Higher-order Functional Language Research Articles

A Pure Demand Operational Semantics with Applications to Program Analysis

This paper develops a novel minimal-state operational semantics for higher-order functional languages that uses only the call stack and a source program point or a lexical level as the complete state information: there is no environment, no substitution, no continuation, etc. We prove this form of operational semantics equivalent to standard presentations. We then show how this approach can open the door to potential new applications: we define a program analysis as a direct finitization of this operational semantics. The program analysis that naturally emerges has a number of novel and interesting properties compared to standard program analyses for higher-order programs: for example, it can infer recurrences and does not need value widening. We both give a formal definition of the analysis and describe our current implementation.

Explicit Effects and Effect Constraints in ReML

An important aspect of building robust systems that execute on dedicated hardware and perhaps in constrained environments is to control and manage the effects performed by program code. We present ReML, a higher-order statically-typed functional language, which allows programmers to be explicit about the effects performed by program code and in particular effects related to memory management. Allowing programmers to be explicit about effects, the regions in which values reside, and the constraints under which code execute, makes programs robust to changes in the program source code and to compiler updates, including compiler optimisations. ReML is integrated with a polymorphic inference system that builds on top of region-inference, as it is implemented in the MLKit, a Standard ML compiler that uses region-based memory management as its primary memory management scheme.

Alice or Bob?: Process polymorphism in choreographies

Abstract We present PolyChor $\lambda$ , a language for higher-order functional choreographic programming—an emerging paradigm for concurrent programming. In choreographic programming, programmers write the desired cooperative behaviour of a system of processes and then compile it into an implementation for each process, a translation called endpoint projection. Unlike its predecessor, Chor $\lambda$ , PolyChor $\lambda$ has both type and process polymorphism inspired by System F $_\omega$ . That is, PolyChor $\lambda$ is the first (higher-order) functional choreographic language which gives programmers the ability to write generic choreographies and determine the participants at runtime. This novel combination of features also allows PolyChor $\lambda$ processes to communicate distributed values, leading to a new and intuitive way to write delegation. While some of the functional features of PolyChor $\lambda$ give it a weaker correspondence between the semantics of choreographies and their endpoint-projected concurrent systems than some other choreographic languages, we still get the hallmark end result of choreographic programming: projected programmes are deadlock-free by design.

Analyzing binding extent in 3CPS

To date, the most effective approach to compiling strict, higher-order functional languages (such as OCaml, Scheme, and SML) has been to use whole-program techniques to convert the program to a first-order monomorphic representation that can be optimized using traditional compilation techniques. This approach, popularized by MLton, has limitations, however. We are interested in exploring a different approach to compiling such languages, one that preserves the higher-order and polymorphic character of the program throughout optimization. To enable such an approach, we must have effective analyses that both provide precise information about higher-order programs and that scale to larger units of compilation. This paper describes one such analysis for determining the extent of variable bindings. We classify the extent of variables as either register (only one binding instance can be live at any time), stack (the lifetimes of binding instances obey a LIFO order), or heap (binding lifetimes are arbitrary). These extents naturally connect variables to the machine resources required to represent them. We believe that precise information about binding extents will enable efficient management of environments, which is a key problem in the efficient compilation of higher-order programs. At the core of the paper is the 3CPS intermediate representation, which is a factored CPS-based intermediate representation (IR) that statically marks variables to indicate their binding extent. We formally specify the management of this binding structure by means of a small-step operational semantics and define a static analysis that determines the extents of the variables in a program. We evaluate our analysis using a standard suite of SML benchmark programs. Our implementation gets surprisingly high yield and exhibits scalable performance. While this paper uses a CPS-based IR, the algorithm and results are easily transferable to other λ-calculus IRs, such as ANF.

EVL: A Typed Higher-order Functional Language for Events

We define EVL, a minimal higher-order functional language for dealing with generic events. The notion of generic event extends the well-known notion of event traditionally used in a variety of areas, such as database management, concurrency, reactive systems and cybersecurity. Generic events were introduced in the context of a metamodel to deal with obligations in access control systems. Event specifications are represented as records and we use polymorphic record types to type events in our language. We show how the higher-order capabilities of EVL can be used in the context of Complex Event Processing (CEP), to define higher-order parameterised functions that deal with the usual CEP techniques.

Seminaïve evaluation for a higher-order functional language

One of the workhorse techniques for implementing bottom-up Datalog engines is seminaïve evaluation. This optimization improves the performance of Datalog's most distinctive feature: recursively defined predicates. These are computed iteratively, and under a naïve evaluation strategy, each iteration recomputes all previous values. Seminaïve evaluation computes a safe approximation of the difference between iterations. This can asymptotically improve the performance of Datalog queries. Seminaïve evaluation is defined partly as a program transformation and partly as a modified iteration strategy, and takes advantage of the first-order nature of Datalog code. This paper extends the seminaïve transformation to higher-order programs written in the Datafun language, which extends Datalog with features like first-class relations, higher-order functions, and datatypes like sum types.

