Abstract One of the natural problems of operational semantics is to characterise the relationship between eager and lazy evaluation. In the context of $\lambda$ -calculus, this is expressed by the classic theorem that call-by-value evaluation of a program to (weak-head) normal form can always be simulated by a call-by-name evaluation. While the statement and intuition behind it are simple and clear, naive attempts at proof famously fail: the result is usually established as a consequence of the more complex standardisation theorem. In this work, we develop and formalise a novel and lightweight inductive approach to tackle the problem of simulation between two semantics for a single calculus, but with different evaluation orders. We exercise our method on the classic call-by-value and call-by-name example and report on methodological takeaways suggested by our approach, in particular what effect the flavour of semantics chosen has on the proof.

This paper considers direct encodings of generalized algebraic data types (GADTs) in a minimal suitable lambda-calculus. To this end, we develop an extension of System F ω with recursive types and internalized type equalities with injective constant type constructors. We show how GADTs and associated pattern-matching constructs can be directly expressed in the calculus, thus showing that it may be treated as a highly idealized modern functional programming language. We prove that the internalized type equalities in conjunction with injectivity rules increase the expressive power of the calculus by establishing a non-macro-expressibility result in F ω , and prove the system type-sound via a syntactic argument. Finally, we build two relational models of our calculus: a simple, unary model that illustrates a novel, two-stage interpretation technique, necessary to account for the equational constraints; and a more sophisticated, binary model that relaxes the construction to allow, for the first time, formal reasoning about data-abstraction in a calculus equipped with GADTs.

Effect handlers are a modern and increasingly popular approach to structuring computational effects in functional programming languages. However, while their traditional operational semantics is well-suited to implementation tasks, it is less ideal as a reduction theory. We therefore introduce a fine-grained reduction theory for deep effect handlers, inspired by our existing reduction theory for shift0, along with a standard reduction strategy. We relate this strategy to the traditional, non-local operational semantics via a simulation argument, and show that the reduction theory preserves observational equivalence with respect to the classical semantics of handlers, thus allowing its use as a rewriting theory for handler-equipped programming languages -- this rewriting system mostly coincides with previously studied type-based optimisations. In the process, we establish theoretical properties of our reduction theory, including confluence and standardisation theorems, adapting and extending existing techniques. Finally, we demonstrate the utility of our semantics by providing the first normalisation-by-evaluation algorithm for effect handlers, and prove its soundness and completeness. Additionally, we establish non-expressibility of the lift operator, found in some effect-handler calculi, by the other constructs.

The delimited-control operator shift0 has been formally shown to capture the operational semantics of deep handlers for algebraic effects. Its CPS translation generates λ-terms in which continuation composition is not expressed in terms of nested function calls, as is typical of other delimited-control operators, e.g. shift, but with function applications consuming a sequence of continuations one at a time, as if they formed a stack.

We present a study of the continuation-composing style (CCS) that describes the image of the CPS translation of Danvy and Filinski’s shift and reset delimited-control operators. In CCS continuations are composable rather than abortive as in the traditional CPS, and, therefore, the structure of terms is considerably more complex. We show that the CPS translation from Moggi’s computational lambda calculus extended with shift and reset has a right inverse and that the two translations form a reflection i.e., a Galois connection in which the target is isomorphic to a subset of the source (the orders are given by the reduction relations). Furthermore, we use this result to show that Plotkin’s call-by-value lambda calculus extended with shift and reset is isomorphic to the image of the CPS translation. This result, in particular, provides a first direct-style transformation for delimited continuations that is an inverse of the CPS transformation up to syntactic identity.

Handlers of algebraic effects aspire to be a practical and robust programming construct that allows one to define, use, and combine different computational effects. Interestingly, a critical problem that still bars the way to their popular adoption is how to combine different uses of the same effect in a program, particularly in a language with a static type-and-effect system. For example, it is rudimentary to define the “mutable memory cell” effect as a pair of operations, put and get , together with a handler, but it is far from obvious how to use this effect a number of times to operate a number of memory cells in a single context. In this paper, we propose a solution based on lexically scoped effects in which each use (an “instance”) of an effect can be singled out by name, bound by an enclosing handler and tracked in the type of the expression. Such a setting proves to be delicate with respect to the choice of semantics, as it depends on the explosive mixture of effects, polymorphism, and reduction under binders. Hence, we devise a novel approach to Kripke-style logical relations that can deal with open terms, which allows us to prove the desired properties of our calculus. We formalise our core results in Coq, and introduce an experimental surface-level programming language to show that our approach is applicable in practice.

