compositional probabilistic verification

This paper addresses the challenge of temporal reasoning in quantitative models—specifically, computing the likelihood that program execution traces satisfy quantitative temporal properties such as probabilistic constraints, reward accumulation, or resource bounds. We propose the first unified semantic framework grounded in category theory. Our method models both systems and temporal properties as coalgebras, and introduces, for the first time, a formalized product construction via distributive laws, accompanied by sufficient conditions ensuring soundness of inference. The framework uniformly captures heterogeneous quantitative models—including probabilistic programs, weighted automata, resource-sensitive reachability, and quantitative temporal logics. Experimentally, we reproduce several classical algorithms and construct novel product instances between weighted programs and weighted temporal properties, thereby demonstrating the framework’s expressive power, scalability, and theoretical rigor.

This paper addresses the challenge of modularly reasoning about error probability bounds in higher-order concurrent probabilistic programs. To this end, it introduces Coneris—the first separation logic supporting such reasoning. Methodologically, it pioneers the concept of “randomized logical atomicity,” extending linearizability to probabilistic semantics by introducing pre-sampled traces and a probabilistic update modality to capture probabilistic state evolution at linearization points. Built upon the Rocq and Iris frameworks, Coneris integrates higher-order separation logic, probabilistic semantic modeling, and formal verification techniques. Its contributions are threefold: (1) the first modular verification framework for error probability bounds of higher-order concurrent probabilistic modules; (2) a fully mechanized metatheory and validation across multiple case studies, including large-scale systems; and (3) rigorous guarantees of correctness and composability for derived error upper bounds.

This work addresses the scalability and correctness challenges in compositional verification of stochastic automata with uncertain transition probabilities by proposing a assume-guarantee (AG) framework that supports compositional reasoning for parametric and robust stochastic automata. The framework accommodates multi-objective queries—such as probabilistic reachability and parametric expected total reward—and handles uncertainty under history-dependent semantics. Key contributions include the design of asymmetric, cyclic, and interleaved proof rules tailored to parametric automata; the introduction of specialized AG rules leveraging parameter monotonicity; the definition of strong and robust strong simulation relations; and the first extension of AG reasoning to convex uncertainty sets within history-dependent contexts. Experimental evaluation demonstrates the approach’s effectiveness across a broad range of properties while also revealing limitations concerning non-convex robust automata, memoryless semantics, and interval relaxations.

This work addresses the automatic verification of expected output bounds for probabilistic programs featuring general loops, continuous distributions, and conditional branching—where the integral semantics induced by continuous sampling impede conventional invariant-based reasoning. We propose a Riemann-sum-based approximation of the expected semantics, transforming integral bounds into quantitative invariants expressible in SMT logic. This constitutes the first systematic integration of Riemann integration into probabilistic program verification, accompanied by formal convergence guarantees for the approximation and a proof that the verification problem is coRE-complete. We implement a prototype within the Caesar verification framework, supporting intermediate-language encoding and SMT-driven inference; it successfully verifies multiple benchmarks involving continuous sampling and loops. Our approach bridges discrete program verifiers with continuous probabilistic analysis, enabling existing discrete verification tools to scale to programs with continuous distributions.

Existing program logics cannot fully characterize the output distribution of probabilistic concurrent programs, and no distribution-level formal verification methodology exists for such programs. Method: We propose the first distributional verification logic supporting programs combining probabilistic and concurrent features, systematically integrating independence, conditional distributions, and invariants into Outcome Logic, and introducing the first probabilistic concurrent separation principle. By unifying probabilistic separation logic, concurrent separation logic, and Outcome Logic, we design distribution-aware assertions, randomized resource models, and context-compositional proof rules to enable modular, compositional distributional verification. Results: Our logic is the first to precisely model independent execution, conditional dependencies, and concurrency invariants at the distribution level. It supports fully automated formal verification of representative probabilistic concurrent programs, thereby filling a fundamental theoretical gap in distributional verification of concurrent probabilistic programs.

This work addresses the formal specification and verification of probabilistic hyperproperties—such as probabilistic noninterference and perfect indistinguishability—that relate multiple execution traces through probabilistic dependencies. To this end, the paper introduces DTL, a novel probabilistic temporal logic grounded in measure decomposition theory, which for the first time incorporates measure-theoretic decomposition into temporal logic to enable reasoning about conditional probabilities over infinite execution sequences. By integrating measure theory, linear algebra, and automata theory, the authors develop a rigorous logical semantics and provide decision procedures for two decidable fragments: one supporting polynomial-time model checking and the other enabling automata-based verification of qualitative properties. Both fragments effectively verify a range of significant probabilistic hyperproperties over Markov chains.

This work addresses the challenge that existing probabilistic program verification frameworks are confined to idealized languages and cannot directly verify practical probabilistic programs written in mainstream languages such as Rust. Building upon the Verus verification tool, this paper presents the first integration of probabilistic weakest preconditions from Eris logic with Verus’s separation-logic–style reasoning, augmented by a lightweight probabilistic failure credit mechanism to extend support for probabilistic programs. The approach features a mechanized soundness proof formalized in Rocq and seamlessly combines SMT automation, separation logic, and probabilistic reasoning. Experimental evaluation demonstrates successful verification of several discrete distribution sampling algorithms—including discrete Gaussian sampling, the alias method, and fast loaded dice rollers—highlighting the framework’s applicability and effectiveness on real-world systems code.

Existing property-based testing frameworks, such as Hedgehog, lack compositional semantics, making it difficult to formally verify the correctness of generator optimizations. This work develops a formal semantic model for such frameworks, revealing that their distributional semantics are inherently non-compositional. To address this, we propose Hedgehog→, a restricted variant based on arrow calculus, which trades modest expressiveness for compositional distributional semantics. This design enables, for the first time in property-based testing, compositional formal proofs of generator equivalence. We implement a Haskell prototype of Hedgehog→ and demonstrate that it retains sufficient expressiveness to encode practical test generators while providing a rigorous, compositional foundation for reasoning about generator optimizations.

Existing approaches to verifying probabilistic programs suffer from severe scalability limitations due to the combinatorial explosion in the size of weakest pre-expectation (WPE) representations caused by loop unrolling. This work proposes a novel method based on Typed Extended Decision Diagrams (TEDDs), which, for the first time, leverages TEDDs to compactly represent WPEs. By integrating SMT-based pruning with tailored deductive proof rules, the approach enables direct symbolic reasoning at the TEDD level. This strategy dramatically compresses the logical representation, circumventing the state-space explosion inherent in conventional unrolling techniques. As a result, verification efficiency improves by several orders of magnitude, enabling the scalable and effective analysis of complex discrete probabilistic programs.