On Equivalence and Canonical Forms in the LF Type Theory

Decidability of definitional equality and conversion of terms into canonical form play a central role in the meta-theory of a type-theoretic logical framework. Most studies of definitional equality are based on a confluent, strongly-normalizing notion of reduction. Coquand has considered a different approach, directly proving the correctness of a practical equivalance algorithm based on the shape of terms. Neither approach appears to scale well to richer languages with unit types or subtyping, and neither directly addresses the problem of conversion to canonical. In this paper we present a new, type-directed equivalence algorithm for the LF type theory that overcomes the weaknesses of previous approaches. The algorithm is practical, scales to richer languages, and yields a new notion of canonical form sufficient for adequate encodings of logical systems. The algorithm is proved complete by a Kripke-style logical relations argument similar to that suggested by Coquand. Crucially, both the algorithm itself and the logical relations rely only on the shapes of types, ignoring dependencies on terms.Comment: 41 page

Internal Parametricity for Cubical Type Theory

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity

\u27\u27Shots from the Pulpit:\u27\u27 An Ethnographic Content Analysis of United States Anti-Gambling Social Movement Documents from 1816-2010

The history of anti-gambling impulses is perhaps as old as the gambling impulse itself, but academic research has thus far neglected the topic of anti-gambling social movements. Using social movement literature as a theoretical guide and ethnographic content analysis as a methodological tool, this paper examines anti-gambling documents produced in the United States over nearly two hundred years. During this period, three distinct periods emerge: first, an early ( 1816-1915) period framed the gambling act on strict religious grounds as an individual sin. This religious framing was then challenged by the rise of more rational and scientifically-based medical discourses on problem gambling (1915-1980). From 1980 through the present, gambling opponents have modified (and in some cases reversed) their arguments- and now incorporate both moral and scientific rhetoric into their claims. Drawing from sociological research and theory, we identify a process of frame inversion in which problem gamblers were once cast as villains to be scorned, but now are characterized as sympathetic victims of the gaming industry. In this first academic study of anti-gambling social movement rhetoric, we develop an illustrative example of how social movements\u27 rhetorical tactics can change over time, and of the kinds of opponents the global gaming industry has faced - and might still face in the future