An Unboxed Operational Semantics for ML Polymorphism

Extensions of the ML type system, based on constrained type schemes, have been proposed for languages with overloading. Type inference in these systems requires solving the following satisfiability problem. Given a set of type assumptions C over finite types and a type basis A, is there is a substitution S that satisfies C in that A implies that CS is derivable? Under arbitrary overloading, the problem is undecidable. Haskell limits overloading to a form similar to that proposed by Kaes (1988) called parametric overloading. We formally characterize parametric overloading in terms of a regular tree language and prove that although decidable, satisfiability is NP-hard when overloading is parametric. >

Recent research has demonstrated the effectiveness of extending the Hindley-Milner (HM) type system with Boolean kinds to support type inference for a wide variety of features. However, the means to support classic type system provisions such as local let generalization and polymorphic recursion is either limited or unknown for such extensions. This paper contributes procedures for equational generalization and semiunification in arbitrary Boolean rings, enabling let generalization and polymorphic recursion in Boolean-kinded type inference. Additionally, methods to minimize the number of bound Boolean type variables are developed to keep types small in these systems. Boolean-kinded HM extensions are exemplified with nullable reference types , and how to use the developed procedures to support let generalization and polymorphic recursion is outlined.

We investigate finite-rank intersection type systems, analyzing the complexity of their type inference problems and their relation to the problem of recognizing semantically equivalent terms. Intersection types allow something of type τ 1 Λ τ 2 to be used in some places at type τ 1 and in other places at type τ 2 . A finite-rank intersection type system bounds how deeply the Λ can appear in type expressions. Such type systems enjoy strong normalization, subject reduction, and computable type inference, and they support a pragmatics for implementing parametric polymorphism. As a consequence, they provide a conceptually simple and tractable alternative to the impredicative polymorphism of System F and its extensions, while typing many more programs than the Hindley-Milner type system found in ML and Haskell.While type inference is computable at every rank, we show that its complexity grows exponentially as rank increases. Let K (0, n ) = n and K ( t + 1, n ) = 2 K ( t,n ) ; we prove that recognizing the pure λ-terms of size n that are typable at rank k is complete for DTIME [ K ( k −1, n )]. We then consider the problem of deciding whether two λ-terms typable at rank k have the same normal form, generalizing a well-known result of Statman from simple types to finite-rank intersection types. We show that the equivalence problem is DTIME [ K ( K ( k − 1, n ), 2)]-complete. This relationship between the complexity of typability and expressiveness is identical in wellknown decidable type systems such as simple types and Hindley-Milner types, but seems to fail for System F and its generalizations. The correspondence gives rise to a conjecture that if Τ is a predicative type system where typability has complexity t ( n ) and expressiveness has complexity e ( n ), then t ( n ) = Ω(log* e ( n )).

Hindley and Milner's type system, in its purest form, allows what one might call full type inference. Even when a program contains no explicit type information, its very structure gives rise to a conjunction of type equations (or, more generally, to a constraint) whose resolution allows reconstructing the type of every variable and of every sub-expression. Actual programming languages based on this type system, such as Standard ML, Objective Caml, and Haskell, do allow users to provide explicit type information via optional type annotations. Yet, this extra information does not make type inference any easier: it simply gives rise to extra equations, leading to a more specific constraint.There are extensions of Hindley and Milner's type system where full type inference becomes undecidable or intractable. Then, it is common to require explicit type information, via mandatory type annotations. Polymorphic recursion is a simple and well-known instance of this phenomenon. This talk presents two more elaborate instances, commonly known as arbitrary-rank polymorphism and generalized algebraic data types.In both cases, type inference is made tractable thanks to mandatory type annotations. There is a twist, though: these required annotations are often numerous and redundant. So, it seems desirable to make them optional again, and to set up a distinct mechanism to guess some of the elided annotations. This mechanism, referred to as elaboration, can be heuristic. It must be predictable. In the two cases considered in this talk, it is a form of local type inference.This approach leads to a so-called stratified type inference system, where type information is propagated first in a local, ad hoc manner during elaboration, then in a possibly nonlocal, but well-specified manner during constraint solving. This appears to be one of the less displeasing ways of describing type inference systems that are able to exploit optional type annotations to infer better (or different) types.

