Typability and Type Checking in the Second-Order Λ-Calculus Are Equivalent and Undecidable (Preliminary Draft)

We consider the problems of typability[1] and type checking[2] in the Girard/Reynolds second-order polymorphic typed λ-calculus, for which we use the short name "System F" and which we use in the "Curry style" where types are assigned to pure λ -terms. These problems have been considered and proven to be decidable or undecidable for various restrictions and extensions of System F and other related systems, and lower-bound complexity results for System F have been achieved, but they have remained "embarrassing open problems"[3] for System F itself. We first prove that type checking in System F is undecidable by a reduction from semi-unification. We then prove typability in System F is undecidable by a reduction from type checking. Since the reverse reduction is already known, this implies the two problems are equivalent. The second reduction uses a novel method of constructing λ-terms such that in all type derivations, specific bound variables must always be assigned a specific type. Using this technique, we can require that specific subterms must be typable using a specific, fixed type assignment in order for the entire term to be typable at all. Any desired type assignment may be simulated. We develop this method, which we call "constants for free", for both the λK and λI calculi.

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.

Stochastic Refinement of the Visual Hull to Satisfy Photometric and Silhouette Consistency Constraints

An iterative method for reconstructing a 3D polygonal mesh and color texture map from multiple views of an object is presented. In each iteration, the method first estimates a texture map given the current shape estimate. The texture map and its associated residual error image are obtained via maximum a posteriori estimation and reprojection of the multiple views into texture space. Next, the surface shape is adjusted to minimize residual error in texture space. The surface is deformed towards a photometrically-consistent solution via a series of 1D epipolar searches at randomly selected surface points. The texture space formulation has improved computational complexity over standard image-based error approaches, and allows computation of the reprojection error and uncertainty for any point on the surface. Moreover, shape adjustments can be constrained such that the recovered model's silhouette matches those of the input images. Experiments with real world imagery demonstrate the validity of the approach.

A Linearization of the Lambda-Calculus and Consequences

If every lambda-abstraction in a lambda-term M binds at most one variable occurrence, then M is said to be "linear". Many questions about linear lambda-terms are relatively easy to answer, e.g. they all are beta-strongly normalizing and all are simply-typable. We extend the syntax of the standard lambda-calculus L to a non-standard lambda-calculus L^ satisfying a linearity condition generalizing the notion in the standard case. Specifically, in L^ a subterm Q of a term M can be applied to several subterms R1,...,Rk in parallel, which we write as (Q. R1 \wedge ... \wedge Rk). The appropriate notion of beta-reduction beta^ for the calculus L^ is such that, if Q is the lambda-abstraction (\lambda x.P) with m\geq 0 bound occurrences of x, the reduction can be carried out provided k = max(m,1). Every M in L^ is thus beta^-SN. We relate standard beta-reduction and non-standard beta^-reduction in several different ways, and draw several consequences, e.g. a new simple proof for the fact that a standard term M is beta-SN iff M can be assigned a so-called "intersection" type ("top" type disallowed).

Some Considerations on a Calculus with Weak References

Weak references are references that do not prevent the object they point to from being garbage collected. Most realistic languages, including Java, SML/NJ, and OCaml to name a few, have some facility for programming with weak references. Weak references are used in implementing idioms like memoizing functions and hash-consing in order to avoid potential memory leaks. However, the semantics of weak references in many languages are not clearly specified. Without a formal semantics for weak references it becomes impossible to prove the correctness of implementations making use of this feature. Previous work by Hallett and Kfoury extends λgc, a language for modeling garbage collection, to λweak, a similar language with weak references. Using this previously formalized semantics for weak references, we consider two issues related to well-behavedness of programs. Firstly, we provide a new, simpler proof of the well-behavedness of the syntactically restricted fragment of λweak defined previously. Secondly, we give a natural semantic criterion for well-behavedness much broader than the syntactic restriction, which is useful as principle for programming with weak references. Furthermore we extend the result, proved in previously of λgc, which allows one to use type-inference to collect some reachable objects that are never used. We prove that this result holds of our language, and we extend this result to allow the collection of weakly-referenced reachable garbage without incurring the computational overhead sometimes associated with collecting weak bindings (e.g. the need to recompute a memoized function). Lastly we use extend the semantic framework to model the key/value weak references found in Haskell and we prove the Haskell is semantics equivalent to a simpler semantics due to the lack of side-effects in our language.

