Confluence in Probabilistic Rewriting

Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of uniqueness of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence which is shown to imply the desired uniqueness (even for infinite sequences of reduction) and further properties. We then carry over several criteria from the classical case, such as Newman's lemma, to simplify proving confluence in concrete languages. Using these criteria, we obtain simple proofs of confluence for λ1, an affine probabilistic λ-calculus, and for Q*, a quantum programming language for which a related property has already been proven in the literature.Fil: Díaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaFil: Martínez, Guido. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentin

Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory

We describe categorical models of a circuit-based (quantum) functional programming language. We show that enriched categories play a crucial role. Following earlier work on QWire by Paykin et al., we consider both a simple first-order linear language for circuits, and a more powerful host language, such that the circuit language is embedded inside the host language. Our categorical semantics for the host language is standard, and involves cartesian closed categories and monads. We interpret the circuit language not in an ordinary category, but in a category that is enriched in the host category. We show that this structure is also related to linear/non-linear models. As an extended example, we recall an earlier result that the category of W*-algebras is dcpo-enriched, and we use this model to extend the circuit language with some recursive types

The Geometry of Concurrent Interaction: Handling Multiple Ports by Way of Multiple Tokens (Long Version)

We introduce a geometry of interaction model for Mazza's multiport interaction combinators, a graph-theoretic formalism which is able to faithfully capture concurrent computation as embodied by process algebras like the $\pi$-calculus. The introduced model is based on token machines in which not one but multiple tokens are allowed to traverse the underlying net at the same time. We prove soundness and adequacy of the introduced model. The former is proved as a simulation result between the token machines one obtains along any reduction sequence. The latter is obtained by a fine analysis of convergence, both in nets and in token machines

Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory

Inductive datatypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive datatypes may be added to the quantum programming language QPL. We construct a sound categorical model for the language and by doing so we provide the first detailed semantic treatment of user-defined inductive datatypes in quantum programming. We also show our denotational interpretation is invariant with respect to big-step reduction, thereby establishing another novel result for quantum programming. Compared to classical programming, this property is considerably more difficult to prove and we demonstrate its usefulness by showing how it immediately implies computational adequacy at all types. To further cement our results, our semantics is entirely based on a physically natural model of von Neumann algebras, which are mathematical structures used by physicists to study quantum mechanics

Decomposing Probabilistic Lambda-Calculi

International audienc

The Dynamic Geometry of Interaction Machine: A Token-Guided Graph Rewriter

In implementing evaluation strategies of the lambda-calculus, both correctness and efficiency of implementation are valid concerns. While the notion of correctness is determined by the evaluation strategy, regarding efficiency there is a larger design space that can be explored, in particular the trade-off between space versus time efficiency. Aiming at a unified framework that would enable the study of this trade-off, we introduce an abstract machine, inspired by Girard's Geometry of Interaction (GoI), a machine combining token passing and graph rewriting. We show soundness and completeness of our abstract machine, called the \emph{Dynamic GoI Machine} (DGoIM), with respect to three evaluations: call-by-need, left-to-right call-by-value, and right-to-left call-by-value. Analysing time cost of its execution classifies the machine as ``efficient'' in Accattoli's taxonomy of abstract machines.Comment: arXiv admin note: text overlap with arXiv:1802.0649

Probabilistic Rewriting: On Normalization, Termination, and Unique Normal Forms

While a mature body of work supports the study of rewriting systems, even infinitary ones, abstract tools for Probabilistic Rewriting are still limited. Here, we investigate questions such as uniqueness of the result (unique limit distribution) and we develop a set of proof techniques to analyze and compare reduction strategies. The goal is to have tools to support the operational analysis of probabilistic calculi (such as probabilistic lambda-calculi) whose evaluation is also non-deterministic, in the sense that different reductions are possible. In particular, we investigate how the behavior of different rewrite sequences starting from the same term compare w.r.t. normal forms, and propose a robust analogue of the notion of "unique normal form". Our approach is that of Abstract Rewrite Systems, i.e. we search for general properties of probabilistic rewriting, which hold independently of the specific structure of the objects.Comment: Extended version of the paper in FSCD 2019, International Conference on Formal Structures for Computation and Deductio

Game semantics for quantum programming

Quantum programming languages permit a hardware independent, high-level description of quantum algo rithms. In particular, the quantum lambda-calculus is a higher-order programming language with quantum primitives, mixing quantum data and classical control. Giving satisfactory denotational semantics to the quantum lambda-calculus is a challenging problem that has attracted significant interest in the past few years. Several models have been proposed but for those that address the whole quantum λ-calculus, they either do not represent the dynamics of computation, or they lack the compositionality one often expects from denotational models. In this paper, we give the first compositional and interactive model of the full quantum lambda-calculus, based on game semantics. To achieve this we introduce a model of quantum games and strategies, combining quantum data with a representation of the dynamics of computation inspired from causal models of concurrent systems. In this model we first give a computationally adequate interpretation of the affine fragment. Then, we extend the model with a notion of symmetry, allowing us to deal with replication. In this refined setting, we interpret and prove adequacy for the full quantum lambda-calculus. We do this both from a sequential and a parallel interpretation, the latter representing faithfully the causal independence between sub-computations

Transparent Synchronous Dataflow

Dataflow programming is a popular and convenient programming paradigm in systems modelling, optimisation, and machine learning. It has a number of advantages, for instance the lacks of control flow allows computation to be carried out in parallel as well as in distributed machines. More recently the idea of dataflow graphs has also been brought into the design of various deep learning frameworks. They facilitate an easy and efficient implementation of automatic differentiation, which is the heart of modern deep learning paradigm. [abstract abridged