J. B. Wells

Typability and type checking in system F are equivalent and undecidable

Abstract Girard and Reynolds independently invented System F (a.k.a. the second-order polymorphically typed lambda calculus) to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking. Typability asks for a term whether there exists some type it can be given. Type checking asks, for a particular term and type, whether the term can be given that type. The decidability of these problems has been settled for restrictions and extensions of F and related systems and complexity lower-bounds have been determined for typability in F, but this report is the first to resolve whether these problems are decidable for System F. This report proves that type checking in F is undecidable, by a reduction from semi-unification, and that typability in F is undecidable, by a reduction from type checking. Because there is an easy reduction from typability to type checking, the two problems are equivalent. The reduction from type checking to typability uses a novel method of constructing lambda terms that simulate arbitrarily chosen type environments. All of the results also hold for the λI-calculus.

Principality and decidable type inference for finite-rank intersection types

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 ¿-terms. More interestingly, every finite-rank restriction of this system (using Leivants 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. This is also in contrast to earlier presentations of intersection types where the status (decidable or undecidable) of these properties is unknown for the finite-rank restrictions at 3 and above. 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, ß-unification.

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 Leivants 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.

Equational Reasoning for Linking with First-Class Primitive Modules

Modules and linking are usually formalized by encodings which use the λ-calculus, records (possibly dependent), and possibly some construct for recursion. In contrast, we introduce the m-calculus, a calculus where the primitive constructs are modules, linking, and the selection and hiding of module components. The m-calculus supports smooth encodings of software structuring tools such as functions (λ-calculus), records, objects (ζ-calculus), and mutually recursive definitions. The m-calculus can also express widely varying kinds of module systems as used in languages like C, Haskell, and ML. We prove the m-calculus is confluent, thereby showing that equational reasoning via the m-calculus is sensible and well behaved.

A direct algorithm for type inference in the rank-2 fragment of the second-order λ-calculus

We examine the problem of type inference for a family of polymorphic type systems containing the power of Core-ML. This family comprises the levels of the stratification of the second-order λ-calculus (system F) by “rank” of types. We show that typability is an undecidable problem at every rank k≥3. While it was already known that typability is decidable at rank 2, no direct and easy-to-implement algorithm was available. We develop a new notion of λ-term reduction and use it to prove that the problem of typability at rank 2 is reducible to the problem of acyclic semi-unification. We also describe a simple procedure for solving acyclic semi-unification. Issues related to principle types are discussed.

A calculus with polymorphic and polyvariant flow types

We present λCIL, a typed λ-calculus which serves as the foundation for a typed intermediate language for optimizing compilers for higher-order polymorphic programming languages. The key innovation of λCIL is a novel formulation of intersection and union types and flow labels on both terms and types. These flow types can encode polyvariant control and data flow information within a polymorphically typed program representation. Flow types can guide a compiler in generating customized data representations in a strongly typed setting. Since λCIL enjoys confluence, standardization, and subject reduction properties, it is a valuable tool for reasoning about programs and program transformations.

Computerizing Mathematical Text with MathLang

Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerizing mathematical texts and knowledge which is flexible enough to connect the different approaches to computerization, which allows various degrees of formalization, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Three Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), and Robert Lamar (since 2006)) and over a dozen masters degree and undergraduate students have worked on MathLang. The projects progress and design choices are driven by the needs for computerizing real representative mathematical texts chosen from various branches of mathematics. Currently, MathLang supports entry of mathematical text either in an XML format or using the editor. Methods are provided for adding, checking, and displaying various information aspects. One aspect is a kind of weak type system that assigns categories (term, statement, noun (class), adjective (class modifier), etc.) to parts of the text, deals with binding names to meanings, and checks that a kind of grammatical sense is maintained. Another aspect allows weaving together mathematical meaning and visual presentation and can associate natural language text with its mathematical meaning. Another aspect allows identifying chunks of text, marking their roles (theorem, definition, explanation, example, section, etc.), and indicating relationships between the chunks (A uses B, A contradicts B, A follows from B, etc.). Software tool support can use this aspect to check and explain the overall logical structure of a text. Further aspects are being designed to allow adding additional formality to a text such as proof structure and details of how a human-readable proof is encoded into a fully formalized version (so far this has only been done for Mizar and started for Isabelle). A number of mathematical texts have been computerized, helping with the development of these aspects, and indicating what additional work is needed for the future. This paper surveys the past and future work of the MathLang project.

Restoring Natural Language as a Computerised Mathematics Input Method

Methods for computerised mathematics have found little appeal among mathematicians because they call for additional skills which are not available to the typical mathematician. We herein propose to reconcile computerised mathematics to mathematicians by restoring natural language as the primary medium for mathematical authoring. Our method associates portions of text with grammatical argumentation roles and computerises the informal mathematical style of the mathematician. Typical abbreviations like the aggregation of equations a = b > c, are not usually accepted as input to computerised languages. We propose specific annotations to explicate the morphology of such natural language style, to accept input in this style, and to expand this input in the computer to obtain the intended representation (i.e., a = b and b > c). We have named this method syntax souringin contrast to the usual syntax sugaring. All results have been implemented in a prototype editor developed on top of

Compilation of extended recursion in call-by-value functional languages

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