Ranking Sets of Objects: The Complexity of Avoiding Impossibility Results

The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating properties the lifted order should have. Well-known impossibility results by Kannai and Peleg and by Barbera and Pattanaik tell us that some desirable axioms – namely dominance and (strict) independence – are not jointly satisfiable for any linear order on the objects if all non-empty sets of objects are to be ordered. On the other hand, if not all non-empty sets of objects are to be ordered, the axioms are jointly satisfiable for all linear orders on the objects for some families of sets. Such families are very important for applications as they allow for the use of lifted orders, for example, in combinatorial voting. In this paper, we determine the computational complexity of recognizing such families. We show that it is \Pi_2^p-complete to decide for a given family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for all linear orders on the objects if the lifted order needs to be total. Furthermore, we show that the problem remains coNP-complete if the lifted order can be incomplete. Additionally, we show that the complexity of these problems can increase exponentially if the family of sets is not given explicitly but via a succinct domain restriction. Finally, we show that it is NP-complete to decide for a family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for at least one linear order on the objects.

The Grammar of Prominence

This chapter draws theoretical conclusions and outlines directions for future developments. It summarizes the key theoretical and philosophical upshots of the account developed in the book and discusses further extensions of this framework. It discusses how the account can be applied to model context-sensitivity of situated utterances, in a way that can offer insights into puzzles concerning disagreement in discourse and communication under ignorance, which have plagued standard accounts of context and content. Further, it outlines the way the account is to be extended and applied to various types of context-sensitive items, including relational expressions, gradable adjectives, and domain restriction.

Domain restriction in child Mandarin: Implications for quantifier spreading

This study explores Mandarin children’s competence with quantifier domain restriction. We present results from two experiments in which adults and four- to five-year old children evaluated two possible candidates for the domain selection associated with the distributive operator dou ‘all’ in Mandarin Chinese. In the first experiment, we investigated whether children and adults are capable of selecting an appropriate domain when two candidate NPs both appear inside dou’s quantification scope; i.e., both of the NPs c-command dou. In the second experiment, still two candidate NPs were presented, but one within dou’s scope and the other outside its scope. Our results indicate that four- to five-year-old children are capable of basic distributive computation associated with dou, but they may choose an NP that adults do not usually choose as the domain of dou, resulting in non-adult interpretations of distributive computation in certain cases. Based on the results, we propose that four- to five-year-old children are less certain about the domain restriction associated with dou-quantification. This proposal has important implications for the current debate on the acquisition of universal quantifiers, specifically, the problem of quantifier spreading. We explain children’s uncertainty about the domain restriction with a universal grammar-based statistical acquisition model.

On the semantics of classifier reduplication in Cantonese

This paper studies prenominal reduplicative classifier in Cantonese, which has been argued to be a distributive quantifier on a par with English every/each and Mandarin mei ‘every’, and a plural classifier giving the ‘many’ reading. The analysis I propose draws heavily on ideas introduced in the cover theory proposed by Schwarzschild (1996) and Brisson (1998, 2003), and ideas introduced by Partee (2004) and others on quantifying determiner many. I argue that prenominal reduplicative classifier is a quantifying determiner which is ambiguous between a quantifier type and a modifier type. When it occurs with the distributive quantifier dou1 ‘all’, it serves as a modifier, regulating the domain of dou1-quantification by imposing a maximizing effect on the nominal it modifies (see e.g. Link 1983; Gillon 1987; Schwarzschild 1996; Brisson 1998, 2003). Without the presence of a distributive quantifier, prenominal reduplicative classifier serves either as a modifier or as a quantifier, giving its NP a weak cardinal reading or a strong proportional reading, respectively. The proposed analysis implies that domain restriction in Chinese is overtly realized in grammatical form by means of the reduplicative classifier (when combined with a distributive quantifier) and that Chinese may have determiners, which is at least true in Cantonese.

Lifting Preferences over Alternatives to Preferences over Sets of Alternatives: The Complexity of Recognizing Desirable Families of Sets

The problem of lifting a preference order on a set of objects to a preference order on a family of subsets of this set is a fundamental problem with a wide variety of applications in AI. The process is often guided by axioms postulating properties the lifted order should have. Well-known impossibility results by Kannai and Peleg and by Barberà and Pattanaik tell us that some desirable axioms – namely dominance and (strict) independence – are not jointly satisfiable for any linear order on the objects if all non-empty sets of objects are to be ordered. On the other hand, if not all non-empty sets of objects are to be ordered, the axioms are jointly satisfiable for all linear orders on the objects for some families of sets. Such families are very important for applications as they allow for the use of lifted orders, for example, in combinatorial voting. In this paper, we determine the computational complexity of recognizing such families. We show that it is Π2p-complete to decide for a given family of subsets whether dominance and independence or dominance and strict independence are jointly satisfiable for all linear orders on the objects if the lifted order needs to be total. Furthermore, we show that the problem remains coNP-complete if the lifted order can be incomplete. Additionally, we show that the complexity of these problem can increase exponentially if the family of sets is not given explicitly but via a succinct domain restriction.

Quantificational Weather Reports

A major strand to the ‘standard view’ is that meteorological predicates must contain a locative position because we can quantify into it. If there were no such position, then the quantifications would be, contrary to fact, illicit. In this chapter I shall reject this line of reasoning. The kind of binding relations into which weather reports may enter do not militate for, let alone entail, the syntactic or lexical occurrence of a locative variable. The quantification at issue is analysed as a free relative construction. It will also be shown how domain restriction fails to have a linguistic licence.

Linguistic Pragmatism and Weather Reporting

Linguistic pragmatism claims that what we literally say goes characteristically beyond what the linguistic properties themselves mandate. In this book, John Collins provides a novel defence of this doctrine, arguing that linguistic meaning alone fails to fix truth conditions. While this position is supported by a range of theorists, Collins shows that it naturally follows from a syntactic thesis concerning the relative sparseness of what language alone can provide to semantic interpretation. Language–and by extension meaning–provides constraints upon what a speaker can literally say, but does not characteristically encode any definite thing to say. Collins then defends this doctrine against a range of alternatives and objections, focusing in particular on an analysis of weather reports: ‘it is raining/snowing/sunny’. Such reporting is mostly location-sensitive in the sense that the utterance is true or not depending upon whether it is raining/snowing/sunny at the location of the utterance, rather than some other location. Collins offers a full analysis of the syntax, semantics, and pragmatics of weather reports, including many novel data. He shows that the constructions lack the linguistic resources to support the common literal locative readings. Other related phenomena are discussed such as the Saxon genitive, colour predication, quantifier domain restriction, and object deletion.

The Morphosyntax of Arabic Exceptives: A minimalist approach

Arabic exceptive constructions involve an intricate interaction among word order, negation, and case assignment resulting from syntactic restrictions imposed on the argumental, appositional, and adjunctive functions of exceptive phrases. The morphosyntactic complexities in Arabic exceptives cast doubt on the adequacy of previous analyses that exceptive particles are prepositions, focal adverbs, or coordinating conjunctions, and they also lead the paper to argue for more principled accounts in which exceptive particles are analyzed as functional heads that project into an ExP ‘exceptive phrase’ which exists in two distinct configurations. The first includes an exceptive marker carrying [Acc-Case] ‘accusative case’ and [DS] ‘domain subtraction’ features when the ExP is an adjunct introduced by late Merge. The second includes a negative determiner which selects and c-commands the exceptive particle and deactivates/ suspends its [Acc-Case], consequently, the case feature of the ExP-complement is valued by percolation from D which combines with the exceptive marker to form a discontinuous focus particle with a [DR] ‘domain restriction’ feature.