Decidable Type Inference Research Articles

The best of both worlds: linear functional programming without compromise

We present a linear functional calculus with both the safety guarantees expressible with linear types and the rich language of combinators and composition provided by functional programming. Unlike previous combinations of linear typing and functional programming, we compromise neither the linear side (for example, our linear values are first-class citizens of the language) nor the functional side (for example, we do not require duplicate definitions of compositions for linear and unrestricted functions). To do so, we must generalize abstraction and application to encompass both linear and unrestricted functions. We capture the typing of the generalized constructs with a novel use of qualified types. Our system maintains the metatheoretic properties of the theory of qualified types, including principal types and decidable type inference. Finally, we give a formal basis for our claims of expressiveness, by showing that evaluation respects linearity, and that our language is a conservative extension of existing functional calculi.

Implicit self-adjusting computation for purely functional programs

AbstractComputational problems that involve dynamic data, such as physics simulations and program development environments, have been an important subject of study in programming languages. Building on this work, recent advances in self-adjusting computation have developed techniques that enable programs to respond automatically and efficiently to dynamic changes in their inputs. Self-adjusting programs have been shown to be efficient for a reasonably broad range of problems, but the approach still requires an explicit programming style, where the programmer must use specific monadic types and primitives to identify, create, and operate on data that can change over time. We describe techniques for automatically translating purely functional programs into self-adjusting programs. In this implicit approach, the programmer need only annotate the (top-level) input types of the programs to be translated. Type inference finds all other types, and a type-directed translation rewrites the source program into an explicitly self-adjusting target program. The type system is related to information-flow type systems and enjoys decidable type inference via constraint solving. We prove that the translation outputs well- typed self-adjusting programs and preserves the source program's input–output behavior, guaranteeing that translated programs respond correctly to all changes to their data. Using a cost semantics, we also prove that the translation preserves the asymptotic complexity of the source program.

Secure information flow by self-composition

Information flow policies are confidentiality policies that control information leakage through program execution. A common way to enforce secure information flow is through information flow type systems. Although type systems are compositional and usually enjoy decidable type checking or inference, their extensibility is very poor: type systems need to be redefined and proved sound for each new variation of security policy and programming language for which secure information flow verification is desired.In contrast, program logics offer a general mechanism for enforcing a variety of safety policies, and for this reason are favoured in Proof Carrying Code, which is a promising security architecture for mobile code. However, the encoding of information flow policies in program logics is not straightforward because they refer to a relation between two program executions.The purpose of this paper is to investigate logical formulations of secure information flow based on the idea of self-composition, which reduces the problem of secure information flow of a program P to a safety property for a program derived from P by composing P with a renaming of itself. Self-composition enables the use of standard techniques for information flow policy verification, such as program logics and model checking, that are suitable in Proof Carrying Code infrastructures.We illustrate the applicability of self-composition in several settings, including different security policies such as non-interference and controlled forms of declassification, and programming languages including an imperative language with parallel composition, a non-deterministic language and, finally, a language with shared mutable data structures.

Implicit self-adjusting computation for purely functional programs

Computational problems that involve dynamic data, such as physics simulations and program development environments, have been an important subject of study in programming languages. Building on this work, recent advances in self-adjusting computation have developed techniques that enable programs to respond automatically and efficiently to dynamic changes in their inputs. Self-adjusting programs have been shown to be efficient for a reasonably broad range of problems but the approach still requires an explicit programming style, where the programmer must use specific monadic types and primitives to identify, create and operate on data that can change over time. We describe techniques for automatically translating purely functional programs into self-adjusting programs. In this implicit approach, the programmer need only annotate the (top-level) input types of the programs to be translated. Type inference finds all other types, and a type-directed translation rewrites the source program into an explicitly self-adjusting target program. The type system is related to information-flow type systems and enjoys decidable type inference via constraint solving. We prove that the translation outputs well-typed self-adjusting programs and preserves the source program's input-output behavior, guaranteeing that translated programs respond correctly to all changes to their data. Using a cost semantics, we also prove that the translation preserves the asymptotic complexity of the source program.

Type Inference for Bimorphic Recursion

This paper proposes bimorphic recursion, which is restricted polymorphic recursion such that every recursive call in the body of a function definition has the same type. Bimorphic recursion allows us to assign two different types to a recursively defined function: one is for its recursive calls and the other is for its calls outside its definition. Bimorphic recursion in this paper can be nested. This paper shows bimorphic recursion has principal types and decidable type inference. Hence bimorphic recursion gives us flexible typing for recursion with decidable type inference. This paper also shows that its typability becomes undecidable because of nesting of recursions when one removes the instantiation property from the bimorphic recursion.