Modeling the Possible Influences of Eye Movements on the Refinement of Cortical Direction Selectivity

The second-order statistics of neural activity was examined in a model of the cat LGN and V1 during free-viewing of natural images. In the model, the specific patterns of thalamocortical activity required for a Bebbian maturation of direction-selective cells in VI were found during the periods of visual fixation, when small eye movements occurred, but not when natural images were examined in the absence of fixational eye movements. In addition, simulations of stroboscopic reming that replicated the abnormal pattern of eye movements observed in kittens chronically exposed to stroboscopic illumination produced results consistent with the reported loss of direction selectivity and preservation of orientation selectivity. These results suggest the involvement of the oculomotor activity of visual fixation in the maturation of cortical direction selectivity.

An Examination of a Church-Based Recovery Program on the Quality of Physical and Spiritual Life of Mentally Ill Persons in Korea

This study explores the effectiveness of a Church-based recovery program for the mentally ill in Korea where many Christian communities view mental illness as evidence of sin. Building on theological and psychological literature, an empirical study was conducted with participants in the alternative program of the Han-ma-um community. Data analysis revealed that this program, which views mental disorders as illness rather than sin, helps participants build self-respect and enables families to provide support as they move toward recovery. Based on this empirical examination, recommendations for refinement and expansion of the program and avenues for future research are proposed.

An Algorithm for Inferring Quasi-Static Types

This report presents an algorithm, and its implementation, for doing type inference in the context of Quasi-Static Typing (QST) ["Quasy-static Typing." Satish Thatte Proc. ACM Symp. on Principles of Programming Languages, 1988]. The package infers types a la "QST" for the simply typed λ-calculus.

New Notions of Reduction and Non-Semantic Proofs of β-Strong Normalization in Typed λ-Calculi

Two new notions of reduction for terms of the λ-calculus are introduced and the question of whether a λ-term is beta-strongly normalizing is reduced to the question of whether a λ-term is merely normalizing under one of the new notions of reduction. This leads to a new way to prove beta-strong normalization for typed λ-calculi. Instead of the usual semantic proof style based on Girard's "candidats de réductibilité'', termination can be proved using a decreasing metric over a well-founded ordering in a style more common in the field of term rewriting. This new proof method is applied to the simply-typed λ-calculus and the system of intersection types.

Safe Compositional Specification of Networking Systems

The Science of Network Service Composition has clearly emerged as one of the grand themes driving many of our research questions in the networking field today [NeXtworking 2003]. This driving force stems from the rise of sophisticated applications and new networking paradigms. By "service composition" we mean that the performance and correctness properties local to the various constituent components of a service can be readily composed into global (end-to-end) properties without re-analyzing any of the constituent components in isolation, or as part of the whole composite service. The set of laws that would govern such composition is what will constitute that new science of composition. The combined heterogeneity and dynamic open nature of network systems makes composition quite challenging, and thus programming network services has been largely inaccessible to the average user. We identify (and outline) a research agenda in which we aim to develop a specification language that is expressive enough to describe different components of a network service, and that will include type hierarchies inspired by type systems in general programming languages that enable the safe composition of software components. We envision this new science of composition to be built upon several theories (e.g., control theory, game theory, network calculus, percolation theory, economics, queuing theory). In essence, different theories may provide different languages by which certain properties of system components can be expressed and composed into larger systems. We then seek to lift these lower-level specifications to a higher level by abstracting away details that are irrelevant for safe composition at the higher level, thus making theories scalable and useful to the average user. In this paper we focus on services built upon an overlay management architecture, and we use control theory and QoS theory as example theories from which we lift up compositional specifications.

