Universal Equivalence and Majority of Probabilistic Programs over Finite Fields

We study decidability problems for equivalence of probabilistic programs for a core probabilistic programming language over finite fields of fixed characteristic. The programming language supports uniform sampling, addition, multiplication, and conditionals and thus is sufficiently expressive to encode Boolean and arithmetic circuits. We consider two variants of equivalence: The first one considers an interpretation over the finite field F q , while the second one, which we call universal equivalence, verifies equivalence over all extensions F q k of F q . The universal variant typically arises in provable cryptography when one wishes to prove equivalence for any length of bitstrings, i.e., elements of F 2 k for any k . While the first problem is obviously decidable, we establish its exact complexity, which lies in the counting hierarchy. To show decidability and a doubly exponential upper bound of the universal variant, we rely on results from algorithmic number theory and the possibility to compare local zeta functions associated to given polynomials. We then devise a general way to draw links between the universal probabilistic problems and widely studied problems on linear recurrence sequences. Finally, we study several variants of the equivalence problem, including a problem we call majority, motivated by differential privacy. We also define and provide some insights about program indistinguishability, proving that it is decidable for programs always returning 0 or 1.

Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

Metareasoning can be achieved in probabilistic programming languages (PPLs) using agent models that recursively nest inference queries inside inference queries. However, the semantics of this powerful, reflection-like language feature has defied an operational treatment, much less reasoning principles for contextual equivalence. We give formal semantics to a core PPL with continuous distributions, scoring, general recursion, and nested queries. Unlike prior work, the presence of nested queries and general recursion makes it impossible to stratify the definition of a sampling-based operational semantics and that of a measure-theoretic semantics—the two semantics must be defined mutually recursively. A key yet challenging property we establish is that probabilistic programs have well-defined meanings: limits exist for the step-indexed measures they induce. Beyond a semantics, we offer relational reasoning principles for probabilistic programs making nested queries. We construct a step-indexed, biorthogonal logical-relations model. A soundness theorem establishes that logical relatedness implies contextual equivalence. We demonstrate the usefulness of the reasoning principles by proving novel equivalences of practical relevance—in particular, game-playing and decisionmaking agents. We mechanize our technical developments leading to the soundness proof using the Coq proof assistant. Nested queries are an important yet theoretically underdeveloped linguistic feature in PPLs; we are first to give them semantics in the presence of general recursion and to provide them with sound reasoning principles for contextual equivalence.

Semantics for variational Quantum programming

We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify how classical probabilistic effects are induced by quantum measurements, and conversely, how classical (probabilistic) programs can influence the quantum dynamics. We also describe a sound and computationally adequate denotational semantics for the language. Classical probabilistic effects are interpreted using a recently-described commutative probabilistic monad on DCPO. Quantum effects and resources are interpreted in a category of von Neumann algebras that we show is enriched over (continuous) domains. This strong sense of enrichment allows us to develop novel semantic methods that we use to interpret the relationship between the quantum and classical probabilistic effects. By doing so we provide a very detailed denotational analysis that relates domain-theoretic models of classical probabilistic programming to models of quantum programming.

Quantum Hoare Logic with Classical Variables

Hoare logic provides a syntax-oriented method to reason about program correctness and has been proven effective in the verification of classical and probabilistic programs. Existing proposals for quantum Hoare logic either lack completeness or support only quantum variables, thus limiting their capability in practical use. In this article, we propose a quantum Hoare logic for a simple while language that involves both classical and quantum variables. Its soundness and relative completeness are proven for both partial and total correctness of quantum programs written in the language. Remarkably, with novel definitions of classical-quantum states and corresponding assertions, the logic system is quite simple and similar to the traditional Hoare logic for classical programs. Furthermore, to simplify reasoning in real applications, auxiliary proof rules are provided that support standard logical operation in the classical part of assertions and super-operator application in the quantum part. Finally, a series of practical quantum algorithms, in particular the whole algorithm of Shor’s factorisation, are formally verified to show the effectiveness of the logic.