Higher-order Functional Programs Research Articles

Higher-order functional languages and intensional logic

In this paper we demonstrate that a broad class of higher-order functional programs can be transformed into semantically equivalent multidimensional intensional programs that contain only nullary variable definitions. The proposed algorithm systematically eliminates user-defined functions from the source program, by appropriately introducing context manipulation (i.e. intensional) operators. The transformation takes place in M steps, where M is the order of the initial functional program. During each step the order of the program is reduced by one, and the final outcome of the algorithm is an M-dimensional intensional program of order zero. As the resulting intensional code can be executed in a purely tagged-dataflow way, the proposed approach offers a promising new technique for the implementation of higher-order functional languages.

Higher-order narrowing with definitional trees

Functional logic languages with a sound and complete operational semantics are mainly based on an inference rule called narrowing. Narrowing extends functional evaluation by goal solving capabilities, as in logic programming. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for first-order functional logic programs due to its optimality properties wrt the length of derivations and the number of computed solutions. In this paper, we extend the needed narrowing strategy to higher-order functions and λ-terms as data structures. By the use of definitional trees, our strategy computes only independent solutions. Thus, it is the first calculus for higher-order functional logic programming which provides for such an optimality result. Since we allow higher-order logical variables denoting λ-terms, applications go beyond current functional and logic programming languages. We show soundness and completeness of our strategy with respect to LNT reductions, a particular form of higher-order reductions defined via definitional trees. A general completeness result is only provided for terminating rewrite systems due to the lack of an overall theory of higher-order reduction which is outside the scope of this paper.

Linear-time hierarchies for a functional language machine model

In STOC 1993, Jones showed the existence of a hierarchy within problems decidable in linear time by a canonical first-order functional language based on tree-structured data (F), as well as for an extension of that language based on graph-structured data maintained through selective updating (F su). In this paper, we prove the existence of a linear-time hierarchy for an authentic and realistic intermediate “machine” language featuring higher order constructs: the Categorical Abstract Machine. We show the existence of such a hierarchy for the Categorical Abstract Machine based on tree-structured data (CAM) as well as on graph-structured data (CAM su). The existence is shown by constructing mutually efficient interpreters between CAM and F, and CAM su and F su, two robustness results establishing that first-order functional programs written in F, and in F su, define the same class of linear-time decidable problems as the higher-order functional programs run in CAM, and in CAM su, respectively.

A type-based escape analysis for functional languages

An important issue faced by implementors of higher-order functional programming languages is the allocation and deallocation of storage for variables. The possibility of variables escaping their scope during runtime makes traditional stack allocation inadequate. We consider the problem of detecting when variables in such languages do not escape their scope, and thus can have their bindings allocated in an efficient manner. We use an annotated type system to infer information about the use of variables in a higher-order, strict functional language and combine this system with a translation to an annotated language which explicitly indicates which variables do not escape. The type system uses a notion of annotated types which extends the traditional simple type system with information about the extent of variables. To illustrate the use of this information we define an operational semantics for the annotated language which supports both stack and environment allocation of variable bindings. Only the stack allocated bindings need follow the protocol for stacks: their extent may not exceed their scope. Environment allocated bindings can have any extent, and their allocation has no impact on the stack allocated ones. We prove the analysis and translation correct with respect to this operational semantics by adapting a traditional type consistency proof to our setting. We have encoded the proof into the Elf programming language and typechecked it, providing a partially machine-checked proof.

Monadic Type Systems: Pure Type Systems for Impure Settings (Preliminary Report)

Pure type systems and computational monads are two parameterized frameworks that have proved to be quite useful in both theoretical and practical applications. We join the foundational concepts of both of these to obtain monadic type systems. Essentially, monadic type systems inherit the parameterized higher-order type structure of pure type systems and the monadic term and type structure used to capture computational effects in the theory of computational monads. We demonstrate that monadic type systems nicely characterize previous work and suggest how they can support several new theoretical and practical applications.A technical foundation for monadic type systems is laid by recasting and scaling up the main results from pure type systems (confluence, subject reduction, strong normalisation for particular classes of systems, etc.) and from operational presentations of computational monads (notions of operational equivalence based on applicative similarity, co-induction proof techniques).We demonstrate the use of monadic type systems with case studies of several call-by-value and call-by-name systems. Our framework allows to capture the restriction to value polymorphism in the type structure and is flexible enough to accommodate extensions of the type system, e.g., with higher-order polymorphism. The theoretical foundations make monadic type systems well-suited as a typed intermediate language for compilation and specialization of higher-order, strict and non-strict functional programs. The monadic structure guarantees sound compile-time optimizations and the parameterized type structure guarantees sufficient expressiveness.

Cogen in six lines

We have designed and implemented a program-generator generator (PGG) for an untyped higher-order functional programming language. The program generators perform continuation-based multi-level offline specialization and thus combine the most powerful and general offline partial evaluation techniques. The correctness of the PGG is ensured by deriving it from a multi-level specialize. Our PGG is extremely simple to implement due to the use of multi-level techniques and higher-order abstract syntax.

