Type Inference Algorithm Research Articles

Improving Error Messages in Type System

We propose a type inference algorithm for a polymorphic type system which provides improved error messages. While the standard type inference algorithms often produce unnecessarily long or incomplete error messages, our algorithm provides relevant and complete information for type errors. It is relevant in the sense that all the program points and types in the output of our algorithm contribute to some type error, and is complete in the sense that, for each type error, our algorithm identifies not only two conflicting types, but also all types which conflict with each other. The latter property is particularly useful for debugging programs with lists or case branches. Our algorithm keeps track of the set of program points that are relevant to each type. To achieve completeness, we introduce a new type variable which represents a conflict among two or more incompatible types, and extend the unification algorithm to handle the special type variable appropriately. Finally, we argue that our algorithm is more efficient than those in the literature when there are more than two conflicting types in the given expression.

Flexible types

We present HML, a type inference system that supports full first-class polymorphism where few annotations are needed: only function parameters with a polymorphic type need to be annotated. HML is a simplification of MLF where only flexibly quantified types are used. This makes the types easier to work with from a programmers perspective, and simplifies the implementation of the type inference algorithm. Still, HML retains much of the expressiveness of MLF, it is robust with respect to small program transformations, and has a simple specification of the type rules with an effective type inference algorithm that infers principal types. A small reference implementation with many examples is available at: http://research.microsoft.com/users/daan/pubs.html.

Java type inference is broken

Java 5, the most recent major update to the Java Programming Language, introduced a number of sophisticated features, including a major extension to the type system. While the technical details of these new features are complex, much of this complexity is hidden from the typical Java developer by an ambitious type inference mechanism. Unfortunately, the extensions to the Java 5 type system were so novel that their technical details had not yet been thoroughly investigated in the research literature. As a result, the Java 5 compiler includes a pragmatic but flawed type inference algorithm that is, by design, neither sound nor locally complete. The language specification points out that neither of these failures is catastrophic: the correctness of potentially-unsound results must be verified during type checking; and incompleteness can usually be worked around by manually providing the method type parameter bindings for a given call site. This paper dissects the type inference algorithm of Java 5 and proposes a signficant revision that is sound and able to calculate correct results where the Java 5 algorithm fails. The new algorithm is locally complete with the exception of a difficult corner case. Moreover, the new algorithm demonstrates that several arbitrary restrictions in the Java type system---most notably the ban on lower-bounded type parameter declarations and the limited expressibility of intersection types---are unnecessary. We hope that this work will spur the evolution of a more coherent, more comprehensive generic type system for Java.

FPH

Languages supporting polymorphism typically have ad-hoc restrictions on where polymorphic types may occur. Supporting "firstclass" polymorphism, by lifting those restrictions, is obviously desirable, but it is hard to achieve this without sacrificing type inference. We present a new type system for higher-rank and impredicative polymorphism that improves on earlier proposals: it is an extension of Damas-Milner; it relies only on System F types; it has a simple, declarative specification; it is robust to program transformations; and it enjoys a complete and decidable type inference algorithm.

HMF

HMF is a conservative extension of Hindley-Milner type inference with first-class polymorphism. In contrast to other proposals, HML uses regular System F types and has a simple type inference algorithm that is just a small extension of the usual Damas-Milner algorithm W. Given the relative simplicity and expressive power, we feel that HMF can be an attractive type system in practice. There is a reference implementation of the type system available online together with a technical report containing proofs (Leijen 2007a,b).

Types and trace effects for object orientation

Trace effects are statically generated program abstractions, that can be model checked for verification of assertions in a temporal program logic. In this paper we develop a type and effect analysis for obtaining trace effects of Object Oriented programs in Featherweight Java. We observe that the analysis is significantly complicated by the interaction of trace behavior with inheritance and other Object Oriented features, particularly overridden methods, dynamic dispatch, and downcasting. We propose an expressive type and effect inference algorithm combining polymorphism and subtyping/subeffecting constraints to obtain a flexible trace effect analysis in this setting, and show how these techniques are applicable to Object Oriented features. We also extend the basic language model with exceptions and stack-based event contexts, and show how trace effects scale to these extensions by structural transformations.

Coercions in a polymorphic type system

We incorporate the idea of coercive subtyping, a theory of abbreviation for dependent type theories, into the polymorphic type system in functional programming languages. The traditional type system with let-polymorphism is extended with argument coercions and function coercions, and a corresponding type inference algorithm is presented and proved to be sound and complete.

