Simply Typed Lambda Calculus with First-Class Environments

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.

A common approach to improve software quality is to use programming guidelines to avoid common kinds of errors. In this paper, we consider the problem of enforcing guidelines for Featherweight Java (FJ). We formalize guidelines as sets of finite or infinite execution traces and develop a region-based type and effect system for FJ that can enforce such guidelines. We build on the work by Erbatur, Hofmann and Z\u{a}linescu, who presented a type system for verifying the finite event traces of terminating FJ programs. We refine this type system, separating region typing from FJ typing, and use ideas of Hofmann and Chen to extend it to capture also infinite traces produced by non-terminating programs. Our type and effect system can express properties of both finite and infinite traces and can compute information about the possible infinite traces of FJ programs. Specifically, the set of infinite traces of a method is constructed as the greatest fixed point of the operator which calculates the possible traces of method bodies. Our type inference algorithm is realized by working with the finitary abstraction of the system based on B\"uchi automata.

With the introduction of parametric types in Java, the type system becomes more complex to handle. The Java generics requires more programming efforts to instantiate appropriate types. In such a situation, a sound type inference algorithm may reduce programming load and ensure safety. Java type inference algorithm has eased programming efforts by reducing explicit instantiation of types. The Java type inference algorithm is being expanded. Still, the impact of inference in mainstream programming is limited. Moreover, the unsound approach for inference causes unpredictable behavior in program code. The limitations and unsoundness of Java type inference need to be discussed to avoid the failure of codes. In this paper, we discuss limitations and issues of type inference algorithm, such as the incapability to infer wildcards as return type in generic methods, and limitations of local variable type inference. Further, we discuss how the machine learning approaches can help to develop a safe type inference algorithm.

Three versions of the lambda calculus are discussed: typed lambda calculus, untyped lambda calculus, and polymorphically typed lambda calculus. It is shown that the syntax for these calculi can be described as initial algebras for suitable endofunctors. Models of the first two calculi are described as in [Lambek and Scott 1985] and the Per model of the third is recalled. Algebraic types are described in terms of sketches. A main contribution of the paper is the internalization of the theory of sketches in the category Per. Sketches themselves, models of sketches, categories of models of sketches and the category of all models of all sketches can all be realized in this setting. Next, dinatural transformations are reviewed in the context of the category Per and the relation between certain end formulas and initial algebras for finite signatures is discussed. The point of the theory of algebraic semantics in Per is to provide a setting for thinking about algebraic types and algebraic operations in conjunction with logical types and polymorphic operations. Two specific examples are discussed and a general principle for combining algebraic and polymorphic operations is proposed.KeywordsNatural TransformationCategory TheoryLogical TypeDenotational SemanticGraph ObjectThese 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.

We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified proof of standardization for beta reduction that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.

Many programming languages provides non-local exit. In C language, it is implemented by setjump and longjump functions in its standard library. In Java, a try-catch-finally statement is equipped as non-local exit with control capturing. The try-catch-finally mechanism can be categorized into two parts: global jump and control capturing. The non-local jump has been studied well for a long time by many researchers. On the other hand, control capturing has not yet been researched well enough. In this paper, we propose a lambda calculus with non-local jump and control capturing and its operational semantics based on small-step transition. We provide continuation-passing style translation of the calculus into the usual lambda calculus. The continuation-passing style translation is known as a translation style semantics of a control structure such as a non-local jump or first-class continuation. We extend the continuation passing style translation in order to formalize the control capturing in the framework of a functional programming language paradigm. We develop a type system for the calculus with control capturing and show conformity of the typed version of the CPS translation with the type system of the calculus.

This paper presents a polymorphic type system for a language with delimited control operators, shift and reset. Based on the monomorphic type system by Danvy and Filinski, the proposed type system allows pure expressions to be polymorphic. Thanks to the explicit presence of answer types, our type system satisfies various important properties, including strong type soundness, existence of principal types and an inference algorithm, and strong normalization. Relationship to CPS translation as well as extensions to impredicative polymorphism are also discussed. These technical results establish the foundation of polymorphic delimited continuations.KeywordsType SystemDelimited ContinuationControl OperatorCPS TranslationPredicative/Impredicative Polymorphism

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 )).

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.

In this paper we discuss usability, and propose to take that notion as a formalisation of (un)definedness in typed lambda calculus, especially in calculi based on PCF. We discuss some important properties that make usability attractive as a formalisation of (un)definedness. There is a remarkable difference between usability and solvability: in the untyped lambda calculus the solvable terms are precisely the terms with a head normal form, whereas in typed lambda calculus the usable terms are between the terms with a normal form and the terms with a (weak) head normal form.

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.

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

A completeness result for a realisability semantics for an intersection type system

Subtyping appears in a variety of programming languages, in the form of the ‘automatic coercion’ of integers to reals, Pascal subranges, and subtypes arising from class hierarchies in languages with inheritance. A general framework based on untyped lambda calculus provides a simple semantic model of subtyping and is used to demonstrate that an extension of Curry's type inference rules are semantically complete. An algorithm G for computing the most general typing associated with any given expression, and a restricted, optimized algorithm GA using only atomic subtyping hypotheses are developed. Both algorithms may be extended to insert type conversion functions at compile time or allow polymorphic function declarations as in ML.