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Kleene algebra with tests (KAT) is a foundational equational framework for reasoning about programs, which has found applications in program transformations, networking and compiler optimizations, among many other areas. In his seminal work, Kozen proved that KAT subsumes propositional Hoare logic, showing that one can reason about the (partial) correctness of while programs by means of the equational theory of KAT. In this work, we investigate the support that KAT provides for reasoning about incorrectness, instead, as embodied by O'Hearn's recently proposed incorrectness logic. We show that KAT cannot directly express incorrectness logic. The main reason for this limitation can be traced to the fact that KAT cannot express explicitly the notion of codomain, which is essential to express incorrectness triples. To address this issue, we study Kleene Algebra with Top and Tests (TopKAT), an extension of KAT with a top element. We show that TopKAT is powerful enough to express a codomain operation, to express incorrectness triples, and to prove all the rules of incorrectness logic sound. This shows that one can reason about the incorrectness of while-like programs by means of the equational theory of TopKAT. 

Formal reasoning about hashing-based probabilistic data structures often requires reasoning about random variables where when one variable gets larger (such as the number of elements hashed into one bucket), the others tend to be smaller (like the number of elements hashed into the other buckets). This is an example of negative dependence , a generalization of probabilistic independence that has recently found interesting applications in algorithm design and machine learning. Despite the usefulness of negative dependence for the analyses of probabilistic data structures, existing verification methods cannot establish this property for randomized programs. To fill this gap, we design LINA, a probabilistic separation logic for reasoning about negative dependence. Following recent works on probabilistic separation logic using separating conjunction to reason about the probabilistic independence of random variables, we use separating conjunction to reason about negative dependence. Our assertion logic features two separating conjunctions, one for independence and one for negative dependence. We generalize the logic of bunched implications (BI) to support multiple separating conjunctions, and provide a sound and complete proof system. Notably, the semantics for separating conjunction relies on a non-deterministic , rather than partial, operation for combining resources. By drawing on closure properties for negative dependence, our program logic supports a Frame-like rule for negative dependence and monotone operations. We demonstrate how LINA can verify probabilistic properties of hash-based data structures and balls-into-bins processes. 

Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed λ-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples. 

Differential privacy offers a formal framework for reasoning about the privacy and accuracy of computations on private data. It also offers a rich set of building blocks for constructing private data analyses. When carefully calibrated, these analyses simultaneously guarantee the privacy of the individuals contributing their data, and the accuracy of the data analysis results, inferring useful properties about the population. The compositional nature of differential privacy has motivated the design and implementation of several programming languages to ease the implementation of differentially private analyses. Even though these programming languages provide support for reasoning about privacy, most of them disregard reasoning about the accuracy of data analyses. To overcome this limitation, we present DPella, a programming framework providing data analysts with support for reasoning about privacy, accuracy, and their trade-offs. The distinguishing feature of DPella is a novel component that statically tracks the accuracy of different data analyses. To provide tight accuracy estimations, this component leverages taint analysis for automatically inferring statistical independence of the different noise quantities added for guaranteeing privacy. We evaluate our approach by implementing several classical queries from the literature and showing how data analysts can calibrate the privacy parameters to meet the accuracy requirements, and vice versa. 

This paper presents λ-amor, a new type-theoretic framework for amortized cost analysis of higher-order functional programs and shows that existing type systems for cost analysis can be embedded in it. λ-amor introduces a new modal type for representing potentials – costs that have been accounted for, but not yet incurred, which are central to amortized analysis. Additionally, λ-amor relies on standard type-theoretic concepts like affineness, refinement types and an indexed cost monad. λ-amor is proved sound using a rather simple logical relation. We embed two existing type systems for cost analysis in λ-amor showing that, despite its simplicity, λ-amor can simulate cost analysis for different evaluation strategies (call-by-name and call-by-value), in different styles (effect-based and coeffect-based), and with or without amortization. One of the embeddings also implies that λ-amor is relatively complete for all terminating PCF programs.