Modern network management requires dynamically reacting to a broad set of events occurring in the network. Detecting the traffic behaviors associated with these events requires identifying patterns and computing metrics over high-velocity streams of network packets flowing through a link, router, or switch. To facilitate performing computations, a number of traffic query languages have been proposed in the network systems community. However, existing traffic query languages primarily focus on expressing computations that can be efficiently implemented on particular hardware or software platforms (such as programmable switch ASICs or multi-core CPUs) and do not support the full range of behaviors required by modern network management tasks. For example, languages like Sonata efficiently support count-based metrics on switch ASICs, but fail to support complex inter-packet pattern identification, whereas a language like FLM can identify patterns on switch ASICs but cannot compute quantitative metrics.

Abstract Structural induction is pervasively used by functional programmers and researchers for both informal reasoning as well as formal methods for program verification and semantics. In this paper, we promote its dual—structural coinduction—as a technique for understanding corecursive programs only in terms of the logical structure of their context. We illustrate this technique as an informal method of proofs which closely match the style of informal inductive proofs, where it is straightforward to check that all cases are covered and the coinductive hypotheses are used correctly. This intuitive idea is then formalized through a syntactic theory for deriving program equalities, which is justified purely in terms of the computational behavior of abstract machines and proved sound with respect to observational equivalence.

Closure conversion, an essential step in compiling functional programs, is traditionally presented as a global transformation from a language with higher-order functions to one without. Optimizing this transformation can be done at any of its three stages with various tradeoffs. After conversion, optimizations will be in the target language at the cost of a weaker equational theory. During conversion, optimizations may be embedded into the transformation itself at the cost of increasing its complexity and correctness. And before conversion, optimizations may be planned and anticipated in a specialized intermediate language (IL) where all the properties of the source program are still known, with the hope that they will survive the rest of the compilation process.

Abstract Recursion is a mature, well-understood topic in the theory and practice of programming. Yet its dual, corecursion is underappreciated and still seen as exotic. We aim to put them both on equal footing by giving a foundation for primitive corecursion based on computation, giving a terminating calculus analogous to the original computational foundation of recursion. We show how the implementation details in an abstract machine strengthens their connection, syntactically deriving corecursion from recursion via logical duality. We also observe the impact of evaluation strategy on the computational complexity of primitive (co)recursive combinators: call-by-name allows for more efficient recursion, but call-by-value allows for more efficient corecursion.

The duality between "true" and "false" is a hallmark feature of logic. We show how this duality can be put to use in the theory and practice of programming languages and their implementations, too. Starting from a foundation of constructive logic as dialogues, we illustrate how it describes a symmetric language for computation, and survey several applications of the dualities found therein.

Recursion and induction are mature, well-understood topics in programming. Yet their duals, corecursion and coinduction, are still exotic and underdeveloped programming features. We aim to put them on equal footing by giving a foundation for corecursion based on computation, analogous to the original computational foundation of recursion. At the lower level, we show how the connection between the two can be strengthened through their implementation details in an abstract machine. At the higher level, we develop a syntactic equational theory for inductive and coinductive reasoning based on control flow. We also observe the impact of evaluation strategy: call-by-name has efficient recursion and strong coinductive reasoning, but call-by-value has efficient corecursion and strong inductive reasoning.

A language supporting polymorphism is a boon to programmers: they can express complex ideas once and reuse functions in a variety of situations. However, polymorphism is pain for compilers tasked with producing efficient code that manipulates concrete values. This paper presents a new intermediate language that allows for efficient static compilation, while still supporting flexible polymorphism. Specifically, it permits polymorphism over not only the types of values, but also the representation of values, the arity of primitive machine functions, and the evaluation order of arguments---all three of which are useful in practice. The key insight is to encode information about a value's calling convention in the kind of its type, rather than in the type itself.

Computer scientists are well-versed in dealing with data structures. The same cannot be said about their dual: codata. Even though codata is pervasive in category theory, universal algebra, and logic, the use of codata for programming has been mainly relegated to representing infinite objects and processes. Our goal is to demonstrate the benefits of codata as a general-purpose programming abstraction independent of any specific language: eager or lazy, statically or dynamically typed, and functional or object-oriented. While codata is not featured in many programming languages today, we show how codata can be easily adopted and implemented by offering simple inter-compilation techniques between data and codata. We believe codata is a common ground between the functional and object-oriented paradigms; ultimately, we hope to utilize the Curry-Howard isomorphism to further bridge the gap.

