Fast incremental discovery of pointwise order dependencies

Pointwise order dependencies (PODs) are dependencies that specify ordering semantics on attributes of tuples. POD discovery refers to the process of identifying the set Σ of valid and minimal PODs on a given data set D. In practice D is typically large and keeps changing, and it is prohibitively expensive to compute Σ from scratch every time. In this paper, we make a first effort to study the incremental POD discovery problem, aiming at computing changes ΔΣ to Σ such that Σ ⊕ ΔΣ is the set of valid and minimal PODs on D with a set Δ D of tuple insertion updates. (1) We first propose a novel indexing technique for inputs Σ and D. We give algorithms to build and choose indexes for Σ and D , and to update indexes in response to Δ D. We show that POD violations w.r.t. Σ incurred by Δ D can be efficiently identified by leveraging the proposed indexes, with a cost dependent on log (| D |). (2) We then present an effective algorithm for computing ΔΣ, based on Σ and identified violations caused by Δ D. The PODs in Σ that become invalid on D + Δ D are efficiently detected with the proposed indexes, and further new valid PODs on D + Δ D are identified by refining those invalid PODs in Σ on D + Δ D. (3) Finally, using both real-life and synthetic datasets, we experimentally show that our approach outperforms the batch approach that computes from scratch, up to orders of magnitude.

Ranked ordering sources and embedded modality

I present an ordering semantics for modality in which possible worlds are ordered by ordering sources augmented with a partial order structure. This extension of Kratzer’s (1991) ordering semantics allows propositions to contribute to the ideal defined by an ordering source with differing degrees of priority and allows this priority relation to vary with the world of evaluation. Although the * operator of Katz et al. (2012) also allows ordering sources to be combined with different degrees of priority, I show that it does not account for a variant of Goble’s (1996) Medicine Problem in which a modal is embedded under an attitude verb. I also extend the investigation by Katz et al. (2012) into the combinatorial structure of complex ordering sources by proposing a generalization of their * operator for partially ordered ordering sources.

Ordering combination for modal comparison

The well-known ordering semantics for modality has recently been challenged by a number of puzzles which may cast doubt on the viability of this approach. We investigate the properties of the ordering relations used in ordering semantics with a focus on how to derive appropriate orderings based on intuitively correct premises. We use these tools to propose solutions to two of the puzzles and discuss how they relate to other puzzles that have been raised in the literature, and to modal semantics more generally.

Ordering combination for modal comparison

The well-known ordering semantics for modality has recently been challenged by a number of puzzles which may cast doubt on the viability of this approach. We investigate the properties of the ordering relations used in ordering semantics with a focus on how to derive appropriate orderings based on intuitively correct premises. We use these tools to propose solutions to two of the puzzles and discuss how they relate to other puzzles that have been raised in the literature, and to modal semantics more generally.

EXPRESSIBLE SEMANTICS FOR EXPRESSIBLE COUNTERFACTUALS

Lewis (1981) showed the equivalence between two dominant semantic frameworks for counterfactuals: ordering semantics, which relies on orders between possible worlds, and premise semantics, which relies on sets of propositions (so-called ordering sources). I define a natural, restricted version of premise semantics, expressible premise semantics, which is based on ordering sources containing only expressible propositions. First, I extend Lewis’ (1981) equivalence result to expressible premise semantics and some corresponding expressible version of ordering semantics. Second, I show that expressible semantics are strictly less powerful than their nonexpressible counterparts, even when attention is restricted to the truth values of expressible counterfactuals. Assuming that the expressibility constraint is natural for premise semantics, this result breaks the equivalence between ordering semantics and (expressible) premise semantics. Finally, I show that these results cast doubt on various desirable conjectures, and in particular on a particular defense of the so-called limit assumption.