Typability is Undecidable for F+Eta

System F is the well-known polymorphically-typed λ-calculus with universal quantifiers ("∀"). F+η is System F extended with the eta rule, which says that if term M can be given type τ and M η-reduces to N, then N can also be given the type τ. Adding the eta rule to System F is equivalent to adding the subsumption rule using the subtyping ("containment") relation that Mitchell defined and axiomatized [Mit88]. The subsumption rule says that if M can be given type τ and τ is a subtype of type σ, then M can be given type σ. Mitchell's subtyping relation involves no extensions to the syntax of types, i.e., no bounded polymorphism and no supertype of all types, and is thus unrelated to the system F≤("F-sub"). Typability for F+η is the problem of determining for any term M whether there is any type τ that can be given to it using the type inference rules of F+η. Typability has been proven undecidable for System F [Wel94] (without the eta rule), but the decidability of typability has been an open problem for F+η. Mitchell's subtyping relation has recently been proven undecidable [TU95, Wel95b], implying the undecidability of "type checking" for F+η. This paper reduces the problem of subtyping to the problem of typability for F+η, thus proving the undecidability of typability. The proof methods are similar in outline to those used to prove the undecidability of typability for System F, but the fine details differ greatly.

Type Inference For Recursive Definitions

We consider type systems that combine universal types, recursive types, and object types. We study type inference in these systems under a rank restriction, following Leivant's notion of rank. To motivate our work, we present several examples showing how our systems can be used to type programs encountered in practice. We show that type inference in the rank-k system is decidable for k ≤ 2 and undecidable for k ≥ 3. (Similar results based on different techniques are known to hold for System F, without recursive types and object types.) Our undecidability result is obtained by a reduction from a particular adaptation (which we call "regular") of the semi-unification problem and whose undecidability is, interestingly, obtained by methods totally different from those used in the case of standard (or finite) semi-unification.

Deciding Isomorphisms of Simple Types in Polynomial Time

The isomorphisms holding in all models of the simply typed lambda calculus with surjective and terminal objects are well studied - these models are exactly the Cartesian closed categories. Isomorphism of two simple types in such a model is decidable by reduction to a normal form and comparison under a finite number of permutations (Bruce, Di Cosmo, and Longo 1992). Unfortunately, these normal forms may be exponentially larger than the original types so this construction decides isomorphism in exponential time. We show how using space-sharing/hash-consing techniques and memoization can be used to decide isomorphism in practical polynomial time (low degree, small hidden constant). Other researchers have investigated simple type isomorphism in relation to, among other potential applications, type-based retrieval of software modules from libraries and automatic generation of bridge code for multi-language systems. Our result makes such potential applications practically feasible.

What Are Polymorphically-Typed Ambients?

Abstract: The Ambient Calculus was developed by Cardelli and Gordon as a formal framework to study issues of mobility and migrant code. 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; the time complexity of our decision procedure is exponential (this is a worst-case in theory, arguably not encountered in practice). Our polymorphically-typed calculus is a conservative extension of the typed Ambient Calculus originally proposed by Cardelli and Gordon.

Typed Abstraction of Complex Network Compositions

The heterogeneity and open nature of network systems make analysis of compositions of components quite challenging, making the design and implementation of robust network services largely inaccessible to the average programmer. We propose the development of a novel type system and practical type spaces which reflect simplified representations of the results and conclusions which can be derived from complex compositional theories in more accessible ways, essentially allowing the system architect or programmer to be exposed only to the inputs and output of compositional analysis without having to be familiar with the ins and outs of its internals. Toward this end we present the TRAFFIC (Typed Representation and Analysis of Flows For Interoperability Checks) framework, a simple flow-composition and typing language with corresponding type system. We then discuss and demonstrate the expressive power of a type space for TRAFFIC derived from the network calculus, allowing us to reason about and infer such properties as data arrival, transit, and loss rates in large composite network applications.