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Almost-sure termination is an important correctness property for probabilistic programs, and a number of program logics have been developed for establishing it. However, these logics have mostly been developed for first-order programs written in languages with specific syntactic patterns for looping. In this paper, we consider almost-sure termination for higher-order probabilistic programs with general references. This combination of features allows for recursion and looping to be encoded through a variety of patterns. Therefore, rather than developing proof rules for reasoning about particular recursion patterns, we instead propose an approach based on proving refinement between a higher-order program and a simpler probabilistic model, in such a way that the refinement preserves termination behavior. By proving a refinement, almost-sure termination behavior of the program can then be established by analyzing the simpler model. We present this approach in the form of Caliper, a higher-order separation logic for proving termination-preserving refinements. Caliper uses probabilistic couplings to carry out relational reasoning between a program and a model. To handle the range of recursion patterns found in higher-order programs, Caliper uses guarded recursion, in particular the principle of Löb induction. A technical novelty is that Caliper does not require the use of transfinite step indexing or other technical restrictions found in prior work on guarded recursion for termination-preservation refinement. We demonstrate the flexibility of this approach by proving almost-sure termination of several examples, including first-order loop constructs, a random list generator, treaps, and a sampler for Galton-Watson trees that uses higher-order store. All the results have been mechanized in the Coq proof assistant. 

Probabilistic programs often trade accuracy for efficiency, and thus may, with a small probability, return an incorrect result. It is important to obtain precise bounds for the probability of these errors, but existing verification approaches have limitations that lead to error probability bounds that are excessively coarse, or only apply to first-order programs. In this paper we present Eris, a higher-order separation logic for proving error probability bounds for probabilistic programs written in an expressive higher-order language. Our key novelty is the introduction of error credits, a separation logic resource that tracks an upper bound on the probability that a program returns an erroneous result. By representing error bounds as a resource, we recover the benefits of separation logic, including compositionality, modularity, and dependency between errors and program terms, allowing for more precise specifications. Moreover, we enable novel reasoning principles such as expectation-preserving error composition, amortized error reasoning, and error induction. We illustrate the advantages of our approach by proving amortized error bounds on a range of examples, including collision probabilities in hash functions, which allow us to write more modular specifications for data structures that use them as clients. We also use our logic to prove correctness and almost-sure termination of rejection sampling algorithms. All of our results have been mechanized in the Coq proof assistant using the Iris separation logic framework and the Coquelicot real analysis library. 

Probabilistic couplings are the foundation for many probabilistic relational program logics and arise when relating random sampling statements across two programs. In relational program logics, this manifests as dedicated coupling rules that, e.g., say we may reason as if two sampling statements return the same value. However, this approach fundamentally requires aligning or synchronizing the sampling statements of the two programs which is not always possible.

In this paper, we develop Clutch, a higher-order probabilistic relational separation logic that addresses this issue by supporting asynchronous probabilistic couplings. We use Clutch to develop a logical step-indexed logical relation to reason about contextual refinement and equivalence of higher-order programs written in a rich language with a probabilistic choice operator, higher-order local state, and impredicative polymorphism. Finally, we demonstrate our approach on a number of case studies.

All the results that appear in the paper have been formalized in the Coq proof assistant using the Coquelicot library and the Iris separation logic framework. 

K2 is a new architecture and verification approach for hardware security modules (HSMs). The K2 architecture's rigid separation between I/O, storage, and computation over secret state enables modular proofs and allows for software development and verification independent of hardware development and verification while still providing correctness and security guarantees about the composed system. For a key step of verification, K2 introduces a new tool called Chroniton that automatically proves timing properties of software running on a particular hardware implementation, ensuring the lack of timing side channels at a cycle-accurate level. 

Grove is a concurrent separation logic library for verifying distributed systems. Grove is the first to handle time-based leases, including their interaction with reconfiguration, crash recovery, thread-level concurrency, and unreliable networks. This paper uses Grove to verify several distributed system components written in Go, including GroveKV, a realistic distributed multi-threaded key-value store. GroveKV supports reconfiguration, primary/backup replication, and crash recovery, and uses leases to execute read-only requests on any replica. GroveKV achieves high performance (67-73% of Redis on a single core), scales with more cores and more backup replicas (achieving about 2× the throughput when going from 1 to 3 servers), and can safely execute reads while reconfiguring. 

This paper presents ProbCompCert, a compiler for a subset of the Stan probabilistic programming language (PPL), in which several key compiler passes have been formally verified using the Coq proof assistant. Because of the probabilistic nature of PPLs, bugs in their compilers can be difficult to detect and fix, making verification an interesting possibility. However, proving correctness of PPL compilation requires new techniques because certain transformations performed by compilers for PPLs are quite different from other kinds of languages. This paper describes techniques for verifying such transformations and their application in ProbCompCert. In the course of verifying ProbCompCert, we found an error in the Stan language reference manual related to the semantics and implementation of a key language construct. 

Multi-version concurrency control (MVCC) is a widely used, sophisticated approach for handling concurrent transactions. vMVCC is the first MVCC-based transaction library that comes with a machine-checked proof of correctness, providing clients with a guarantee that it will correctly handle all transactions despite a complicated design and implementation that might otherwise be error-prone. vMVCC is implemented in Go, stores data in memory, and uses several optimizations, such as RDTSC-based timestamps, to achieve high performance (25–96% the throughput of Silo, a state-of-the-art in-memory database, for YCSB and TPC-C workloads). Formally specifying and verifying vMVCC required adopting advanced proof techniques, such as logical atomicity and prophecy variables, owing to the fact that MVCC transactions can linearize at timestamp generation prior to transaction execution. 

This paper develops a new approach to verifying a performant file system that isolates crash safety and concurrency reasoning to a transaction system that gives atomic access to the disk, so that the rest of the file system can be verified with sequential reasoning. We demonstrate this approach in DaisyNFS, a Network File System (NFS) server written in Go that runs on top of a disk. DaisyNFS uses GoTxn, a new verified, concurrent transaction system that extends GoJournal with two-phase locking and an allocator. The transaction system's specification formalizes under what conditions transactions can be verified with only sequential reasoning, and comes with a mechanized proof in Coq that connects the specification to the implementation. As evidence that proofs enjoy sequential reasoning, DaisyNFS uses Dafny, a sequential verification language, to implement and verify all the NFS operations on top of GoTxn. The sequential proofs helped achieve a number of good properties in DaisyNFS: easy incremental development (for example, adding support for large files), a relatively short proof (only 2x as many lines of proof as code), and a performant implementation (at least 60% the throughput of the Linux NFS server exporting ext4 across a variety of benchmarks).