We analyze the relationship between ordinal ranking and binary classification with a new technique called reverse reduction. In particular, we prove that the regret can be transformed between ordinal ranking and binary classification. The proof allows us to establish a general equivalence between the two in terms of hardness. Furthermore, we use the technique to design a novel boosting approach...

Ordinary computations can be characterised by register machines working with natural numbers. We study ordinal register machines where the registers can hold arbitrary ordinal numbers. The class of sets of ordinals which are computable by such machines has strong closure properties and satisfies the set theoretic axiom system SO. This implies that ordinal computability is equivalent to Gödel’s ...

A solution to bargaining problems is ordinal when it is covariant with respect to order-preserving transformations of utility. Shapley has constructed an ordinal, symmetric, efficient solution to threeplayer problems. Here, we extend Shapley’s solution in two directions. First, we extend it to more than three players. Second, we show that this extension lends itself to the construction of a con...

We specify the frontier of decidability for fragments of the firstorder theory of ordinal multiplication. We give a NEXPTIME lower bound for the complexity of the existential fragment of 〈ωωλ ;×, ω, ω+ 1, ω + 1〉 for every ordinal λ. Moreover, we prove (by reduction from Hilbert Tenth Problem) that the ∃∗∀6-fragment of 〈ωωλ ;×〉 is undecidable for every ordinal λ.

We provide a positive solution for Post’s Problem for ordinal register machines, and also prove that these machines and ordinal Turing machines compute precisely the same partial functions on ordinals. To do so, we construct ordinal register machine programs which compute the necessary functions. In addition, we show that any set of ordinals solving Post’s Problem must be unbounded in the writa...

Perception of the ordinal position of a sequence element is critical to many cognitive and motor functions. Here, the prediction that this ability is based on a domain-general perceptual mechanism and, thus, that it emerges prior to the emergence of language was tested. Infants were habituated with sequences of moving/sounding objects and then tested for the ability to perceive the invariant or...

We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The wellknown range voting scheme can be thought of as a non-truthful mechanism for exact social welfare maximization in this setting. With m being the number of alternatives, we exhibit a randomized truthfulin-e...

Iris recognition provides a reliable method for personal identification. Inspired by recent achievements in the field of visual neuroscience, we encode the non-local image comparisons qualitatively for iris recognition. In this scheme, each bit iris code corresponds to the sign of an inequality across several distant image regions. Compared with local ordinal measures, the relationships of diss...

This paper describes an R package, rpartOrdinal, that implements alternative splitting functions for fitting a classification tree when interest lies in predicting an ordinal response. This includes the generalized Gini impurity function, which was introduced as a method for predicting an ordinal response by including costs of misclassification into the impurity function, as well as an alternat...

Abstraci—What is learned during mastery of a serial task: associations between adjacent and remote items, associations between an item and its ordinal position, or both? A dear answer to this question is lacking in the literature on human serial memory because it is difficult to control for a "naive" subject's linguistic competence and extensive experience with serial tasks. In this article, we...