System F with Constraint Types

System F is a type system that can be seen as both a proof system for second-order propositional logic and as a polymorphic programming language. In this work we explore several extensions of System F by types which express subtyping constraints. These systems include terms which represent proofs of subtyping relationships between types. Given a proof that one type is a subtype of another, one may use a coercion term constructor to coerce terms from the first type to the second. The ability to manipulate type constraints as first-class entities gives these systems a lot of expressive power, including the ability to encode generalized algebraic data types and intensional type analysis. The main contributions of this work are in the formulation of constraint types and a proof of strong normalization for an extension of System F with constraint types.

A Direct Algorithm for the Type Interference in the Rank 2 Fragment of the Second--Order λ-Calculus

We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML. This family comprises all levels of the stratification of the second-order lambda-calculus by "rank" of types. We show that typability is an undecidable problem at every rank k ≥ 3 of this stratification. While it was already known that typability is decidable at rank ≤ 2, no direct and easy-to-implement algorithm was available. To design such an algorithm, we develop a new notion of reduction and show how to use it to reduce the problem of typability at rank 2 to the problem of acyclic semi-unification. A by-product of our analysis is the publication of a simple solution procedure for acyclic semi-unification.

A Type System For Safe SN Resource Allocation

snBench is a platform on which novice users compose and deploy distributed Sense and Respond programs for simultaneous execution on a shared, distributed infrastructure. It is a natural imperative that we have the ability to (1) verify the safety/correctness of newly submitted tasks and (2) derive the resource requirements for these tasks such that correct allocation may occur. To achieve these goals we have established a multi-dimensional sized type system for our functional-style Domain Specific Language (DSL) called Sensor Task Execution Plan (STEP). In such a type system data types are annotated with a vector of size attributes (e.g., upper and lower size bounds). Tracking multiple size aspects proves essential in a system in which Images are manipulated as a first class data type, as image manipulation functions may have specific minimum and/or maximum resolution restrictions on the input they can correctly process. Through static analysis of STEP instances we not only verify basic type safety and establish upper computational resource bounds (i.e., time and space), but we also derive and solve data and resource sizing constraints (e.g., Image resolution, camera capabilities) from the implicit constraints embedded in program instances. In fact, the static methods presented here have benefit beyond their application to Image data, and may be extended to other data types that require tracking multiple dimensions (e.g., image "quality", video frame-rate or aspect ratio, audio sampling rate). In this paper we present the syntax and semantics of our functional language, our type system that builds costs and resource/data constraints, and (through both formalism and specific details of our implementation) provide concrete examples of how the constraints and sizing information are used in practice.

Type Inference for Variant Object Types

Existing type systems for object calculi are based on invariant subtyping. Subtyping invariance is required for soundness of static typing in the presence of method overrides, but it is often in the way of the expressive power of the type system. Flexibility of static typing can be recovered in different ways: in first-order systems, by the adoption of object types with variance annotations, in second-order systems by resorting to Self types. Type inference is known to be P-complete for first-order systems of finite and recursive object types, and NP-complete for a restricted version of Self types. The complexity of type inference for systems with variance annotations is yet unknown. This paper presents a new object type system based on the notion of Split types, a form of object types where every method is assigned two types, namely, an update type and a select type. The subtyping relation that arises for Split types is variant and, as a result, subtyping can be performed both in width and in depth. The new type system generalizes all the existing first-order type systems for objects, including systems based on variance annotations. Interestingly, the additional expressive power does not affect the complexity of the type inference problem, as we show by presenting an O(n^3) inference algorithm.

Safe Compositional Specification of Networking Systems: A Compositional Analysis Approach

We present a type inference algorithm, in the style of compositional analysis, for the language TRAFFIC—a specification language for flow composition applications proposed in [2]—and prove that this algorithm is correct: the typings it infers are principal typings, and the typings agree with syntax-directed type checking on closed flow specifications. This algorithm is capable of verifying partial flow specifications, which is a significant improvement over syntax-directed type checking algorithm presented in [3]. We also show that this algorithm runs efficiently, i.e., in low-degree polynomial time.