For a long time, intersection types have been admired for their surprising ability to complete the simply typed lambda calculus. Intersection types are an example of an implicit typing feature which can describe program behavior without manifesting itself within the syntax of a program. Dual to int ersections, union types are another implicit typing feature which extends the completeness property of intersection types in the lambda calculus to full-fledged programming languages. However, the formalization of union types can easily break other desirable meta-theoretical properties of the type system. But why should unions be troublesome when their dual, intersections, are not? We look at the issues surrounding the design of type systems for both intersection and union types through the lens of duality by formalizing them within the symmetric language of the classical sequent calculus. In order to formulate type systems which have all of our properties of interest—soundness, completeness, and type safety—we also look at the impact of evaluation strategy on typing. As a result, we present two dual type systems—one for call-by-value and one for call-by-name evaluation—which have all three properties. We also consider the possibility of classical non-deterministic evaluation, for which there is a choice between two different systems depending on which properties are desired: a full type system which is complete, and a simplified type system which is sound and type safe.

The study of polarity in computation has revealed that an "ideal" programming language combines both call-by-value and call-by-name evaluation; the two calling conventions are each ideal for half the types in a programming language. But this binary choice leaves out call-by-need which is used in practice to implement lazy-by-default languages like Haskell. We show how the notion of polarity can be extended beyond the value/name dichotomy to include call-by-need by only adding a mechanism for sharing and the extra polarity shifts to connect them, which is enough to compile a Haskell-like functional language with user-defined types.

Abstract We present a model of computation that heavily emphasizes the concept of duality and the interaction between opposites–production interacts with consumption. The symmetry of this framework naturally explains more complicated features of programming languages through relatively familiar concepts. For example, binding a value to a variable is dual to manipulating the flow of control in a program. By looking at the computational interpretation of the sequent calculus, we find a language that lets us speak about duality, control flow, and evaluation order in programs as first-class concepts. We begin by reviewing Gentzen's LK sequent calculus and show how the Curry–Howard isomorphism still applies to give us a different basis for expressing computation. We then illustrate how the fundamental dilemma of computation in the sequent calculus gives rise to a duality between evaluation strategies : strict languages are dual to lazy languages. Finally, we discuss how the concept of focusing , developed in the setting of proof search, is related to the idea of type safety for computation expressed in the sequent calculus. In this regard, we compare and contrast two different methods of focusing that have appeared in the literature, static and dynamic focusing, and illustrate how they are two means to the same end.

Many fields of study in compilers give rise to the concept of a join point—a place where different execution paths come together. Join points are often treated as functions or continuations, but we believe it is time to study them in their own right. We show that adding join points to a direct-style functional intermediate language is a simple but powerful change that allows new optimizations to be performed, including a significant improvement to list fusion. Finally, we report on recent work on adding join points to the intermediate language of the Glasgow Haskell Compiler.

Abstract Designing rewriting systems that respect functional extensionality for call-by-name languages with effects turns out to be surprisingly challenging. Simply interpreting extensional laws like η as reduction rules easily breaks confluence. We explore these issues in the setting of a sequent calculus. Building on an insight that appears in different aspects of the theory of call-by-name functional languages—confluent rewriting for two independent control calculi and sound continuation-passing style transformations—we give a confluent reduction system for lazy extensional functions. Finally, we consider limitations to this approach when used for strict evaluation and types beyond functions.

The λ-calculus is popular as an intermediate language for practical compilers. But in the world of logic it has a lesser-known twin, born at the same time, called the sequent calculus. Perhaps that would make for a good intermediate language, too? To explore this question we designed Sequent Core, a practically-oriented core calculus based on the sequent calculus, and used it to re-implement a substantial chunk of the Glasgow Haskell Compiler.

Our goal is to develop co-induction from our understanding of induction, putting them on level ground as equal partners for reasoning about programs. We investigate several structures which represent well-founded forms of recursion in programs. These simple structures encapsulate reasoning by primitive and noetherian induction principles, and can be composed together to form complex recursion schemes for programs operating over a wide class of data and co-data types. At its heart, this study is guided by duality: each structure for recursion has a dual form, giving perfectly symmetric pairs of equal and opposite data and co-data types for representing recursion in programs. Duality is brought out through a framework presented in sequent style, which inherently includes control effects that are interpreted logically as classical reasoning principles. To accommodate the presence of effects, we give a calculus parameterized by a notion of strategy, which is strongly normalizing for a wide range of strategies. We also present a more traditional calculus for representing effect-free functional programs, but at the cost of losing some of the founding dualities.