Total correctness by local improvement in the transformation of functional programs

The goal of program transformation is to improve efficiency while preserving meaning. One of the best-known transformation techniques is Burstall and Darlington's unfold-fold method. Unfortunately the unfold-fold method itself guarantees neither improvement in efficiency nor total correctness. The correctness problem for unfold-fold is an instance of a strictly more general problem: transformation by locally equivalence-preserving steps does not necessarily preserve (global) equivalence. This article presents a condition for the total correctness of transformations on recursive programs, which, for the first time, deals with higher-order functional languages (both strict and nonstrict) including lazy data structures. The main technical result is an improvement theorem which says that if the local transformation steps are guided by certain optimization concerns (a fairly natural condition for a transformation), then correctness of the transformation follows. The improvement theorem makes essential use of a formalized improvement theory; as a rather pleasing corollary it also guarantees that the transformed program is a formal improvement over the original. The theorem has immediate practical consequences: it is a powerful tool for proving the correctness of existing transformation methods for higher-order functional programs, without having to ignore crucial factors such as memoization or folding , and it yields a simple syntactic method for guiding and constraining the unfold-fold method in the general case so that total correctness (and improvement) is always guaranteed.

Proving the correctness of recursion-based automatic program transformations

This paper shows how the Improvement Theorem — a semantic condition for establishing the total correctness of program transformation on higher-order functional programs — has practical value in proving the correctness of automatic techniques. To this end we develop and study a family of automatic program transformations. The root of this family is a well-known and widely studied transformation called deforestation; descendants include generalisations to richer input languages (e.g. higher-order functions), and more powerful transformations, including a source-level representation of some of the techniques known from Turchin's super compiler.

Lexical profiling: theory and practice

AbstractThis paper addresses the issue of analysing the run-time behaviour of lazy, higher-order functional programs. We examine the difference between the way that functional programmers and functional language implementors view program behaviour. Existing profiling techniques are discussed and a new technique is proposed which produces results that are straightforward for programmers to assimilate. The new technique, which we call lexical profiling, collects information about the run-time behaviour of functional programs, and reports the results with respect to the original source code rather than simply listing the actions performed at run-time. Lexical profiling complements implementation-specific profiling and is important because it provides a view of program activity which is largely independent of the underlying evaluation mechanism. Using the lexical profiler, programmers may easily relate results back to the source program. We give a full implementation of the lexical profiling technique for a sequential, interpretive graph reduction engine, and extensions for compiled and parallel graph reduction are discussed.

Closure analysis in constraint form

Flow analyses of untyped higher-order functional programs have in the past decade been presented by Ayers, Bondorf, Consel, Jones, Heintze, Sestoft, Shivers, Steckler, Wand, and others. The analyses are usually defined as abstract interpretations and are used for rather different tasks such as type recovery, globalization, and binding-time analysis. The analyses all contain a global closure analysis that computes information about higher-order control-flow. Sestoft proved in 1989 and 1991 that closure analysis is correct with respect to call-by-name and call-by-value semantics, but it remained open if correctness holds for arbitrary beta-reduction. This article answers the question; both closure analysis and others are correct with respect to arbitrary beta-reduction. We also prove a subject-reduction result: closure information is still valid after beta-reduction. The core of our proof technique is to define closure analysis using a constraint system. The constraint system is equivalent to the closure analysis of Bondorf, which in turn is based on Sestoft's.

Conjunctive Type Systems and Abstract Interpretation of Higher-order Functional Programs

Conjunctive Type Systems and Abstract Interpretation of Higher-order Functional Programs

Safe fusion of functional expressions II: Further improvements

AbstractLarge functional programs are often constructed by decomposing each big task into smaller tasks which can be performed by simpler functions. This hierarchical style of developing programs has been found to improve programmers' productivity because smaller functions are easier to construct and reuse. However, programs written in this way tend to be less efficient. Unnecessary intermediate data structures may be created. More function invocations may be required.To reduce such performance penalties, Phil Wadler proposed a transformation algorithm, called deforestation, which could automatically fuse certain composed expressions together to eliminate intermediate tree-like data structures. However, his technique is currently safe (terminates with no loss of efficiency) for only a subset of first-order expressions.This paper will generalise the deforestation technique to make it safe for all first-order and higher-order functional programs. Our generalisation is explained using a model for safe fusion which views each function as a producer and its parameters as consumers. Through this model, syntactic program properties are proposed to classify producers and consumers as either safe or unsafe. This classification is used to identify sub-terms that can be safely fused/eliminated. We present the generalised transformation algorithm, illustrate it with examples and provide a termination proof for the transformation algorithm of first-order programs. This paper also contains a suite of additional techniques to further improve the basic safe fusion method. These improvements could be viewed as enhancements to compensate for some inadequacies of the syntactic analyses used.

Safe fusion of functional expressions

Large functional programs are often constructed by decomposing each big task into smaller tasks which can be performed by simpler functions. This hierarchical style of developing programs has been found to improve programmers' productivity because smaller functions are easier to construct and reuse. However, programs written in this way tend to be less efficient. Unnecessary intermediate data structures may be created. More function invocations may be required. To reduce such performance penalties, Wadler proposed a transformation algorithm, called deforestation , which could automatically fuse certain composed expressions together in order to eliminate intermediate tree-like data structures. However, his technique is only applicable to a subset of first-order expressions. This paper will generalise the deforestation technique to make it applicable to all first-order and higher-order functional programs. Our generalisation is made possible by the adoption of a model for safe fusion which views each function as a producer and its parameters as consumers. Through this model, static program properties are proposed to classify producers and consumers as either safe or unsafe. This classification is used to identify sub-terms that can be safely fused/eliminated. We present the generalised transformation algorithm as a set of syntax-directed rewrite rules, illustrate it with examples, and provide an outline of its termination proof.

The use of combinators in translating a purely functional language to low-level data-flow graphs

FCL is a higher-order functional programming language which consolidates and extends a number of desirable features of existing languages. This paper describes the salient features of FCL and an algorithm for translation to highly parallel data flow graphs. The translation algorithm is based on a set of extended “combinators”. The relationship between functional programming languages and demand-driven or data-driven data flow architectures is established.