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In this functional pearl, we examine the use of definitional interpreters as a basis for abstract interpretation of higher-order programming languages. As it turns out, definitional interpreters, especially those written in monadic style, can provide a nice basis for a wide variety of collecting semantics, abstract interpretations, symbolic executions, and their intermixings.But the real insight of this story is a replaying of an insight from Reynold's landmark paper, Definitional Interpreters for Higher-Order Programming Languages, in which he observes definitional interpreters enable the defined-language to inherit properties of the defining-language. We show the same holds true for definitional abstract interpreters. Remarkably, we observe that abstract definitional interpreters can inherit the so-called "pushdown control flow" property, wherein function calls and returns are precisely matched in the abstract semantics, simply by virtue of the function call mechanism of the defining-language.The first approaches to achieve this property for higher-order languages appeared within the last ten years, and have since been the subject of many papers. These approaches start from a state-machine semantics and uniformly involve significant technical engineering to recover the precision of pushdown control flow. In contrast, starting from a definitional interpreter, the pushdown control flow property is inherent in the metalanguage and requires no further technical mechanism to achieve.

We show how to implement a parser combinator library in Haskell capable of parsing arbitrary context-free grammars. The library is based on the work of Might et al. on parsing with derivatives, [15] of which we derive the required results from a single Galois connection. The library improves upon the previous Haskell implementation [5] with regards to completeness, safety and documentation; we can handle any grammar and use generalized tries for memoization instead of unsafe IO. However, our implementation shows severe performance issues. We describe these issues and possible solutions where applicable.

Recent developments in the systematic construction of abstract interpreters hinted at the possibility of a broad unification of concepts in static analysis. We deliver that unification by showing context-sensitivity, polyvariance, flow-sensitivity, reachabilitypruning, heap-cloning and cardinality-bounding to be independent of any particular semantics. Monads become the unifying agent between these concepts and between semantics. For instance, by plugging the same "context-insensitivity monad" into a monadicallyparameterized semantics for Java or for the lambda calculus, it yields the expected context-insensitive analysis.To achieve this unification, we develop a systematic method for transforming a concrete semantics into a monadically-parameterized abstract machine. Changing the monad changes the behavior of the machine. By changing the monad, we recover a spectrum of machines-from the original concrete semantics to a monovariant, flow-and context-insensitive static analysis with a singly-threaded heap and weak updates.The monadic parameterization also suggests an abstraction over the ubiquitous monotone fixed-point computation found in static analysis. This abstraction makes it straightforward to instrument an analysis with high-level strategies for improving precision and performance, such as abstract garbage collection and widening.While the paper itself runs the development for continuationpassing style, our generic implementation replays it for direct-style lambda-calculus and Featherweight Java to support generality.

During the past decade, differential privacy has become the gold standard for protecting the privacy of individuals. However, verifying that a particular program provides differential privacy often remains a manual task to be completed by an expert in the field. Language-based techniques have been proposed for fully automating proofs of differential privacy via type system design, however these results have lagged behind advances in differentially-private algorithms, leaving a noticeable gap in programs which can be automatically verified while also providing state-of-the-art bounds on privacy.We propose Duet, an expressive higher-order language, linear type system and tool for automatically verifying differential privacy of general-purpose higher-order programs. In addition to general purpose programming, Duet supports encoding machine learning algorithms such as stochastic gradient descent, as well as common auxiliary data analysis tasks such as clipping, normalization and hyperparameter tuning-each of which are particularly challenging to encode in a statically verified differential privacy framework.We present a core design of the Duet language and linear type system, and complete key proofs about privacy for well-typed programs. We then show how to extend Duet to support realistic machine learning applications and recent variants of differential privacy which result in improved accuracy for many practical differentially private algorithms. Finally, we implement several differentially private machine learning algorithms in Duet which have never before been automatically verified by a language-based tool, and we present experimental results which demonstrate the benefits of Duet's language design in terms of accuracy of trained machine learning models. privacy provides a robust solution to this problem, and as a result, a number of differentially private algorithms have been developed for machine learning [3,13,17,28,42,50,51,54].Few practical approaches exist, however, for automatically proving that a general-purpose program satisfies differential privacy-an increasingly desirable goal, since many machine learning pipelines are expressed as programs that combine existing algorithms with custom code. Enforcing differential privacy for a new program currently requires a new, manually-written privacy proof. This process is arduous, error-prone, and must be performed by an expert in differential privacy (and re-performed, each time the program is modified).We present Duet, a programming language, type system and tool for expressing and statically verifying privacy-preserving programs. Duet supports (1) general purpose programming features like compound datatypes and higher-order functions, (2) library functions for matrix-based computations, and (3) multiple state-of-the-art variants of differential privacy-(ϵ, δ )-differential privacy [25], Rényi differential privacy [38], zero-concentrated differential privacy (zCDP) [16], and truncated-concentrated differential privacy (tCDP) [15]-and can be easily extended to...

