Monadic encapsulation of effects: a revised approach (extended version)

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.

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.

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 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 )).

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.

We present an unboxed operational semantics for an ML-style polymorphic language. Different from the conventional formalisms, the proposed semantics accounts for actual representations of run-time objects of various types, and supports a refined notion of polymorphism that allows polymorphic functions to be applied directly to values of various different representations. In particular, polymorphic functions can receive multi-word constants such as floating-point numbers without requiring them to be “boxed” (i.e., heap allocated.) This semantics will serve as an alternative basis for implementing polymorphic languages. The development of the semantics is based on the technique of the type-inference-based compilation for polymorphic record operations [20]. We first develop a lower-level calculus, called a polymorphic unboxed calculus, that accounts for direct manipulation of unboxed values in a polymorphic language. This unboxed calculus supports efficient value binding through integer representation of variables. Different from de Bruijn indexes, our integer representation of a variable corresponds to the actual offset to the value in a run-time environment consisting of objects of various sizes. Polymorphism is supported through an abstraction mechanism over argument sizes. We then develop an algorithm that translates ML into the polymorphic unboxed calculus by using type information obtained through type inference. At the time of polymorphic let binding, the necessary size abstractions are inserted so that a polymorphic function is translated into a function that is polymorphic not only in the type of the argument but also in its size. The ML type system is shown to be sound with respect to the operational semantics realized by the translation.

Multi-stage programming languages provide a convenient notation for explicitly staging programs. Staging a definitional interpreter for a domain specific language is one way of deriving an implementation that is both readable and efficient. In an untyped setting, staging an interpreter removes a complete layer of interpretive overhead, just like partial evaluation. In a typed setting however, Hindley-Milner type systems do not allow us to exploit typing information in the language being interpreted. In practice, this can mean a slowdown cost by a factor of three or mor.Previously, both type specialization and tag elimination were applied to this problem. In this paper we propose an alternative approach, namely, expressing the definitional interpreter in a dependently typed programming language. We report on our experience with the issues that arise in writing such an interpreter and in designing such a language. .To demonstrate the soundness of combining staging and dependent types in a general sense, we formalize our language (called Meta-D) and prove its type safety. To formalize Meta-D, we extend Shao, Saha, Trifonov and Papaspyrou's λH language to a multi-level setting. Building on λH allows us to demonstrate type safety in a setting where the type language contains all the calculus of inductive constructions, but without having to repeat the work needed for establishing the soundness of that system.

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. >

Pure Pattern Type Systems (P 2 TS) combine in a unified setting the capabilities of rewriting and λ-calculus. Their type systems, adapted from Barendregt’s λ-cube are especially interesting from a logical point of view.Strong normalization, an essential property for logical soundness, had only been conjectured so far: in this paper, we give a positive answer for the simply-typed system.The proof is based on a translation of terms and types from (P 2 TS) into the λ-calculus. First, we deal with untyped terms, ensuring that reductions are faithfully mimicked in the λ-calculus. For this, we rely on an original encoding of the pattern matching capability of (P 2 TS) into the λ-calculus.Then we show how to translate types: the expressive power of System Fω is needed in order to fully reproduce the original typing judgments of (P 2 TS) We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalization of simply-typed (P 2 TS) terms.KeywordsType SystemPattern MatchFree VariableProduct RuleExpressive PowerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Finite subtype inference occupies a middle ground between Hindley-Milner type inference (as in ML) and subtype inference with recursively constrained types. It refers to subtype inference where only finite types are allowed as solutions. This approach avoids some open problems with general subtype inference, and has practical motivation where recursively constrained types are not appropriate. This paper presents algorithms for finite subtype inference, including checking for entailment of inferred types against explicitly declared polymorphic types. This resolves for finite types a problem that is still open for recursively constrained types. Some motivation for this work, particularly for finite types and explicit polymorphism, is in providing subtype inference for first-class container objects with polymorphic methods.

