Polymorphic Higher-order Functions Research Articles

A relational framework for higher-order shape analysis

We propose the integration of a relational specification framework within a dependent type system capable of verifying complex invariants over the shapes of algebraic datatypes. Our approach is based on the observation that structural properties of such datatypes can often be naturally expressed as inductively-defined relations over the recursive structure evident in their definitions. By interpreting constructor applications (abstractly) in a relational domain, we can define expressive relational abstractions for a variety of complex data structures, whose structural and shape invariants can be automatically verified. Our specification language also allows for definitions of parametric relations for polymorphic data types that enable highly composable specifications and naturally generalizes to higher-order polymorphic functions. We describe an algorithm that translates relational specifications into a decidable fragment of first-order logic that can be efficiently discharged by an SMT solver. We have implemented these ideas in a type checker called CATALYST that is incorporated within the MLton SML compiler. Experimental results and case studies indicate that our verification strategy is both practical and effective.

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.

Quantum Data and Control Made Easier

In this paper we define nQML, a functional quantum programming language that follows the “quantum data and control” paradigm. In comparison to Altenkirch and Grattage's QML, the control constructs of nQML are simpler and can implement quantum algorithms more directly and naturally. We avoid the unnecessary complexities of a linear type system by using types that carry the address of qubits in the quantum state. We provide a denotational semantics over density matrices and unitary transformations, inspired by Selinger's semantics for QPL. Our semantics leads naturally to an interpreter for nQML, written in Haskell. We also explore the extension of nQML with polymorphic higher-order functions.

Autonomous mobility skeletons

To manage load on large and dynamic networks we have developed Autonomous Mobile Programs (AMPs) that periodically use a cost model to decide where to execute. A disadvantage of directly programming AMPs is that the cost model, mobility decision function, and network interrogation are all explicit in the program. This paper proposes autonomous mobility skeletons that encapsulate self-aware mobile coordination for common patterns of computation over collections. Autonomous mobility skeletons are akin to algorithmic skeletons in being polymorphic higher order functions, but where algorithmic skeletons abstract over parallel coordination, autonomous mobility skeletons abstract over autonomous mobile coordination. We present the automap, autofold and autoiter autonomous mobility skeletons, together with performance measurements of Jocaml, Java Voyager, and JavaGo implementations on small networks. autoiter is an unusual skeleton, abstracting over the Iterator interface commonly used with Java collections.

A Type-theoretic Reconstruction of the Visitor Pattern

In object-oriented languages, the Visitor pattern can be used to traverse tree-like data structures: a visitor object contains some operations, and the data structure objects allow themselves to be traversed by accepting visitors. In the polymorphic lambda calculus (System F), tree-like data structures can be encoded as polymorphic higher-order functions. In this paper, we reconstruct the Visitor pattern from the polymorphic encoding by way of generics in Java. We sketch how the quantified types in the polymorphic encoding can guide reasoning about visitors in general.

More dynamic imperative languages

We pursue entirely dynamic constructs for imperative object-oriented programming languages by shifting responsibilities from compile-time handling to runtime management. This accounts to the full range of language elements, going beyond statically defined: types, classes, inheritance, and function / operator overloading. The accommodation of the dynamic features in a language named Delta is discussed whose design emphasizes a minimal set of special-purpose language constructs. We demonstrate the way polymorphic higher-order functions, and polymorphic software patterns are directly programmable in a dynamic language context.

Functional programming with the FC++ library

We describe the FC++ library, a rich library supporting functional programming in C++. Prior approaches to encoding higher order functions in C++ have suffered with respect to polymorphic functions from either lack of expressiveness or high complexity. In contrast, FC++ offers full and concise support for higher-order polymorphic functions through a novel use of C++ type inference. The FC++ library has a number of useful features, including a generalized mechanism to implement currying in C++, a “lazy list” class which enables the creation of “infinite data structures”, a subtype polymorphism facility, and an extensive library of useful functions, including a large part of the Haskell Standard Prelude. The FC++ library has an efficient implementation. We show the results of a number of experiments which demonstrate the value of optimizations we have implemented. These optimizations have improved the run-time performance by about an order of magnitude for some benchmark programs that make heavy use of FC++ lazy lists. We also make an informal performance comparison with similar programs written in Haskell.

Short cut fusion is correct

Fusion is the process of removing intermediate data structures from modularly constructed functional programs. Short cut fusion is a particular fusion technique which uses a single, local transformation rule to fuse compositions of list-processing functions. Short cut fusion has traditionally been treated purely syntactically, and justifications for it have appealed either to intuition or to “free theorems” – even though the latter have not been known to hold in languages supporting higher-order polymorphic functions and fixpoint recursion. In this paper we use Pitts' recent demonstration that contextual equivalence in such languages is parametric to provide the first formal proof of the correctness of short cut fusion for them. In particular, we show that programs which have undergone short cut fusion are contextually equivalent to their unfused counterparts.

A Generalization of Short-Cut Fusion and its Correctness Proof

Short-cut fusion is a program transformation technique that uses a single, local transformation—called the \tt foldr-build rule—to remove certain intermediate lists from modularly constructed functional programs. Arguments that short-cut fusion is correct typically appeal either to intuition or to “free theorems”—even though the latter have not been known to hold for the languages supporting higher-order polymorphic functions and fixed point recursion in which short-cut fusion is usually applied. In this paper we use Pitts' recent demonstration that contextual equivalence in such languages is relationally parametric to prove that programs in them which have undergone short-cut fusion are contextually equivalent to their unfused counterparts. For each algebraic data type we then define a generalization of \tt build which constructs substitution instances of its associated data structures, and use Pitts' techniques to prove the correctness of a contextual equivalence-preserving fusion rule which generalizes short-cut fusion. These rules optimize compositions of functions that uniformly consume algebraic data structures with functions that uniformly produce substitution instances of those data sructures.

Functional programming in C++

This paper describes FC++: a rich library supporting functional programming in C++. Prior approaches to encoding higher order functions in C++ have suffered with respect to polymorphic functions from either lack of expressiveness or high complexity. In contrast, FC++ offers full and concise support for higher-order polymorphic functions through a novel use of C++ type inference.Another new element in FC++ is that it implements a subtype polymorphism policy for functions, in addition to the more common parametric polymorphism facilities. Subtype polymorphism is common in object oriented languages and ensures that functions in FC++ fit well within the C++ object model.Apart from these conceptual differences, FC++ is also an improvement in technical terms over previous efforts in the literature. Our function objects are reference-counted and can be aliased without needing to be copied, resulting in an efficient implementation. The reference-counting mechanism is also exported to the user as a general-purpose replacement of native C++ pointers. Finally, we supply a number of useful functional operators (a large part of the Haskell Standard Prelude) to facilitate programming with FC++. The end result is a library that is usable and efficient, while requiring no extensions to the base C++ language.

Efficient high-level parallel programming

Algorithmic skeletons are polymorphic higher-order functions that represent common parallelization patterns and that are implemented in parallel. They can be used as the building blocks of parallel and distributed applications by embedding them into a sequential language. In this paper, we present a new approach to programming with skeletons. We integrate the skeletons into an imperative host language enhanced with higher-order functions and currying, as well as with a polymorphic type system. We thus obtain a high-level programming language, which can be implemented very efficiently. We then present a compile-time technique for the implementation of the functional features which has an important positive impact on the efficiency of the language. After describing a series of skeletons which work with distributed arrays, we give two examples of parallel algorithms implemented in our language, namely matrix multiplication and Gaussian elimination. Run-time measurements for these and other applications show that we approach the efficiency of message-passing C up to a factor between 1 and 1.5.