Developing a static type system suitable for Erlang has been of ongoing interest for almost two decades now. The challenge with retrofitting a static type system onto a dynamically typed language, such as Erlang, is the loss of flexibility in programming offered by the language. In light of this, many attempts to type Erlang trade sound type checking for the ability to retain flexibility. Hence, simple type errors which would be caught by the type checker of a statically typed language are easily missed in these developments. This has us wishing for a way to avoid such errors in Erlang programs. In this paper, we develop a static type system for Erlang which strives to remain sound without being too restrictive. Our type system is based on Hindley-Milner type inference, however it---unlike contemporary implementations of Hindley-Milner---is flexible enough to allow overloading of data constructors, branches of different types etc. Further, to allow Erlang's dynamic type behaviour, we employ a program specialization technique called partial evaluation. Partial evaluation simplifies programs prior to type checking, and hence enables the type system to type such behaviour under certain restricted circumstances.

We show how to combine a general purpose type system for an existing language with support for programming with binders and contexts by refining the type system of ML with a restricted form of dependent types where index objects are drawn from contextual LF. This allows the user to specify formal systems within the logical framework LF and index ML types with contextual LF objects. Our language design keeps the index language generic only requiring decidability of equality of the index language providing a modular design. To illustrate the elegance and effectiveness of our language, we give programs for closure conversion and normalization by evaluation. Our three key technical contribution are: 1) We give a bi-directional type system for our core language which is centered around refinement substitutions instead of constraint solving. As a consequence, type checking is decidable and easy to trust, although constraint solving may be undecidable. 2) We give a big-step environment based operational semantics with environments which lends itself to efficient implementation. 3) We prove our language to be type safe and have mechanized our theoretical development in the proof assistant Coq using the fresh approach to binding.

The type systems of most typed functional programming languages are based on the Hindley-Milner type system. A practical problem with these type systems is that it is often hard to understand why a program is not type correct or a function does not have the intended type. We suggest that at the core of this problem is the difficulty of explaining why a given expression has a certain type. The type system is not defined compositionally. We propose to explain types using a variant of the Hindley-Milner type system that defines a compositional type explanation graph of principal typing. We describe how the programmer understands types by interactive navigation through the explanation graph. Furthermore, the explanation graph can be the foundation for algorithmic debugging of type errors, that is, semi-automatic localisation of the source of a type error without even having to understand the type inference steps. We implemented a prototype of a tool to explore the usefulness of the proposed methods.

Article Free Access Share on Refinement types for ML Authors: Tim Freeman School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania School of Computer Science, Carnegie Mellon University, Pittsburgh, PennsylvaniaView Profile , Frank Pfenning School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania School of Computer Science, Carnegie Mellon University, Pittsburgh, PennsylvaniaView Profile Authors Info & Claims PLDI '91: Proceedings of the ACM SIGPLAN 1991 conference on Programming language design and implementationMay 1991 Pages 268–277https://doi.org/10.1145/113445.113468Online:01 May 1991Publication History 293citation1,256DownloadsMetricsTotal Citations293Total Downloads1,256Last 12 Months93Last 6 weeks10 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

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.

Types are a good starting point for various software reengineering tasks. Unfortunately, programs requiring reengineering most desperately are written in languages without an adequate type system (such as COBOL). To solve this problem, we propose a method of automated type inference for these languages. The main ingredients are that if variables are compared using some relational operator their types must be the same; likewise if an expression is assigned to a variable, the type of the expression must be a subtype of that of the variable. We present the formal type system and inference rules for this approach, show their effect on various real life COBOL fragments, describe the implementation of our ideas in a prototype type inference tool for COBOL, and discuss a number of applications.

In this paper, we give a general framework for the foundation of an operational (small step) semantics of object-based languages with an emphasis on functional and imperative issues. The framework allows classifying very naturally many object-based calculi according to their main implementation techniques of inheritance, namely delegation and embedding, and their particular strategies. This distinction comes easily from a choice in the rules. Our framework is founded on two previous works: λObj/+, a version of the Lambda Calculus of Objects of Fischer, Honsell, and Mitchell, for the object aspects, and λσaw of Benaissa, Lescanne, and Rose, for the description of the operational semantics and sharing. The former is the formalization of a small delegation-based language which contains both lambda calculus and object primitives to create, update, and send messages to objects, while the latter is designed to provide a generic description of functional language implementations and is based on a calculus of explicit substitution extended with addresses to deal with memory management. The framework is presented as a set of modules, each of which captures a particular aspect of object-calculi (functional vs. imperative, delegation vs. embedding, and any combination of them). Above all, it introduces and illustrates a new promising approach to formally reason about the operational semantics of languages with (possibly) mutable states.KeywordsDesign of functional and imperative object-oriented languagesoperational semanticsimplementation issuesmemory management

