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.

Ergodicity and Mixing Rate of One-Dimensional Cellular Automata

One-and two-dimensional cellular automata which are known to be fault-tolerant are very complex. On the other hand, only very simple cellular automata have actually been proven to lack fault-tolerance, i.e., to be mixing. The latter either have large noise probability ε or belong to the small family of two-state nearest-neighbor monotonic rules which includes local majority voting. For a certain simple automaton L called the soldiers rule, this problem has intrigued researchers for the last two decades since L is clearly more robust than local voting: in the absence of noise, L eliminates any finite island of perturbation from an initial configuration of all 0's or all 1's. The same holds for a 4-state monotonic variant of L, K, called two-line voting. We will prove that the probabilistic cellular automata Kε and Lε asymptotically lose all information about their initial state when subject to small, strongly biased noise. The mixing property trivially implies that the systems are ergodic. The finite-time information-retaining quality of a mixing system can be represented by its relaxation time Relax(⋅), which measures the time before the onset of significant information loss. This is known to grow as (1/ε)^c for noisy local voting. The impressive error-correction ability of L has prompted some researchers to conjecture that Relax(Lε) = 2^(c/ε). We prove the tight bound 2^(c1log^21/ε) ＜ Relax(Lε) ＜ 2^(c2log^21/ε) for a biased error model. The same holds for Kε. Moreover, the lower bound is independent of the bias assumption. The strong bias assumption makes it possible to apply sparsity/renormalization techniques, the main tools of our investigation, used earlier in the opposite context of proving fault-tolerance.

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.

BRITE: Universal Topology Generation from a User's Perspective

Effective engineering of the Internet is predicated upon a detailed understanding of issues such as the large-scale structure of its underlying physical topology, the manner in which it evolves over time, and the way in which its constituent components contribute to its overall function. Unfortunately, developing a deep understanding of these issues has proven to be a challenging task, since it in turn involves solving difficult problems such as mapping the actual topology, characterizing it, and developing models that capture its emergent behavior. Consequently, even though there are a number of topology models, it is an open question as to how representative the topologies they generate are of the actual Internet. Our goal is to produce a topology generation framework which improves the state of the art and is based on design principles which include representativeness, inclusiveness, and interoperability. Representativeness leads to synthetic topologies that accurately reflect many aspects of the actual Internet topology (e.g. hierarchical structure, degree distribution, etc.). Inclusiveness combines the strengths of as many generation models as possible in a single generation tool. Interoperability provides interfaces to widely-used simulation and visualization applications such as ns and SSF. We call such a tool a universal topology generator. In this paper we discuss the design, implementation and usage of the BRITE universal topology generation tool that we have built. We also describe the BRITE Analysis Engine, BRIANA, which is an independent piece of software designed and built upon BRITE design goals of flexibility and extensibility. The purpose of BRIANA is to act as a repository of analysis routines along with a user–friendly interface that allows its use on different topology formats.

Smooth Multirate Multicast Congestion Control

A significant impediment to deployment of multicast services is the daunting technical complexity of developing, testing and validating congestion control protocols fit for wide-area deployment. Protocols such as pgmcc and TFMCC have recently made considerable progress on the single rate case, i.e. where one dynamic reception rate is maintained for all receivers in the session. However, these protocols have limited applicability, since scaling to session sizes beyond tens of participants necessitates the use of multiple rate protocols. Unfortunately, while existing multiple rate protocols exhibit better scalability, they are both less mature than single rate protocols and suffer from high complexity. We propose a new approach to multiple rate congestion control that leverages proven single rate congestion control methods by orchestrating an ensemble of independently controlled single rate sessions. We describe SMCC, a new multiple rate equation-based congestion control algorithm for layered multicast sessions that employs TFMCC as the primary underlying control mechanism for each layer. SMCC combines the benefits of TFMCC (smooth rate control, equation-based TCP friendliness) with the scalability and flexibility of multiple rates to provide a sound multiple rate multicast congestion control policy.

Validating Arbitrarily Large Network Protocol Compositions with Finite Computation

Formal tools like finite-state model checkers have proven useful in verifying the correctness of systems of bounded size and for hardening single system components against arbitrary inputs. However, conventional applications of these techniques are not well suited to characterizing emergent behaviors of large compositions of processes. In this paper, we present a methodology by which arbitrarily large compositions of components can, if sufficient conditions are proven concerning properties of small compositions, be modeled and completely verified by performing formal verifications upon only a finite set of compositions. The sufficient conditions take the form of reductions, which are claims that particular sequences of components will be causally indistinguishable from other shorter sequences of components. We show how this methodology can be applied to a variety of network protocol applications, including two features of the HTTP protocol, a simple active networking applet, and a proposed web cache consistency algorithm. We also doing discuss its applicability to framing protocol design goals and to representing systems which employ non-model-checking verification methodologies. Finally, we briefly discuss how we hope to broaden this methodology to more general topological compositions of network applications.