Efficient differentiable programming in a functional array-processing language

We present a system for the automatic differentiation (AD) of a higher-order functional array-processing language. The core functional language underlying this system simultaneously supports both source-to-source forward-mode AD and global optimisations such as loop transformations. In combination, gradient computation with forward-mode AD can be as efficient as reverse mode, and that the Jacobian matrices required for numerical algorithms such as Gauss-Newton and Levenberg-Marquardt can be efficiently computed.

Concolic Testing of Higher-order Functional Languages

Concolic Testing of Higher-order Functional Languages

Nominal unification with atom-variables

The problem of nominal unification where variables are allowed for atoms, and computing a complete set of unifiers is considered. The complexity is shown to be NP-complete, while for special cases there are polynomial time algorithms. The main result is a novel algorithm to compute a complete set of unifiers which performs lazy guessing of equality or disequality of atom-variables, runs in NP time, and the collecting variant has more chances to keep the complete set of unifiers small. Applications of this algorithm are in reasoning about program transformations in higher order functional languages. We also present a variant of the unification algorithm that delays guessing and checking solvability, and produces a single most general unifier.

1ML – Core and modules united

AbstractML is two languages in one: there is the core, with types and expressions, and there are modules, with signatures, structures, and functors. Modules form a separate, higher-order functional language on top of the core. There are both practical and technical reasons for this stratification; yet, it creates substantial duplication in syntax and semantics, and it imposes seemingly unnecessary limits on expressiveness because it makes modules second-class citizens of the language. For example, selecting one among several possible modules implementing a given interface cannot be made a dynamic decision. Language extensions allowing modules to be packaged up as first-class values have been proposed and implemented in different variations. However, they remedy expressiveness only to some extent and tend to be even more syntactically heavyweight than using second-class modules alone. We propose a redesign of ML in which modules are truly first-class values, and core and module layers are unified into one language. In this “1ML”, functions, functors, and even type constructors are one and the same construct; likewise, no distinction is needed between structures, records, or tuples. Or viewed the other way round, everything is just (“a mode of use of”) modules. Yet, 1ML does not require dependent types: its type structure is expressible in terms of plain System Fω, with a minor variation of our F-ing modules approach. We introduce both an explicitly typed version of 1ML and an extension with Damas–Milner-style implicit quantification. Type inference for this language is not complete, but, we argue, not substantially worse than for Standard ML.

A logical relation for monadic encapsulation of state: proving contextual equivalences in the presence of runST

We present a logical relations model of a higher-order functional programming language with impredicative polymorphism, recursive types, and a Haskell-style ST monad type with runST. We use our logical relations model to show that runST provides proper encapsulation of state, by showing that effectful computations encapsulated by runST are heap independent. Furthermore, we show that contextual refinements and equivalences that are expected to hold for pure computations do indeed hold in the presence of runST. This is the first time such relational results have been proven for a language with monadic encapsulation of state. We have formalized all the technical development and results in Coq.

Executing and verifying higher-order functional-imperative programs in Maude

We incorporate higher-order functions and state monads in Maude, thereby embedding a higher-order functional language with imperative features in the Maude framework. We illustrate, via simple programs in the resulting language: the concrete and symbolic execution of programs; their verification with respect to properties expressed in Reachability Logic, a language-parametric generalisation of Hoare Logic; and the verification of program-equivalence properties. Our approach is proved sound and is implemented in Full Maude by taking advantage of its reflective features and module system.

Relational cost analysis

Establishing quantitative bounds on the execution cost of programs is essential in many areas of computer science such as complexity analysis, compiler optimizations, security and privacy. Techniques based on program analysis, type systems and abstract interpretation are well-studied, but methods for analyzing how the execution costs of two programs compare to each other have not received attention. Naively combining the worst and best case execution costs of the two programs does not work well in many cases because such analysis forgets the similarities between the programs or the inputs. In this work, we propose a relational cost analysis technique that is capable of establishing precise bounds on the difference in the execution cost of two programs by making use of relational properties of programs and inputs. We develop , a refinement type and effect system for a higher-order functional language with recursion and subtyping. The key novelty of our technique is the combination of relational refinements with two modes of typing-relational typing for reasoning about similar computations/inputs and unary typing for reasoning about unrelated computations/inputs. This combination allows us to analyze the execution cost difference of two programs more precisely than a naive non-relational approach. We prove our type system sound using a semantic model based on step-indexed unary and binary logical relations accounting for non-relational and relational reasoning principles with their respective costs. We demonstrate the precision and generality of our technique through examples.

A lambda-calculus foundation for universal probabilistic programming

We develop the operational semantics of an untyped probabilistic λ-calculus with continuous distributions, and both hard and soft constraints,as a foundation for universal probabilistic programming languages such as Church, Anglican, and Venture. Our first contribution is to adapt the classic operational semantics of λ-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics . To move closer to inference techniques, we also define the sampling-based semantics of a term as a function from a trace of random samples to a value. We show that the distribution induced by integration over the space of traces equals the distribution-based semantics. Our second contribution is to formalize the implementation technique of trace Markov chain Monte Carlo (MCMC) for our calculus and to show its correctness. A key step is defining sufficient conditions for the distribution induced by trace MCMC to converge to the distribution-based semantics. To the best of our knowledge, this is the first rigorous correctness proof for trace MCMC for a higher-order functional language, or for a language with soft constraints.