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.

Understanding functional dependencies via constraint handling rules

AbstractFunctional dependencies are a popular and useful extension to Haskell style type classes. We give a reformulation of functional dependencies in terms of Constraint Handling Rules (CHRs). In previous work, CHRs have been employed for describing user-programmable type extensions in the context of Haskell style type classes. Here, we make use of CHRs to provide for the first time a concise result that under some sufficient conditions, functional dependencies allow for sound, complete and decidable type inference. The sufficient conditions imposed on functional dependencies can be very limiting. We show how to safely relax these conditions and suggest several sound extensions of functional dependencies. Our results allow for a better understanding of functional dependencies and open up the opportunity for new applications.

A theory of overloading

We present a novel approach to allow for overloading of identifiers in the spirit of type classes. Our approach relies on a combination of the HM(X) type system framework with Constraint Handling Rules (CHRs). CHRs are a declarative language for writing incremental constraint solvers, that provide our scheme with a form of programmable type language. CHRs allow us to precisely describe the relationships among overloaded identifiers. Under some sufficient conditions on the CHRs we achieve decidable type inference and the semantic meaning of programs is unambiguous. Our approach provides a common formal basis for many type class extensions such as multiparameter type classes and functional dependencies.

Principality and type inference for intersection types using expansion variables

Principality of typings is the property that for each typable term, there is a typing from which all other typings are obtained via some set of operations. Type inference is the problem of finding a typing for a given term, if possible. We define an intersection type system which has principal typings and types exactly the strongly normalizable λ-terms. More interestingly, every finite-rank restriction of this system (using Leivant's first notion of rank) has principal typings and also has decidable type inference. This is in contrast to System F where the finite rank restriction for every finite rank at 3 and above has neither principal typings nor decidable type inference. Furthermore, the notion of principal typings for our system involves only one operation, substitution, rather than several operations (not all substitution-based) as in earlier presentations of principality for intersection types (without rank restrictions). In our system the earlier notion of expansion is integrated in the form of expansion variables, which are subject to substitution as are ordinary variables. A unification-based type inference algorithm is presented using a new form of unification, β-unification.

Information flow inference for ML

This paper presents a type-based information flow analysis for a call-by-value λ-calculus equipped with references, exceptions and let-polymorphism, which we refer to as ML. The type system is constraint-based and has decidable type inference. Its noninterference proof is reasonably light-weight, thanks to the use of a number of orthogonal techniques. First, a syntactic segregation between values and expressions allows a lighter formulation of the type system. Second, noninterference is reduced to subject reduction for a nonstandard language extension. Lastly, a semi-syntactic approach to type soundness allows dealing with constraint-based polymorphism separately.

A theory of overloading

We present a minimal extension of the Hindley/Milner system to allow for overloading of identifiers. Our approach relies on a combination of the HM(X) type system framework with Constraint Handling Rules (CHRs). CHRs are a declarative language for writing incremental constraint solvers. CHRs allow us to precisely describe the relationships among overloaded identifiers. Under some sufficient conditions on the CHRs we achieve decidable type inference and the semantic meaning of programs is unambiguous. Our approach allows us to combine open and closed world overloading. We also show how to deal with overlapping definitions.

Information flow inference for ML

This paper presents a type-based information flow analysis for a call-by-value λ-calculus equipped with references, exceptions and let-polymorphism, which we refer to as Core ML. The type system is constraint-based and has decidable type inference. Its non-interference proof is reasonably lightweight, thanks to the use of a number of orthogonal techniques. First, a syntactic segregation between values and expressions allows a lighter formulation of the type system. Second, non-interference is reduced to subject reduction for a non-standard language extension. Lastly, a semi-syntactic approach to type soundness allows dealing with constraint-based polymorphism separately.

Category-sorted algebra-based action semantics

In a series of papers, Mosses and Watt define action semantics, a metalanguage for high level, domain-independent formulation of denotational semantics definitions. Action semantics hides details about domain structure (e.g., direct semantics domains vs. continuation semantics domains vs. resumption semantics domains) and coercions (e.g., integers into reals, injections of summands into sum domains) to encourage readability and modifiability. Action semantics notation is of interest as a programming language of itself, for its components (called actions) are polymorphic operators that can be composed in several fundamental ways. We formulate a model for action semantics based on Reynolds' category-sorted algebra. In the model, actions are natural transformations, and the composition operators are compositions in a “category of actions.” We use the model to prove semantic soundness and completeness of a unification-based, decidable type inference algorithm for action semantics expressions.