Some algorithms make critical internal use of updatable state, even though their external specification is purely functional. Based on earlier work on monads, we present a way of securely encapsulating stateful computations that manipulate multiple, named, mutable objects, in the context of a non-strict, purely-functional language. The security of the encapsulation is assured by the type system, using parametricity. Intriguingly, this parametricity requires the provision of a (single) constant with a rank-2 polymorphic type. 1

Monads are becoming an increasingly important tool for functional programming. Different monads can be used to model a wide range of programming language features. However, real programs typically require a combination of different features, so it is important to have techniques for combining several features in a single monad. In practice, it is usually possible to construct a monad that supports some specific combination of features. However, the techniques used are typically ad-hoc and it is very difficult to find general techniques for combining arbitrary monads. This report gives three general constructions for the composition of monads, each of which depends on the existence of an auxiliary function linking the monad structures of the components. In each case, we establish a set of laws that the auxiliary function must satisfy to ensure that the composition is itself a monad. Using the notation of constructor classes, we describe some specific applications of these constructions. These results are used in the development of a simple expression evaluator that combines exceptions, output and an environment of variable bindings using a composition of three corresponding monads. 1

Abstract It has often been suggested that functional languages provide an excellent basis for programming parallel computer systems. This is largely a result of the lack of side effects which makes it possible to evaluate the subexpressions of a given term without any risk of interference. On the other hand, the lack of side-effects has also been seen as a weakness of functional languages since it rules out many features of traditional imperative languages such as state, I/O and exceptions. These ideas can be simulated in a functional language but the resulting programs are sometimes unnatural and inefficient. On the bright side, recent work has shown how many of these features can be naturally incorporated into a functional language without compromising efficiency by expressing computations in terms of monads or continuations. Unfortunately, the "single-threading " implied by these techniques often destroys many opportunities for parallelism. In this paper, we describe a simple extension to the Haskell I/O monad that allows a form of explicit high-level concurrency. It is a simple matter to incorporate these features in a sequential implementation, and genuine parallelism can be obtained on a parallel machine. In addition, the inclusion of constructs for explicit concurrency enhances the use of Haskell as an executable specification language, since some programs are most naturally described as a composition of parallel processes. \Lambda This research was supported by ARPA via a subcontract to Intermetrics, Inc. 1

Launchbury and Peyton Jones came up with an ingenious idea for embedding regions of imperative programming in a pure functional language like Haskell. The key idea was based on a simple modification of Hindley-Milner's type system. Our first contribution is to propose a more natural encapsulation construct exploiting higher-order kinds, which achieves the same encapsulation effect, but avoids the ad hoc type parameter of the original proposal. The second contribution is a type safety result for encapsulation of strict state using both the original encapsulation construct and the newly introduced one. We establish this result in a more expressive context than the original proposal, namely in the context of the higher-order lambda-calculus. The third contribution is a type safety result for encapsulation of lazy state in the higher-order lambda-calculus. This result resolves an outstanding open problem on which previous proof attempts failed. In all cases, we formalize the intended implementations as simple big-step operational semantics on untyped terms, which capture interesting implementation details not captured by the reduction semantics proposed previously. 1

Many object-oriented languages used in practice descend from Algol. With this motivation, we study the theoretical issues underlying such languages via the theory of Algollike languages. It is shown that the basic framework of this theory extends cleanly and elegantly to the concepts of objects and classes. An important idea that comes to light is that classes are abstract data types, whose theory corresponds to that of existential types. Equational and Hoare-like reasoning methods, and relational parametricity provide powerful formal tools for reasoning about Algol-like object-oriented programs. 1

. In an earlier paper, one of the present authors presented a preliminary account of an equational logic called PIM. PIM is intended to function as a "transformational toolkit" to be used by compilers and analysis tools for imperative languages, and has been applied to such problems as program slicing, symbolic evaluation, conditional constant propagation, and dependence analysis. PIM consists of the untyped lambda calculus extended with an algebraic rewriting system that characterizes the behavior of lazy stores and generalized conditionals. A major question left open in the earlier paper was whether there existed a complete equational axiomatization of PIM's semantics. In this paper, we answer this question in the affirmative for PIM's core algebraic component, PIM t , under the assumption of certain reasonable restrictions on term formation. We systematically derive the complete PIM logic as the culmination of a sequence of increasingly powerful equational systems starti...

The extension of Haskell with a built-in state monad combines mathematical elegance with operational efficiency: ffl Semantically, at the source language level, constructs that act on the state are viewed as functions that pass an explicit store data structure around. ffl 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, to...

It is well known that adding side effects to functional languages changes the operational equivalences of the language. We develop a new language construct, encap, that forces imperative pieces of code to behave purely functionally, i.e., without any visible side effects. The coercion operator encap provides a means of extending the simple reasoning principles for equivalences of code in a functional language to a language with side effects. In earlier work [36], similar coercion operators were developed, but their correctness required the underlying functional language to include parallel operations. The coercion operators developed here are simpler and are proven correct for purely sequential languages. The sequential setting requires the construction of fully abstract models for sequential call-by-value languages and the formulation of a weak form of "monad" suitable for expressing the semantics of call-by-value languages with side effects. 1 Introduction Two pieces of code are...

In a programming language with procedures and assignments, it is often important to isolate uses of state to particular program fragments. The frameworks of type, region, and effect inference, and monadic state are technologies that have been used to state and enforce the property that an expression has no visible side-effects. This property has been exploited to justify the deallocation of memory regions despite the presence of dangling pointers. Starting from an idea developed in the context of monadic state in Haskell, we develop an ML-like language with full assignments and an operator that enforces the encapsulation of effects. Using this language, we formalize and prove the folklore connection between effect masking and monadic encapsulation. Then, by employing a novel set of reductions to deal with dangling pointers, we establish the soundness of the type-based encapsulation with a proof based on a standard subject reduction argument. 1 Introduction Two of the recurring theme...

Functional programming languages are informally classified into pure and impure languages. The precise meaning of this distinction has been a matter of controversy. We therefore investigate a formal definition of purity. We begin by showing that some proposed definitions that rely on confluence, soundness of the beta axiom, preservation of pure observational equivalences, and independence of the order of evaluation, do not withstand close scrutiny. We propose instead a definition based on parameter-passing independence. Intuitively, the definition implies that functions are pure mappings from arguments to results; the operational decision of how to pass the arguments is irrelevant. In the context of Haskell, our definition is consistent with the fact that the traditional call-by-name denotational semantics coincides with the traditional call-by-need implementation. Furthermore, our definition is compatible with the stream-based, continuationbased, and monad-based integration of computa...