This thesis proposes a novel way of performing type checking, whose results are incremental, depending on the provided local information. This new way of type checking is called co-contextual, where all context information of expressions, methods, classes, etc., is removed. Instead, we introduce corresponding structures using requirements. Standard type systems are translated to the co-contextual ones systematically using dualism as technique. Type systems play an important role to prevent execution errors from occurring during runtime. They are used to check programs statically for potential errors. Programs are type checked against a given set of rules. Depending on these rules programs are well-typed or not. The set of these rules is called typing rules. Each type rule associates types to the constructs of a program given a certain context. There can be different forms of contexts, depending on the features of the typed programming language. Functional languages use a typing context of variables and their types; object-oriented (OO) languages use additional class tables. Class tables are used for example to ensure that method and class declarations are well-typed. Type checking is performed top-down. While traversing the syntax tree of a program, typing contexts are extended with information on the expressions and their types. In case of OO, class tables are extended with clauses from class declarations, including the class members, i.e., fields, methods, or constructors. Contexts are passed through the nodes of the syntax tree in order to coordinate type checking between them. Therefore, while traversing the syntax tree top-down, the type checker creates dependencies between otherwise independent subexpressions. This way, it inhibits incrementalization and parallelization of type checking. That is, a change to a node of the syntax tree would require to redo the type check of the whole syntax tree. In this thesis a novel formulation of type systems is proposed, in order to remove dependencies between subexpressions. We propose a co-contextual formulation of typing rules that depends only on the local program constructs, e.g., expressions, methods, classes. The co-contextual typing rules have as conclusion a type and sets of requirements. That is, contexts and class tables are replaced by the dual concept of context and class table requirements. In addition, operations on contexts and class tables are replaced by new dual operations on requirements. The co-contextual type checker traverses a syntax tree bottom-up and merges context requirements of independently checked subexpressions. We describe a method for systematically constructing a co-contextual formulation of type rules from a regular context-based formulation and we show how co-contextual type rules give rise to incremental type checking. We derive co-contextual type checkers for functional and OO languages. As a representative of functional languages we consider PCF and extensions of it: records, parametric polymorphism, structural subtyping and let-polymorphism. Also, we investigate featherweight java (FJ) as the basis of OO languages and extensions of it: method overloading and generics. We build a1 co-contextual type checker for FJ enabling key features of OO languages: subtype polymorphism, nominal typing and implementation inheritance. The dualism between the co-contextual and contextual type systems preserves the correctness of the contextual calculus. That is, we prove the correctness of the co-contextual calculus via the equivalence between contextual type rules and their co-contextual formulations. We implemented an incremental type checker for PCF along with a performance evaluation showing that co-contextual type checking has performance comparable to standard context-based type checking, and incrementalization can improve performance significantly. Regarding FJ, we implemented a co-contextual type checker with incrementalization and compared its performance against javac on a number of realistic programs. Our performance evaluation shows significant speedups for the co-contextual type checker with incrementalization in comparison to javac.

We explore the use of dependent types to address the disparity between the theory and the practical hardware description of DSP circuits. After discussing an approach to modeling synchronous circuit behaviour in Idris (a pure functional language with dependent types), two DSP case studies are introduced — an FIR filter with optimal wordlengths and a CIC decimator with register pruning. Both of these scenarios prove difficult to describe in a parameterised fashion using traditional HDLs and, as such, many implementations rely on ad hoc circuit generators which are challenging to test and evaluate. This work demonstrates that such circuits are readily described in an environment with dependent types. Dependent types can also encode various contracts between the IP designer and its user. These contracts are automatically verified by the Idris type checker before compilation, precluding many common mistakes in IP development and evaluation.

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

Asoft type systeminfers types for the procedures and data structures of dynamically typed programs. Like conventional static types, soft types express program invariants and thereby provide valuable information for program optimization and debugging. A soft typecheckeruses the types inferred by a soft type system to eliminate run-time checks that are provably unnecessary; any remaining run-time checks are flagged as potential program errors.Soft Schemeis a practical soft type checker for R4RS Scheme. Its underlying type system generalizes conventional Hindley-Milner type inference by incorporating recursive types and a limited form of union type. Soft Scheme accommodates all of R4RS Scheme including uncurried procedures of fixed and variable arity, assignment, and continuations.

We present a translation of Ada configuration constructs, in a higher order, impredicatively typed, functional language (HOTFUL) with subtypes. The aim of this work is to provide an expressive executable semantics for Ada configuration constructs, and to verify the suitability of the chosen HOTFUL for such a task. In particular, we address the practicability of the approach when dealing with the development of a whole complex system, as well as the description of single modular units. After giving the detailed rules for the translation, we compare our approach with what could be obtained selecting a different typed language as “target”, namely the predicative type system of Standard ML.