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.

Exploiting the Transients of Adaptation for RoQ Attacks on Internet Resources

In this paper, we expose an unorthodox adversarial attack that exploits the transients of a system's adaptive behavior, as opposed to its limited steady-state capacity. We show that a well orchestrated attack could introduce significant inefficiencies that could potentially deprive a network element from much of its capacity, or significantly reduce its service quality, while evading detection by consuming an unsuspicious, small fraction of that element's hijacked capacity. This type of attack stands in sharp contrast to traditional brute-force, sustained high-rate DoS attacks, as well as recently proposed attacks that exploit specific protocol settings such as TCP timeouts. We exemplify what we term as Reduction of Quality (RoQ) attacks by exposing the vulnerabilities of common adaptation mechanisms. We develop control-theoretic models and associated metrics to quantify these vulnerabilities. We present numerical and simulation results, which we validate with observations from real Internet experiments. Our findings motivate the need for the development of adaptation mechanisms that are resilient to these new forms of attacks.