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We propose a symbolic execution method for programs that can draw random samples. In contrast to existing work, our method can verify randomized programs with unknown inputs and can prove probabilistic properties that universally quantify over all possible inputs. Our technique augments standard symbolic execution with a new class of probabilistic symbolic variables , which represent the results of random draws, and computes symbolic expressions representing the probability of taking individual paths. We implement our method on top of the KLEE symbolic execution engine alongside multiple optimizations and use it to prove properties about probabilities and expected values for a range of challenging case studies written in C++, including Freivalds’ algorithm, randomized quicksort, and a randomized property-testing algorithm for monotonicity. We evaluate our method against Psi, an exact probabilistic symbolic inference engine, and Storm, a probabilistic model checker, and show that our method significantly outperforms both tools. 

Modern programmable network switches can implement cus- tom applications using efficient packet processing hardware, and the programming language P4 provides high-level con- structs to program such switches. The increase in speed and programmability has inspired research in dataplane program- ming, where many complex functionalities, e.g., key-value stores and load balancers, can be implemented entirely in network switches. However, dataplane programs may suffer from novel security errors that are not traditionally found in network switches. To address this issue, we present a new information-flow control type system for P4. We formalize our type system in a recently-proposed core version of P4, and we prove a sound- ness theorem: well-typed programs satisfy non-interference. We also implement our type system in a tool, P4BID, which extends the type checker in the p4c compiler, the reference compiler for the latest version of P4. We present several case studies showing that natural security, integrity, and isolation properties in networks can be captured by non-interference, and our type system can detect violations of these properties while certifying correct programs. 

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. 

Differential privacy is a formal, mathematical def- inition of data privacy that has gained traction in academia, industry, and government. The task of correctly constructing differentially private algorithms is non-trivial, and mistakes have been made in foundational algorithms. Currently, there is no automated support for converting an existing, non-private program into a differentially private version. In this paper, we propose a technique for automatically learning an accurate and differentially private version of a given non-private program. We show how to solve this difficult program synthesis problem via a combination of techniques: carefully picking representative example inputs, reducing the problem to continuous optimization, and mapping the results back to symbolic expressions. We demonstrate that our approach is able to learn foundational al- gorithms from the differential privacy literature and significantly outperforms natural program synthesis baselines. 

Sensitivity properties describe how changes to the input of a program affect the output, typically by upper bounding the distance between the outputs of two runs by a monotone function of the distance between the corresponding inputs. When programs are probabilistic, the distance between outputs is a distance between distributions. The Kantorovich lifting provides a general way of defining a distance between distributions by lifting the distance of the underlying sample space; by choosing an appropriate distance on the base space, one can recover other usual probabilistic distances, such as the Total Variation distance. We develop a relational pre-expectation calculus to upper bound the Kantorovich distance between two executions of a probabilistic program. We illustrate our methods by proving algorithmic stability of a machine learning algorithm, convergence of a reinforcement learning algorithm, and fast mixing for card shuffling algorithms. We also consider some extensions: using our calculus to show convergence of Markov chains to the uniform distribution over states and an asynchronous extension to reason about pairs of program executions with different control flow.