Assaf J. Kfoury

Type reconstruction in the presence of polymorphic recursion

We study the problem of type-checking functional programs in three extensions of ML. One distinguishing feature of these extensions is that they allow recursive definitions to be polymorphically typed. Although the motivation for these extensions comes from pragmatic considerations of programming language design, we show that the typability problem for each one of these extensions is polynomial-time equivalent to the Semi-Unification Problem and, therefore, undecidable

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.

An analysis of ML typability

We carry out an analysis of typability of terms in ML. Our main result is that this problem is DEXPTIME-hard, where by DEXPTIME we mean DTIME(2n0(1)). This, together with the known exponential-time algorithm that solves the problem, yields the DEXPTIME-completeness result. This settles an open problem of P. Kanellakis and J. C. Mitchell. Part of our analysis is an algebraic characterization of ML typability in terms of a restricted form of semi-unification, which we identify as acyclic semi-unification. We prove that ML typability and acyclic semi-unification can be reduced to each other in polynomial time. We believe this result is of independent interest.

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.

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.

ML typability is Dexptime-complete

We carry out an analysis of typability of terms in ML. Our main result is that this problem is DEXPTIME-hard, where by DEXPTIME we mean DTIME\((2^{n^{O(1)} } )\). This, together with the known exponential-time algorithm that solves the problem, yields the DEXPTIME-completeness result. This settles an open problem of P. Kanellakis and J.C. Mitchell.

What Are Polymorphically-Typed Ambients?

The Ambient Calculus was developed by Cardelli and Gordon as a formal framework to study issues of mobility and migrant code [6]. We consider an Ambient Calculus where ambients transport and exchange programs rather that just inert data. We propose different senses in which such a calculus can be said to be polymorphically typed, and design accordingly a polymorphic type system for it. Our type system assigns types to embedded programs and what we call behaviors to processes; a denotational semantics of behaviors is then proposed, here called trace semantics, underlying much of the remaining analysis. We state and prove a Subject Reduction property for our polymorphically-typed calculus. Based on techniques borrowed from finite automata theory, type-checking of fully type-annotated processes is shown to be decidable. Our polymorphically-typed calculus is a conservative extension of the typed Ambient Calculus originally proposed by Cardelli and Gordon [7].

Type reconstruction in finite rank fragments of the second-order l-calculus

Abstract The prove that the problem of type reconstruction in the polymorphic λ-calculus of rank 2 is polynomial-time equivalent to the problem of type reconstruction in ML, and is therefore DEXPTIME-complete. We also prove that for every k > 2, the problem of type reconstruction in the polymorphic λ-calculus of rank k, extended with suitably chosen constants with types of rank 1, is undecidable.

Computational consequences and partial solutions of a generalized unification problem

A generalization of first-order unification, called semiunification, is studied with two goals in mind: (1) type-checking functional programs relative to an improved polymorphic type discipline; and (2) deciding the typability of terms in a restricted form of the polymorphic lambda -calculus.<<ETX>>

Verifiably-safe software-defined networks for CPS

Next generation cyber-physical systems (CPS) are expected to be deployed in domains which require scalability as well as performance under dynamic conditions. This scale and dynamicity will require that CPS communication networks be programmatic (i.e., not requiring manual intervention at any stage), but still maintain iron-clad safety guarantees. Software-defined networking standards like Openflow provide a means for scalably building tailor-made network architectures, but there is no guarantee that these systems are safe, correct, or secure. In this work we propose a methodology and accompanying tools for specifying and modeling distributed systems such that existing formal verification techniques can be transparently used to analyze critical requirements and properties prior to system implementation. We demonstrate this methodology by iteratively modeling and verifying an Openflow learning switch network with respect to network correctness, network convergence, and mobility-related properties. We posit that a design strategy based on the complementary pairing of software-defined networking and formal verification would enable the CPS community to build next-generation systems without sacrificing the safety and reliability that these systems must deliver.