Parigot's λμ-calculus, a system for computational reasoning about classical proofs, serves as a foundation for control operations embodied by operators like Scheme's callcc. We demonstrate that the call-by-value theory of the λμ-calculus contains a latent theory of delimited control, and that a known variant of λμ which unshackles the syntax yields a calculus of composable continuations from the existing constructs and rules for classical control. To relate to the various formulations of control effects, and to continuation-passing style, we use a form of compositional program transformations which preserves the underlying structure of equational theories, contexts, and substitution. Finally, we generalize the call-by-name and call-by-value theories of the λμ-calculus by giving a single parametric theory that encompasses both, allowing us to generate a call-by-need instance that defines a calculus of classical and delimited control with lazy evaluation and sharing.

Continuation-passing style (CPS) transforms have long been important tools in the study of programming. They have been shown to correspond to abstract machines and, when combined with a naming transform that expresses shared values, they enjoy a direct correspondence with encodings into process calculi such as the π-calculus. We present our notion of correctness and discuss the sufficient conditions that guarantee the correctness of transforms. We then consider the call-by-value, call-by-name and call-by-need evaluation strategies for the λ-calculus and present their CPS transforms, abstract machines, π-encodings, and proofs of correctness. Our analysis covers a uniform CPS transform, which differentiates the three evaluation strategies only by the treatment of function calls. This leads to a new CPS transform for call-by-need requiring a less expressive form of side effect, which we call constructive update.

Abstract We give a framework for delimited control with multiple prompts, in the style of Parigot's λμ-calculus, through a series of incremental extensions by starting with the pure λ-calculus. Each language inherits the semantics and reduction theory of its parent, giving a systematic way to describe each level of control. For each language of interest, we fully characterize its semantics in terms of a reduction semantics, operational semantics, continuation-passing style transform, and abstract machine. Furthermore, the control operations are expressed in terms of fine-grained primitives that can be used to build well-known, higher-level control operators. In order to illustrate the expressive power provided by various languages, we show how other computational effects can be encoded in terms of these control operators.

We propose a sequent calculus derived from the λ―μμ˜-calculus of Curien and Herbelin that is expressive enough to directly represent the fine details of program evaluation using typical abstract machines. Not only does the calculus easily encode the usual components of abstract machines such as environments and stacks, but it can also simulate the transition steps of the abstract machine with just a constant overhead. Technically this is achieved by ensuring that reduction in the calculus always happens at a bounded depth from the root of the term. We illustrate these properties by providing shallow encodings of the Krivine (call-by-name) and the CEK (call-by-value) abstract machines in the calculus.

Abstract The historical design of the call-by-value theory of control relies on the reification of evaluation contexts as regular functions and on the use of ordinary term application for jumping to a continuation. To the contrary, the control calculus, developed by the authors, distinguishes between jumps and terms . This alternative calculus, which derives from Parigot's λμ-calculus, works by direct structural substitution of evaluation contexts. We review and revisit the legacy theories of control and argue that provides an observationally equivalent but smoother theory. In an additional note contributed by Matthias Felleisen, we review the story of the birth of control calculi during the mid- to late-eighties at Indiana University.

There is a correspondence between classical logic and programming language calculi with first-class continuations. With the addition of control delimiters (prompts), the continuations become composable and the calculi are believed to become more expressive. We formalise that the addition of prompts corresponds to the addition of a single dynamically-scoped variable modelling the special top-level continuation. From a type perspective, the dynamically-scoped variable requires effect annotations. From a logic perspective, the effect annotations can be understood in a standard logic extended with the dual of implication, namely subtraction.