The design and implementation of static analyzers has become increasingly systematic. Yet for a given language or analysis feature, it often requires tedious and error prone work to implement an analyzer and prove it sound. In short, static analysis features and their proofs of soundness do not compose well, causing a dearth of reuse in both implementation and metatheory.We solve the problem of systematically constructing static analyzers by introducing Galois transformers: monad transformers that transport Galois connection properties. In concert with a monadic interpreter, we define a library of monad transformers that implement building blocks for classic analysis parameters like context, path, and heap (in)sensitivity. Moreover, these can be composed together independent of the language being analyzed.Significantly, a Galois transformer can be proved sound once and for all, making it a reusable analysis component. As new analysis features and abstractions are developed and mixed in, soundness proofs need not be reconstructed, as the composition of a monad transformer stack is sound by virtue of its constituents. Galois transformers provide a viable foundation for reusable and composable metatheory for program analysis.Finally, these Galois transformers shift the level of abstraction in analysis design and implementation to a level where non-specialists have the ability to synthesize sound analyzers over a number of parameters.

An oblivious computation is one that is free of direct and indirect information leaks, e.g., due to observable differences in timing and memory access patterns. This paper presents λ obliv , a core language whose type system enforces obliviousness. Prior work on type-enforced oblivious computation has focused on deterministic programs. λ obliv is new in its consideration of programs that implement probabilistic algorithms, such as those involved in cryptography. λ obliv employs a substructural type system and a novel notion of probability region to ensure that information is not leaked via the observed distribution of visible events. Probability regions support reasoning about probabilistic correlation and independence between values, and our use of probability regions is motivated by a source of unsoundness that we discovered in the type system of ObliVM, a language for implementing state of the art oblivious algorithms. We prove that λ obliv 's type system enforces obliviousness and show that it is expressive enough to typecheck advanced tree-based oblivious RAMs. CCS Concepts: • Security and privacy → Logic and verification.

The design and implementation of static analyzers has become increasingly systematic. Yet for a given language or analysis feature, it often requires tedious and error prone work to implement an analyzer and prove it sound. In short, static analysis features and their proofs of soundness do not compose well, causing a dearth of reuse in both implementation and metatheory. We solve the problem of systematically constructing static analyzers by introducing Galois transformers: monad transformers that transport Galois connection properties. In concert with a monadic interpreter, we define a library of monad transformers that implement building blocks for classic analysis parameters like context, path, and heap (in)sensitivity. Moreover, these can be composed together independent of the language being analyzed. Significantly, a Galois transformer can be proved sound once and for all, making it a reusable analysis component. As new analysis features and abstractions are developed and mixed in, soundness proofs need not be reconstructed, as the composition of a monad transformer stack is sound by virtue of its constituents. Galois transformers provide a viable foundation for reusable and composable metatheory for program analysis. Finally, these Galois transformers shift the level of abstraction in analysis design and implementation to a level where non-specialists have the ability to synthesize sound analyzers over a number of parameters.