In the context of probabilistic programs, an oblivious adversary resolves nondeterminism without seeing the outcomes of random draws. Obliviousness is a common assumption in online algorithms and distributed protocols, but the complex interaction between random draws and adversarial choices makes it challenging to reason about correctness. While there has been significant progress toward reasoning about programs that combine randomization with nondeterminism, most of the work has focused on the adaptive model, whose omniscient view of program state is too powerful to establish correctness for certain classes of programs. We introduce Oblivious Probabilistic Outcome Logic (opOL), a new logic for reasoning about probabilistic programs with nondeterminism controlled by an oblivious adversary. Building on Outcome Logic and Probabilistic Separation Logic, opOL models adversarial choice as a resource and uses probabilistic independence to ensure that random outcomes are hidden from the adversary. The opOL proof system provides expressive and compositional rules for case analysis on both random and nondeterministic outcomes, and for proving almost-sure termination. Expressivity is tested through several case studies, including a paging algorithm and a leader election protocol. The opOL metatheory and case studies are mechanized in Lean 4.

We present Foxtrot, the first higher-order separation logic for proving contextual refinement of higherorder concurrent probabilistic programs with higher-order local state. From a high level, Foxtrot inherits various concurrency reasoning principles from standard concurrent separation logic, e.g. invariants and ghost resources, and supports advanced probabilistic reasoning principles for reasoning about complex probability distributions induced by concurrent threads, e.g. tape presampling and induction by error amplification. The integration of these strong reasoning principles is highly non-trivial due to the combination of probability and concurrency in the language and the complexity of the Foxtrot model; the soundness of the logic relies on a version of the axiom of choice within the Iris logic, which is not used in earlier work on Iris-based logics. We demonstrate the expressiveness of Foxtrot on a wide range of examples, including the adversarial von Neumann coin and the randombytes_uniform function of the Sodium cryptography software library. All results have been mechanized in the Rocq proof assistant and the Iris separation logic framework.

Although randomization has long been used in distributed computing, formal methods for reasoning aboutprobabilistic concurrent programs have lagged behind. No existing program logics can express specificationsabout the full distributions of outcomes resulting from programs that are both probabilistic and concurrent. To address this, we introduce Probabilistic Concurrent Outcome Logic ( pcOL ), which incorporates ideas fromconcurrent and probabilistic separation logics into Outcome Logic to introduce new compositional reasoningprinciples. At its core, pcOL reinterprets the rules of Concurrent Separation Logic in a setting where separationmodels probabilistic independence, so as to compositionally describe joint distributions over variables inconcurrent threads. Reasoning about outcomes also proves crucial, as case analysis is often necessary to deriveprecise information about threads that rely on randomized shared state. We demonstrate pcOL on a variety ofexamples, including to prove almost sure termination of unbounded loops.

Programs increasingly rely on randomization in applications such as cryptography and machine learning. Analyzing randomized programs has been a fruitful research direction, but there is a gap when programs also exploit nondeterminism(for concurrency, efficiency, or algorithmic design). In this paper, we introduce Demonic Outcome Logic for reasoning about programs that exploit both randomization and nondeterminism. The logic includes several novel features, such as reasoning about multiple executions in tandem and manipulating pre- and postconditions using familiar equational laws—including the distributive law of probabilistic choices over nondeterministic ones. We also give rules for loops that both establish termination and quantify the distribution of final outcomes from a single premise. We illustrate the reasoning capabilities of Demonic Outcome Logic through several case studies, including the Monty Hall problem, an adversarial protocol for simulating fair coins, and a heuristic based probabilistic SAT solver.

We develop a denotational model for probabilistic and concurrent imperative programs, a class of programs with standard control flow via conditionals and while-loops, as well as probabilistic actions and parallel composition. Whereas semantics for concurrent or randomized programs in isolation is well studied, their combination has not been thoroughly explored and presents unique challenges. The crux of the problem is that interactions between control flow, probabilistic actions, and concurrent execution cannot be captured by straightforward generalizations of prior work on pomsets and convex languages, prominent models for those effects, individually. Our model has good domain theoretic properties, important for semantics of unbounded loops. We also prove two adequacy theorems, showing that the model subsumes typical powerdomain semantics for concurrency and convex powerdomain semantics for probabilistic nondeterminism.