Probabilistic abstract interpretation: From trace semantics to DTMC’s and linear regression

In order to perform probabilistic program analysis we need to consider probabilistic languages or languages with a probabilistic semantics, as well as a corresponding framework for the analysis which is able to accommodate probabilistic properties and properties of probabilistic computations. To this purpose we investigate the relationship between three different types of probabilistic semantics for a core imperative language, namely Kozen’s Fixpoint Semantics, our Linear Operator Semantics and probabilistic versions of Maximal Trace Semantics. We also discuss the relationship between Probabilistic Abstract Interpretation (PAI) and statistical or linear regression analysis. While classical Abstract Interpretation, based on Galois connection, allows only for worst-case analyses, the use of the Moore-Penrose pseudo inverse in PAI opens the possibility of exploiting statistical and noisy observations in order to analyse and identify various system properties

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

Abstract Hidden Markov Models: a monadic account of quantitative information flow

Hidden Markov Models, HMM's, are mathematical models of Markov processes with state that is hidden, but from which information can leak. They are typically represented as 3-way joint-probability distributions. We use HMM's as denotations of probabilistic hidden-state sequential programs: for that, we recast them as `abstract' HMM's, computations in the Giry monad $\mathbb{D}$, and we equip them with a partial order of increasing security. However to encode the monadic type with hiding over some state $\mathcal{X}$ we use $\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ rather than the conventional $\mathcal{X}{\to}\mathbb{D}\mathcal{X}$ that suffices for Markov models whose state is not hidden. We illustrate the $\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ construction with a small Haskell prototype. We then present uncertainty measures as a generalisation of the extant diversity of probabilistic entropies, with characteristic analytic properties for them, and show how the new entropies interact with the order of increasing security. Furthermore, we give a `backwards' uncertainty-transformer semantics for HMM's that is dual to the `forwards' abstract HMM's - it is an analogue of the duality between forwards, relational semantics and backwards, predicate-transformer semantics for imperative programs with demonic choice. Finally, we argue that, from this new denotational-semantic viewpoint, one can see that the Dalenius desideratum for statistical databases is actually an issue in compositionality. We propose a means for taking it into account

Continuous probability distributions in model-based specification languages

PhD ThesisModel-based speci cation languages provide a means for obtaining assurance of dependability of complex computer-based systems, but provide little support for modelling and analysing fault behaviour, which is inherently probabilistic in nature. In particular, the need for a detailed account of the role of continuous probability has been largely overlooked. This thesis addresses the role of continuous probability in model-based speci cation languages. A model-based speci cation language (sGCL) that supports continuous probability distributions is de ned. The use of sGCL and how it interacts with engineering practices is also explored. In addition, a re nement ordering for continuous probability distributions is given, and the challenge of combining non-determinism and continuous probability is discussed in depth. The thesis is presented in three parts. The rst uses two case studies to explore the use of probability in formal methods. The rst case study, on ash memory, is used to present the capabilities of probabilistic formal methods and to determine the kinds of questions that require continuous probability distributions to answer. The second, on an emergency brake system, illustrates the strengths and weaknesses of existing languages and provides a basis for exploring a prototype language that includes continuous probability. The second part of the thesis gives the formal de nition of sGCL's syntax and semantics. The semantics is made up of two parts, the proof theory (transformer semantics) and the underpinning mathematics (relational semantics). The additional language constructs and semantical features required to include non-determinism as well as continuous probability are also discussed. The most challenging aspect lies in proving the consistency of the semantics when non-determinism is also included. The third part uses a nal case study, on an aeroplane pitch monitor, to demonstrate the use of sGCL. The new analysis techniques provided by sGCL, and how they t in with engineering practices, are explored.EPSRC: The School of Computing Science, Newcastle University: DEPLOY project

A partial order on classical and quantum states

In this work we extend the work done by Bob Coecke and Keye Martin in their paper “Partial Order on Classical States and Quantum States (2003)”. We review basic notions involving elementary domain theory, the set of probability measures on a finite set {a1, a2, ..., an}, which we identify with the standard (n-1)-simplex ∆n and Shannon Entropy. We consider partial orders on ∆n, which have the Entropy Reversal Property (ERP) : elements lower in the order have higher (Shannon) entropy or equivalently less information . The ERP property is important because of its applications in quantum information theory. We define a new partial order on ∆n, called Stochastic Order , using the well-known concept of majorization order and show that it has the ERP property and is also a continuous domain. In contrast, the bayesian order on ∆n defined by Coecke and Martin has the ERP property but is not continuous

Quantitative information flow under generic leakage functions and adaptive adversaries

We put forward a model of action-based randomization mechanisms to analyse quantitative information flow (QIF) under generic leakage functions, and under possibly adaptive adversaries. This model subsumes many of the QIF models proposed so far. Our main contributions include the following: (1) we identify mild general conditions on the leakage function under which it is possible to derive general and significant results on adaptive QIF; (2) we contrast the efficiency of adaptive and non-adaptive strategies, showing that the latter are as efficient as the former in terms of length up to an expansion factor bounded by the number of available actions; (3) we show that the maximum information leakage over strategies, given a finite time horizon, can be expressed in terms of a Bellman equation. This can be used to compute an optimal finite strategy recursively, by resorting to standard methods like backward induction.Comment: Revised and extended version of conference paper with the same title appeared in Proc. of FORTE 2014, LNC

Proof rules for the correctness of quantum programs

We apply the notion of quantum predicate proposed by D'Hondt and Panangaden to analyze a simple language fragment which may describe the quantum part of a future quantum computer in Knill's architecture. The notion of weakest liberal precondition semantics, introduced by Dijkstra for classical deterministic programs and by McIver and Morgan for probabilistic programs, is generalized to our quantum programs. To help reasoning about the correctness of quantum programs, we extend the proof rules presented by Morgan for classical probabilistic loops to quantum loops. These rules are shown to be complete in the sense that any correct assertion about the quantum loops can be proved using them. Some illustrative examples are also given to demonstrate the practicality of our proof rules. © 2007 Elsevier Ltd. All rights reserved