Row and Bounded Polymorphism via Disjoint Polymorphism

Polymorphism and subtyping are important features in mainstream OO languages. The most common way to integrate the two is via ?_{< :} style bounded quantification. A closely related mechanism is row polymorphism, which provides an alternative to subtyping, while still enabling many of the same applications. Yet another approach is to have type systems with intersection types and polymorphism. A recent addition to this design space are calculi with disjoint intersection types and disjoint polymorphism. With all these alternatives it is natural to wonder how they are related. This paper provides an answer to this question. We show that disjoint polymorphism can recover forms of both row polymorphism and bounded polymorphism, while retaining key desirable properties, such as type-safety and decidability. Furthermore, we identify the extra power of disjoint polymorphism which enables additional features that cannot be easily encoded in calculi with row polymorphism or bounded quantification alone. Ultimately we expect that our work is useful to inform language designers about the expressive power of those common features, and to simplify implementations and metatheory of feature-rich languages with polymorphism and subtyping

Higher-order subtyping and its decidability

AbstractWe define the typed lambda calculus Fω∧ (F-omega-meet), a natural generalization of Girard's system Fω (F-omega) with intersection types and bounded polymorphism. A novel aspect of our presentation is the use of term rewriting techniques to present intersection types, which clearly splits the computational semantics (reduction rules) from the syntax (inference rules) of the system. We establish properties such as Church-Rosser for the reduction relation on types and terms, and strong normalization for the reduction on types. We prove that types are preserved by computation (subject reduction), and that the system satisfies the minimal types property. We define algorithms for type checking and subtype checking. The development culminates with the proof of decidability of typing in Fω∧, containing the first proof of decidability of subtyping of a higher-order lambda calculus with subtyping

Using Inhabitation in Bounded Combinatory Logic with Intersection Types for Composition Synthesis

We describe ongoing work on a framework for automatic composition synthesis from a repository of software components. This work is based on combinatory logic with intersection types. The idea is that components are modeled as typed combinators, and an algorithm for inhabitation {\textemdash} is there a combinatory term e with type tau relative to an environment Gamma? {\textemdash} can be used to synthesize compositions. Here, Gamma represents the repository in the form of typed combinators, tau specifies the synthesis goal, and e is the synthesized program. We illustrate our approach by examples, including an application to synthesis from GUI-components.Comment: In Proceedings ITRS 2012, arXiv:1307.784

Set-Theoretic Types for Polymorphic Variants

Polymorphic variants are a useful feature of the OCaml language whose current definition and implementation rely on kinding constraints to simulate a subtyping relation via unification. This yields an awkward formalization and results in a type system whose behaviour is in some cases unintuitive and/or unduly restrictive. In this work, we present an alternative formalization of poly-morphic variants, based on set-theoretic types and subtyping, that yields a cleaner and more streamlined system. Our formalization is more expressive than the current one (it types more programs while preserving type safety), it can internalize some meta-theoretic properties, and it removes some pathological cases of the current implementation resulting in a more intuitive and, thus, predictable type system. More generally, this work shows how to add full-fledged union types to functional languages of the ML family that usually rely on the Hindley-Milner type system. As an aside, our system also improves the theory of semantic subtyping, notably by proving completeness for the type reconstruction algorithm.Comment: ACM SIGPLAN International Conference on Functional Programming, Sep 2016, Nara, Japan. ICFP 16, 21st ACM SIGPLAN International Conference on Functional Programming, 201

Mixin Composition Synthesis based on Intersection Types

We present a method for synthesizing compositions of mixins using type inhabitation in intersection types. First, recursively defined classes and mixins, which are functions over classes, are expressed as terms in a lambda calculus with records. Intersection types with records and record-merge are used to assign meaningful types to these terms without resorting to recursive types. Second, typed terms are translated to a repository of typed combinators. We show a relation between record types with record-merge and intersection types with constructors. This relation is used to prove soundness and partial completeness of the translation with respect to mixin composition synthesis. Furthermore, we demonstrate how a translated repository and goal type can be used as input to an existing framework for composition synthesis in bounded combinatory logic via type inhabitation. The computed result is a class typed by the goal type and generated by a mixin composition applied to an existing class

The power of Sherali-Adams relaxations for general-valued CSPs

We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems to optimality. The condition is that of bounded width which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many known valued constraint languages are tractable via the third level of the Sherali-Adams relaxation. For the known languages, this is a significantly simpler algorithm than the previously obtained ones. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Zivny [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301