Abstract We generalise Cayley’s theorem for monoids by providing an explicit formula for a (multi-sorted) equational theory represented by the type $$PX \rightarrow X$$ , where $$P$$ is an arbitrary polynomial endofunctor with natural coefficients. From the computational perspective, examples of effects given by such theories include backtracking nondeterminism (obtained with the original Cayley representation $$X \rightarrow X$$ ), finite mutable state (obtained with $$n \rightarrow X$$ , for a constant n), and their different combinations (via $$n \times X \rightarrow X$$ or $$X^n \rightarrow X$$ ). Moreover, we show that monads induced by such theories are implementable using the type formers available in programming languages based on a polymorphic $$\lambda $$ -calculus, both as compositions of algebraic datatypes and as continuation-like monads. We give a set-theoretic model of the latter in terms of Barr-dinatural transformations. We also introduce , a tool that takes a polynomial as an input and generates the corresponding equational theory together with the two implementations of the induced monad in Haskell.

Proposed originally by Plotkin and Pretnar, algebraic effects and their handlers are a leading-edge approach to computational effects: exceptions, mutable state, nondeterminism, and such. Appreciated for their elegance and expressiveness, they are now progressing into mainstream functional programming languages. In this paper, we introduce and examine programming language constructs that back adoption of programming with algebraic effects on a larger scale in a modular fashion by providing mechanisms for abstraction. We propose two such mechanisms: existential effects (which hide the details of a particular effect from the user) and local effects (which guarantee that no code coming from the outside can interfere with a given effect). The main technical difficulty arises from the dynamic nature of coupling an effectful operation with the right handler during execution, but, as we show in this paper, a carefully designed type system can ensure that this will not break the abstraction. Our main contribution is a novel calculus for algebraic effects and handlers, called λ HEL , equipped with local and existential algebraic effects, in which the dynamic nature of handlers is kept in check by typed runtime coercions. As a proof of concept, we present an experimental programming language based on our calculus, which provides strong abstraction mechanisms via an ML-style module system.

A classic problem in parallel computing is to take a high-level parallel program written, for example, in nested-parallel style with fork-join constructs and run it efficiently on a real machine. The problem could be considered solved in theory, but not in practice, because the overheads of creating and managing parallel threads can overwhelm their benefits. Developing efficient parallel codes therefore usually requires extensive tuning and optimizations to reduce parallelism just to a point where the overheads become acceptable.

Algebraic effects and handlers have received a lot of attention recently, both from the theoretical point of view and in practical language design. This stems from the fact that algebraic effects give the programmer unprecedented freedom to define, combine, and interpret computational effects. This plenty-of-rope, however, demands not only a deep understanding of the underlying semantics, but also access to practical means of reasoning about effectful code, including correctness and program equivalence. In this paper we tackle this problem by constructing a step-indexed relational interpretation of a call-by-value calculus with algebraic effect handlers and a row-based polymorphic type-and-effect system. Our calculus, while striving for simplicity, enjoys desirable theoretical properties, and is close to the cores of programming languages with algebraic effects used in the wild, while the logical relation we build for it can be used to reason about non-trivial properties, such as contextual equivalence and contextual approximation of programs. Our development has been fully formalised in the Coq proof assistant.

Increasing availability of multicore systems has led to greater focus on the design and implementation of languages for writing parallel programs. Such languages support various abstractions for parallelism, such as fork-join, async-finish, futures. While they may seem similar, these abstractions lead to different semantics, language design and implementation decisions, and can significantly impact the performance of end-user applications.

We present Iris, a concurrent separation logic with a simple premise: monoids and invariants are all you need. Partial commutative monoids enable us to express---and invariants enable us to enforce---user-defined *protocols* on shared state, which are at the conceptual core of most recent program logics for concurrency. Furthermore, through a novel extension of the concept of a *view shift*, Iris supports the encoding of *logically atomic specifications*, i.e., Hoare-style specs that permit the client of an operation to treat the operation essentially as if it were atomic, even if it is not.