Lightweight family polymorphism

AbstractFamily polymorphism has been proposed for object-oriented languages as a solution to supporting reusable yet type-safe mutually recursive classes. A key idea of family polymorphism is the notion of families, which are used to group mutually recursive classes. In the original proposal, due to the design decision that families are represented by objects, dependent types had to be introduced, resulting in a rather complex type system. In this article, we propose a simpler solution oflightweightfamily polymorphism, based on the idea that families are represented by classes rather than by objects. This change makes the type system significantly simpler without losing much expressive power of the language. Moreover, “family-polymorphic” methods now take a form of parametric methods; thus, it is easy to apply method type argument inference as in Java 5.0. To rigorously show that our approach is safe, we formalize the set of language features on top of Featherweight Java and prove that the type system is sound. An algorithm for type inference for family-polymorphic method invocations is also formalized and proved to be correct. Finally, a formal translation by erasure to Featherweight Java is presented; it is proved to preserve typing and execution results, showing that our new language features can be implemented in Java by simply extending the compiler.

AspectML

This article defines AspectML, a typed functional, aspect-oriented programming language. The main contribution of AspectML is the seamless integration of polymorphism, run-time type analysis and aspect-oriented programming language features. In particular, AspectML allows programmers to define type-safe polymorphic advice using pointcuts constructed from a collection of polymorphic join points. AspectML also comes equipped with a type inference algorithm that conservatively extends Hindley--Milner type inference. To support first-class polymorphic point-cut designators, a crucial feature for developing aspect-oriented profiling or logging libraries, the algorithm blends the conventional Hindley--Milner type inference algorithm with a simple form of local type inference. We give our language operational meaning via a type-directed translation into an expressive type-safe intermediate language. Many complexities of the source language are eliminated in this translation, leading to a modular specification of its semantics. One of the novelties of the intermediate language is the definition of polymorphic labels for marking control-flow points. When a set of labels is assembled as a pointcut, the type of each label is an instance of the type of the pointcut.

Abstract interpretation and types for systems biology

Abstract interpretation is a theory of abstraction that has been introduced for the analysis of programs. In particular, it has proved useful for organizing the multiple semantics of a given programming language in a hierarchy corresponding to different detail levels, and for defining type systems for programming languages and program analyzers in software engineering. In this paper, we investigate the application of these concepts to systems biology formalisms. More specifically, we consider the Systems Biology Markup Language SBML, and the Biochemical Abstract Machine BIOCHAM with its differential, stochastic, discrete and boolean semantics. We first show how all of these different semantics, except the differential one, can be formally related by simple Galois connections. Then we define three type systems: one for checking or inferring the functions of proteins in a reaction model, one for checking or inferring the activation and inhibition effects of proteins in a reaction model, and another one for checking or inferring the topology of compartments or locations. We show that the framework of abstract interpretation elegantly applies to the formalization of these further abstractions, and to the implementation of linear or quadratic time type checking as well as type inference algorithms. Furthermore, we show a theorem of independence of the graph of activation and inhibition effects from the kinetic expressions in the reaction model, under general conditions. Through some examples, we show that the analysis of biochemical models by type inference provides accurate and useful information. Interestingly, such a mathematical formalization of the abstractions commonly used in systems biology already provides some guidelines for the extensions of biochemical reaction rule languages.

An Optimized Type Inference Algorithm with Lazy Type Substitution

An Optimized Type Inference Algorithm with Lazy Type Substitution

HM(X) type inference is CLP(X) solving

The HM(X) system is a generalization of the Hindley/Milner system parameterized in the constraint domain X. Type inference is performed by generating constraints out of the program text, which are then solved by the domain-specific constraint solver X. The solver has to be invoked at the latest when type inference reaches a let node so that we can build a polymorphic type. A typical example of such an inference approach is Milner's algorithm W. We formalize an inference approach where the HM(X) type inference problem is first mapped to a CLP(X) program. The actual type inference is achieved by executing the CLP(X) program. Such an inference approach supports the uniform construction of type inference algorithms and has important practical consequences when it comes to reporting type errors. The CLP(X) style inference system, where X is defined by Constraint Handling Rules, is implemented as part of the Chameleon system.

Types and trace effects of higher order programs

This paper shows how type effect systems can be combined with model-checking techniques to produce powerful, automatically verifiable program logics for higher order programs. The properties verified are based on the ordered sequence of events that occur during program execution, so-called event traces. Our type and effect systems infer conservative approximations of the event traces arising at run-time, and model-checking techniques are used to verify logical properties of these histories. Our language model is based on the λ-calculus. Technical results include a type inference algorithm for a polymorphic type effect system, and a method for applying known model-checking techniques to the trace effects inferred by the type inference algorithm, allowing static enforcement of history-and stack-based security mechanisms. A type safety result is proven for both unification and subtyping constraint versions of the type system, ensuring that statically well-typed programs do not contain trace event checks that can fail at run-time.

