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.

Parameter Sensitive Detectors

Object detection can be challenging when the object class exhibits large variations. One commonly-used strategy is to first partition the space of possible object variations and then train separate classifiers for each portion. However, with continuous spaces the partitions tend to be arbitrary since there are no natural boundaries (for example, consider the continuous range of human body poses). In this paper, a new formulation is proposed, where the detectors themselves are associated with continuous parameters, and reside in a parameterized function space. There are two advantages of this strategy. First, a-priori partitioning of the parameter space is not needed; the detectors themselves are in a parameterized space. Second, the underlying parameters for object variations can be learned from training data in an unsupervised manner. In profile face detection experiments, at a fixed false alarm number of 90, our method attains a detection rate of 75% vs. 70% for the method of Viola-Jones. In hand shape detection, at a false positive rate of 0.1%, our method achieves a detection rate of 99.5% vs. 98% for partition based methods. In pedestrian detection, our method reduces the miss detection rate by a factor of three at a false positive rate of 1%, compared with the method of Dalal-Triggs.