On choice rules in dependent type theory

In a dependent type theory satisfying the propositions as types correspondence together with the proofs-as-programs paradigm, the validity of the unique choice rule or even more of the choice rule says that the extraction of a computable witness from an existential statement under hypothesis can be performed within the same theory. Here we show that the unique choice rule, and hence the choice rule, are not valid both in Coquand\u2019s Calculus of Constructions with indexed sum types, list types and binary disjoint sums and in its predicative version implemented in the intensional level of the Minimalist Founda- tion. This means that in these theories the extraction of computational witnesses from existential statements must be performed in a more ex- pressive proofs-as-programs theory

Essays in Market Design:

Thesis advisor: M. Utku ÜnverThesis advisor: M. Bumin YenmezThis dissertation consists of two essays in market design. In the first chapter, we study affirmative action policies in college admissions and hiring. A college or firm makes admissions or hiring decisions in which each candidate is characterized by priority ranking and type, which may depend on race, gender, or socioeconomic status. The admissions or hiring committee faces a trade-off between meritocracy and diversity: while a merit-first choice rule may admit candidates of the same type, a diversity-first choice rule may be unfair due to priority violations. To formalize this trade-off, we introduce a measure of meritocracy and a measure of diversity for choice rules. Then, we investigate how to resolve the tension between them. A choice rule that uses both reserves and quotas can be viewed as a compromise and is a generalization of the two extreme rules. The first result is comparative statics for this class of choice rules: we show that as parameters change and the choice rule becomes more meritorious, it also becomes less diverse. The second result is a characterization of the choice rule, which may help admissions or hiring committees to decide their policies. In the second chapter, we introduce a method to measure manipulability of a matching mechanism and use theory and simulation to study constrained mechanisms in school choice. First, we show that the implications from existing measures are strongly dependent on the full preference domain assumption. Our measure is more robust. The implications from existing measures can be carried over as well: while the recent school admissions reforms did not fully eliminate incentives to manipulate, they discouraged manipulation. Second, we use simulations for quantitative analysis. Our results support the recent school admissions reforms quantitatively, as well as qualitatively: they largely eliminated the incentives to manipulate. In addition, while the qualitative implications from theory are parallel to existing measures, the quantitative implications from simulations confirm a significant difference.Thesis (PhD) — Boston College, 2021.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Economics

Homotopy theoretic models of identity types

This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.Comment: 11 page

Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems

We describe the first results of a project of analyzing in which theories formal proofs can be ex- pressed. We use this analysis as the basis of interoperability between proof systems.Comment: In Proceedings PxTP 2017, arXiv:1712.0089

Dependent plural pronouns with Skolemized choice functions

The present paper discusses two interesting phenomena concerning phi-features on plural pronouns: (i) plural pronouns that denote atomic individuals (‘dependent plural pronouns’), and (ii) plural pronouns with more than one binder (‘partial binding’). A novel account of these two phenomena is proposed, according to which all occurrences of phi-features are both semantically and morphologically relevant. For such a ‘uniformly semantic account’ of phi-features, dependent plural pronouns constitute a theoretical challenge, while partial binding is more or less straightforwardly accounted for. In order to make sense of the semantic effects of the phi-features on dependent plural pronouns, the following idea is pursued: the phi-features on a dependent plural pronoun reflect the range of values that the pronoun takes, rather than the particular value it denotes at a time. This idea is implemented in a compositional semantics by making use of (Skolemized) choice functions. An appealing feature of the present account is that, unlike its predecessors, it accounts for dependent plural pronouns without c-commanding antecedents in essentially the same way as for those with c-commanding antecedents. It is also shown how this account of dependent plural pronouns can straightforwardly be augmented with set indices to account for partial binding