Type Systems for Optimizing Stack-based Code

We give a uniform type-systematic account of a number of optimizations and the underlying analyses for a bytecode-like stack-based low-level language, including analysis soundness proofs and strongest analysis (principal type inference) algorithms. Specifically, we treat dead store instructions, load-pop pairs, duplicating load instructions, store-load pairs. The load-pop pairs and store-load pairs elimination optimizations are built on top of bidirectional analyses, facilitating correct elimination of instruction pairs spanning across basic block boundaries. As a result, no assumptions are needed about input code (it need not be the compiled form of a high-level source program, the stack need not be empty at basic block boundaries and not even need it be checked for safety before the analysis). The soundness proofs and strongest analysis algorithms are simple and uniform.

Modular type classes

ML modules and Haskell type classes have proven to be highly effective tools for program structuring. Modules emphasize explicit configuration of program components and the use of data abstraction. Type classes emphasize implicit program construction and ad hoc polymorphism. In this paper, we show how the implicitly-typed style of type class programming may be supported within the framework of an explicitly-typed module language by viewing type classes as a particular mode of use of modules. This view offers a harmonious integration of modules and type classes, where type class features, such as class hierarchies and associated types, arise naturally as uses of existing module-language constructs, such as module hierarchies and type components. In addition, programmers have explicit control over which type class instances are available for use by type inference in a given scope. We formalize our approach as a Harper-Stone-style elaboration relation, and provide a sound type inference algorithm as a guide to implementation.

A static type system for JVM access control

This article presents a static type system for the Java virtual machine (JVM) code that enforces an access control mechanism similar to that found in a Java implementation. In addition to verifying type consistency of a given JVM code, the type system statically verifies whether the code accesses only those resources that are granted by the prescribed access policy. The type system is proved to be sound with respect to an operational semantics that enforces access control dynamically, similar to Java stack inspection. This result ensures that “well-typed code cannot violate access policy.” The authors then develop a type inference algorithm and show that it is sound with respect to the type system. These results allow us to develop a static system for JVM access control, without resorting to costly runtime stack inspection.

Type inference against races

The race condition checker rccjava uses a formal type system to statically identify potential race conditions in concurrent Java programs, but it requires programmer-supplied type annotations. This paper describes a type inference algorithm for rccjava. Due to the interaction of parameterized classes and dependent types, this type inference problem is NP-complete. This complexity result motivates our new approach to type inference, which is via reduction to propositional satisfiability. This paper describes our type inference algorithm and its performance on programs of up to 30,000 lines of code.

OCaml + XDuce

This paper presents the core type system and type inference algorithm of OCamlDuce, a merger between OCaml and XDuce. The challenge was to combine two type checkers of very different natures while preserving the best properties of both (principality and automatic type reconstruction on one side; very precise types and implicit subtyping on the other side). Type inference can be described by two successive passes: the first one is an ML-like unification-based algorithm which also extracts data flow constraints about XML values; the second one is an XDuce-like algorithm which computes XML types in a direct way. An optional preprocessing pass, called strengthening, can be added to allow more implicit use of XML subtyping. This pass is also very similar to an ML type checker.

Simple unification-based type inference for GADTs

Generalized algebraic data types (GADTs), sometimes known as "guarded recursive data types" or "first-class phantom types", are a simple but powerful generalization of the data types of Haskell and ML. Recent works have given compelling examples of the utility of GADTs, although type inference is known to be difficult. Our contribution is to show how to exploit programmer-supplied type annotations to make the type inference task almost embarrassingly easy. Our main technical innovation is wobbly types , which express in a declarative way the uncertainty caused by the incremental nature of typical type-inference algorithms.

Resource Usage Analysis for the Pi-Calculus

We propose a type-based resource usage analysis for the π-calculus extended with resource creation/access primitives. The goal of the resource usage analysis is to statically check that a program accesses resources such as files and memory in a valid manner. Our type system is an extension of previous behavioral type systems for the π-calculus, and can guarantee the safety property that no invalid access is performed, as well as the property that necessary accesses (such as the close operation for a file) are eventually performed unless the program diverges. A sound type inference algorithm for the type system is also developed to free the programmer from the burden of writing complex type annotations. Based on the algorithm, we have implemented a prototype resource usage analyzer for the π-calculus. To the authors' knowledge, ours is the first type-based resource usage analysis that deals with an expressive concurrent language like the pi-calculus.