Verifying relational properties of functional programs by first-order refinement

• We give an automated verification method for relational properties. • The target is a higher-order functional language. • We can use general refinement types as specifications under a certain restriction. • We reduce type checking for general refinement types to that for first-order ones. • We proved the soundness and implemented a prototype verifier. Much progress has been made recently on fully automated verification of higher-order functional programs, based on refinement types and higher-order model checking. Most of those verification techniques are, however, based on first-order refinement types, hence unable to verify certain properties of functions (such as the equality of two recursive functions and the monotonicity of a function, which we call relational properties ). To relax this limitation, we introduce a restricted form of higher-order refinement types where refinement predicates can refer to functions, and formalize a systematic program transformation to reduce type checking/inference for higher-order refinement types to that for first-order refinement types, so that the latter can be automatically solved by using an existing software model checker. We also prove the soundness of the transformation, and report on implementation and experiments.

1ML – core and modules united (F-ing first-class modules)

ML is two languages in one: there is the core, with types and expressions, and there are modules, with signatures, structures and functors. Modules form a separate, higher-order functional language on top of the core. There are both practical and technical reasons for this stratification; yet, it creates substantial duplication in syntax and semantics, and it reduces expressiveness. For example, selecting a module cannot be made a dynamic decision. Language extensions allowing modules to be packaged up as first-class values have been proposed and implemented in different variations. However, they remedy expressiveness only to some extent, are syntactically cumbersome, and do not alleviate redundancy. We propose a redesign of ML in which modules are truly first-class values, and core and module layer are unified into one language. In this "1ML", functions, functors, and even type constructors are one and the same construct; likewise, no distinction is made between structures, records, or tuples. Or viewed the other way round, everything is just ("a mode of use of") modules. Yet, 1ML does not require dependent types, and its type structure is expressible in terms of plain System Fω, in a minor variation of our F-ing modules approach. We introduce both an explicitly typed version of 1ML, and an extension with Damas/Milner-style implicit quantification. Type inference for this language is not complete, but, we argue, not substantially worse than for Standard ML. An alternative view is that 1ML is a user-friendly surface syntax for System Fω that allows combining term and type abstraction in a more compositional manner than the bare calculus.

Supercompilation for Martin-Lof’s type theory

This paper describes the design and implementation of a TT Lite certifying supercompiler, which transforms a source program into a pair consisting of a residual program and a proof that the residual program is equivalent to the source one. As far as we can judge from the presently available literature, it is the first time that certifying supercompilation is implemented for a nontrivial higher-order functional language. Proofs generated by the TT Lite supercompiler can be verified by the type checker that does not depend on the supercompiler and is not based on supercompilation. This is especially important when reliability of results obtained by means of supercompilation is of primary concern.

Polymorphic Functions with Set-Theoretic Types

This article is the second part of a two articles series about the definition of higher order polymorphic functions in a type system with recursive types and set-theoretic type connectives (unions, intersections, and negations). In the first part, presented in a companion paper, we defined and studied the syntax, semantics, and evaluation of the explicitly-typed version of a calculus, in which type instantiation is driven by explicit instantiation annotations. In this second part we present a local type inference system that allows the programmer to omit explicit instantiation annotations for function applications, and a type reconstruction system that allows the programmer to omit explicit type annotations for function definitions. The work presented in the two articles provides the theoretical foundations and technical machinery needed to design and implement higher-order polymorphic functional languages with union and intersection types and/or for semi-structured data processing.

Polymorphic functions with set-theoretic types

This article is the first part of a two articles series about a calculus with higher-order polymorphic functions, recursive types with arrow and product type constructors and set-theoretic type connectives (union, intersection, and negation). In this first part we define and study the explicitly-typed version of the calculus in which type instantiation is driven by explicit instantiation annotations. In particular, we define an explicitly-typed lambda-calculus with intersection types and an efficient evaluation model for it. In the second part, presented in a companion paper, we define a local type inference system that allows the programmer to omit explicit instantiation annotations, and a type reconstruction system that allows the programmer to omit explicit type annotations. The work presented in the two articles provides the theoretical foundations and technical machinery needed to design and implement higher-order polymorphic functional languages for semi-structured data.

Online partial evaluation of sheet-defined functions

We present a spreadsheet implementation, extended with sheet-defined functions, that allows users to define functions using only standard spreadsheet concepts such as cells, formulas and references, requiring no new syntax. This implements an idea proposed by Peyton-Jones and others. As the main contribution of this paper, we then show how to add an online partial evaluator for such sheet-defined functions. The result is a higher-order functional language that is dynamically typed, in keeping with spreadsheet traditions, and an interactive platform for function definition and function specialization. We describe an implementation of these ideas, present some performance data from microbenchmarks, and outline desirable improvements and extensions.