The extension of Haskell with a built-in state monad combines mathematical elegance with operational efficiency: -Semantically, at the source language level, constructs that act on the state are viewed as functions that pass an explicit store data structure around. -Operationally, at the implementation level, constructs that act on the state are viewed as statements whose evaluation has the side-effect of updating the implicit global store in place.There are several unproven conjectures that the two views are consistent.Recently, we have noted that the consistency of the two views is far from obvious: all it takes for the implementation to become unsound is one judiciously-placed beta-step in the optimization phase of the compiler. This discovery motivates the current paper in which we formalize and show the correctness of the implementation of monadic state.For the proof, we first design a typed call-by-need language that models the intermediate language of the compiler, together with a type-preserving compilation map. Second, we show that the compilation is semantics-preserving by proving that the compilation of every source axiom yields an observational equivalence of the target language. Because of the wide semantic gap between the source and target languages, we perform this last step using an additional intermediate language.The imperative call-by-need λ-Calculus is of independent interest for reasoning about system-level Haskell code providing services such as memo-functions, generation of new names, etc., and is the starting point for reasoning about the space usage of Haskell programs.

Plotkin (1975) showed that the lambda calculus is a good model of the evaluation process for call-by-name functional programs. Reducing programs to constants or lambda abstractions according to the leftmost-outermost strategy exactly mirrors execution on an abstract machine like Landin's SECD machine. The machine-based evaluator returns a constant or the token closure if and only if the standard reduction sequence starting at the same program will end in the same constant or in some lambda abstraction. However, the calculus does not capture the sharing of the evaluation of arguments that lazy implementations use to speed up the execution. More precisely, a lazy implementation evaluates procedure arguments only when needed and then only once. All other references to the formal procedure parameter re-use the value of the first argument evaluation. The mismatch between the operational semantics of the lambda calculus and the actual behavior of the prototypical implementation is a major obstacle for compiler writers. Unlike implementors of the leftmost-outermost strategy or of a call-by-value language, implementors of lazy systems cannot easily explain the behavior of their evaluator in terms of source level syntax. Hence, they often cannot explain why a certain syntactic transformation ‘works’ and why another doesn't. In this paper we develop an equational characterization of the most popular lazy implementation technique – traditionally called ‘call-by-need’ – and prove it correct with respect to the original lambda calculus. The theory is a strictly smaller theory than Plotkin's call-by-name lambda calculus. Immediate applications of the theory concern the correctness proofs of a number of implementation strategies, e.g. the call-by-need continuation passing transformation and the realization of sharing via assignments. Some of this material first appeared in a paper presented at the 1995 ACM Conference on the Principles of Programming Languages. The paper was a joint effort with Maraist, Odersky and Wadler, who had independently developed a different equational characterization of call-by-need. We contrast our work with that of Maraist et al. in the body of this paper where appropriate.

We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the μ-rule. Next, orthogonal term graph rewrite systems, also in the presence of copying and hidden nodes, are shown to be confluent.

The mismatch between the operational semantics of the lambda calculus and the actual behavior of implementations is a major obstacle for compiler writers. They cannot explain the behavior of their evaluator in terms of source level syntax, and they cannot easily compare distinct implementations of different lazy strategies. In this paper we derive an equational characterization of call-by-need and prove it correct with respect to the original lambda calculus. The theory is a strictly smaller theory than the lambda calculus. Immediate applications of the theory concern the correctness proofs of a number of implementation strategies, e.g., the call-by-need continuation passing transformation and the realization of sharing via assignments.

Studies cyclic /spl lambda/-graphs. The starting point is to treat a /spl lambda/-graph as a system of recursion equations involving /spl lambda/-terms, and to manipulate such systems in an unrestricted manner, using equational logic, just as is possible for first-order term rewriting. Surprisingly, now the confluence property breaks down in an essential way. Confluence can be restored by introducing a restraining mechanism on the 'copying' operation. This leads to a family of /spl lambda/-graph calculi, which are inspired by the family of /spl lambdaspl sigma/-calculi (/spl lambda/-calculi with explicit substitution). However, these concern acyclic expressions only. In this paper we are not concerned with optimality questions for acyclic /spl lambda/-reduction. We also indicate how Wadsworth's (1978) interpreter can be simulated in the /spl lambda/-graph rewrite rules that we propose.>

Kid, a language for expressing compiler optimizations for functional languages is introduced. The language is λ-calculus based but treats let-blocks as first class objects. Let-blocks and associated rewrite rules provide the basis to capture the sharing of subexpressions precisely. The language goes beyond λ-calculus by including I-structures which are essential to express efficient translations of list and array comprehensions. A calculus and a parallel interpreter for Kid are developed. Many commonly known program transformations are also presented. A partial evaluator for Kid is developed and a notion of correctness of Kid transformations based on the syntactic structure of terms and printable answers is presented.