the guarded and history-free fragment of Probabilistic NetKAT in terms of finite-state, absorbing Markov chains. This view allows the semantics of all programs to be computed exactly, enabling construction of an automatic verification tool. Domain-specific optimizations and a parallelizing backend enable McNetKAT to analyze networks with thousands of nodes, automatically reasoning about general properties such as probabilistic program equivalence and refinement, as well as networking properties such as resilience to failures. We evaluate McNetKAT's scalability using real-world topologies, compare its performance against state-of-the-art tools, and develop an extended case study on a recently proposed data center network design.