For many compiled languages, source-level types are erased very early in the compilation process. As a result, further compiler passes may convert type-safe source into type-unsafe machine code. Type-unsafe idioms in the original source and type-unsafe optimizations mean that type information in a stripped binary is essentially nonexistent. The problem of recovering high-level types by performing type inference over stripped machine code is called type reconstruction, and offers a useful capability in support of reverse engineering and decompilation. In this paper, we motivate and develop a novel type system and algorithm for machine-code type inference. The features of this type system were developed by surveying a wide collection of common source- and machine-code idioms, building a catalog of challenging cases for type reconstruction. We found that these idioms place a sophisticated set of requirements on the type system, inducing features such as recursively-constrained polymorphic types. Many of the features we identify are often seen only in expressive and powerful type systems used by high-level functional languages. Using these type-system features as a guideline, we have developed Retypd: a novel static type-inference algorithm for machine code that supports recursive types, polymorphism, and subtyping. Retypd yields more accurate inferred types than existing algorithms, while also enabling new capabilities such as reconstruction of pointer const annotations with 98% recall. Retypd can operate on weaker program representations than the current state of the art, removing the need for high-quality points-to information that may be impractical to compute.

For many compiled languages, source-level types are erased very early in the compilation process. As a result, further compiler passes may convert type-safe source into type-unsafe machine code. Type-unsafe idioms in the original source and type-unsafe optimizations mean that type information in a stripped binary is essentially nonexistent. The problem of recovering high-level types by performing type inference over stripped machine code is called type reconstruction, and offers a useful capability in support of reverse engineering and decompilation. In this paper, we motivate and develop a novel type system and algorithm for machine-code type inference. The features of this type system were developed by surveying a wide collection of common source- and machine-code idioms, building a catalog of challenging cases for type reconstruction. We found that these idioms place a sophisticated set of requirements on the type system, inducing features such as recursively-constrained polymorphic types. Many of the features we identify are often seen only in expressive and powerful type systems used by high-level functional languages. Using these type-system features as a guideline, we have developed Retypd: a novel static type-inference algorithm for machine code that supports recursive types, polymorphism, and subtyping. Retypd yields more accurate inferred types than existing algorithms, while also enabling new capabilities such as reconstruction of pointer const annotations with 98% recall. Retypd can operate on weaker program representations than the current state of the art, removing the need for high-quality points-to information that may be impractical to compute.

Using type systems, this paper treats heap slicing which is a technique transforming a program into a new one that produces the same result while working on a heap sliced into independent regions. Heap slicing is a common approach to handle the problem of modifying the heap layout without changing the program semantics. Heap slicing has applications in the areas of performance optimization and security.Towards solving the problem of heap slicing, this paper introduces three type systems. The first type system does a pointer analysis and annotates program points with pointer information. This type system is an augmentation of a previously developed type system by the author. The second type system does a region analysis and refines the result of the first type system by augmenting the pointer information with region information. The region information approximately specifies at each program point for each memory cell the region where the cell exists. The third type system uses the information gathered by the region type system to do the principal transformation of heap slicing.The paper also presents two operational semantics; one for single-region heap scenario and the other for multi-regions heap scenario. These semantics are used to prove the soundness of the type systems.Keywordsheap slicingtype systemssemantics of programming languagesoperational semanticsregion analysispointer analysis

A mixin module is a programming abstraction that simultaneously generalizes λ-abstractions, records, and mutually recursive definitions. Although various mixin module type systems have been developed, no one has investigated principal typings or developed type inference for first-class mixin modules, nor has anyone added Milner's let-polymorphism to such a system.This paper proves that typability is NP-complete for the naive approach followed by previous mixin module type systems. Because a λ-calculus extended with record concatenation is a simple restriction of our mixin module calculus, we also prove the folk belief that typability is NP-complete for the naive early type systems for record concatenation.To allow feasible type inference, we present Martini, a new system of simple types for mixin modules with principal typings . Martini is conceptually simple, with no subtyping and a clean and balanced separation between unification-based type inference with type and row variables and constraint solving for safety of linking and field extraction. We have implemented a type inference algorithm and we prove its complexity to be O ( n 2 ), or O ( n ) given a fixed bound on the number of field labels. To prove the complexity, we need to present an algorithm for row unification that may have been implemented by others, but which we could not find written down anywhere. Because Martini has principal typings, we successfully extend it with Milner's let-polymorphism.