Multi-level Contextual Type Theory

Contextual type theory distinguishes between bound variables and meta-variables to write potentially incomplete terms in the presence of binders. It has found good use as a framework for concise explanations of higher-order unification, characterize holes in proofs, and in developing a foundation for programming with higher-order abstract syntax, as embodied by the programming and reasoning environment Beluga. However, to reason about these applications, we need to introduce meta^2-variables to characterize the dependency on meta-variables and bound variables. In other words, we must go beyond a two-level system granting only bound variables and meta-variables. In this paper we generalize contextual type theory to n levels for arbitrary n, so as to obtain a formal system offering bound variables, meta-variables and so on all the way to meta^n-variables. We obtain a uniform account by collapsing all these different kinds of variables into a single notion of variabe indexed by some level k. We give a decidable bi-directional type system which characterizes beta-eta-normal forms together with a generalized substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

Control Flow Analysis for SF Combinator Calculus

Programs that transform other programs often require access to the internal structure of the program to be transformed. This is at odds with the usual extensional view of functional programming, as embodied by the lambda calculus and SK combinator calculus. The recently-developed SF combinator calculus offers an alternative, intensional model of computation that may serve as a foundation for developing principled languages in which to express intensional computation, including program transformation. Until now there have been no static analyses for reasoning about or verifying programs written in SF-calculus. We take the first step towards remedying this by developing a formulation of the popular control flow analysis 0CFA for SK-calculus and extending it to support SF-calculus. We prove its correctness and demonstrate that the analysis is invariant under the usual translation from SK-calculus into SF-calculus.Comment: In Proceedings VPT 2015, arXiv:1512.0221

Refining a Bayesian network using a chain event graph

The search for a useful explanatory model based on a Bayesian Network (BN) now has a long and successful history. However, when the dependence structure between the variables of the problem is asymmetric then this cannot be captured by the BN. The Chain Event Graph (CEG) provides a richer class of models which incorporates these types of dependence structures as well as retaining the property that conclusions can be easily read back to the client. We demonstrate on a real health study how the CEG leads us to promising higher scoring models and further enables us to make more refined conclusions than can be made from the BN. Further we show how these graphs can express causal hypotheses about possible interventions that could be enforced

Information flow analysis for a dynamically typed language with staged metaprogramming

Web applications written in JavaScript are regularly used for dealing with sensitive or personal data. Consequently, reasoning about their security properties has become an important problem, which is made very difficult by the highly dynamic nature of the language, particularly its support for runtime code generation via eval. In order to deal with this, we propose to investigate security analyses for languages with more principled forms of dynamic code generation. To this end, we present a static information flow analysis for a dynamically typed functional language with prototype-based inheritance and staged metaprogramming. We prove its soundness, implement it and test it on various examples designed to show its relevance to proving security properties, such as noninterference, in JavaScript. To demonstrate the applicability of the analysis, we also present a general method for transforming a program using eval into one using staged metaprogramming. To our knowledge, this is the first fully static information flow analysis for a language with staged metaprogramming, and the first formal soundness proof of a CFA-based information flow analysis for a functional programming language