Classical Mathematics for a Constructive World

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.Comment: v2: Final copy for publicatio

On the strength of dependent products in the type theory of Martin-L\"of

One may formulate the dependent product types of Martin-L\"of type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the other constructors of type theory. It is known that the latter rules are at least as strong as the former: we show that they are in fact strictly stronger. We also show, in the presence of the identity types, that the elimination rule for dependent products--which is a "higher-order" inference rule in the sense of Schroeder-Heister--can be reformulated in a first-order manner. Finally, we consider the principle of function extensionality in type theory, which asserts that two elements of a dependent product type which are pointwise propositionally equal, are themselves propositionally equal. We demonstrate that the usual formulation of this principle fails to verify a number of very natural propositional equalities; and suggest an alternative formulation which rectifies this deficiency.Comment: 18 pages; v2: final journal versio

Analyzing the Effects of Transit Network Change on Agency Performance and Riders in a Decentralized, Small-to-Mid-sized US Metropolitan Area: A Case Study of Tallahassee, Florida, MTI Report 12-04

On July 11, 2011, StarMetro, the local public transit agency in Tallahassee, Florida, restructured its entire bus network from a downtown-focused radial system to a decentralized, grid-like system that local officials and agency leaders believed would better serve the dispersed local pattern of population and employment. The new, decentralized network is based on radial routes serving the major arterial roads and new crosstown routes linking the outer parts of the city, where population and employment is growing. Local officials and agency staff hoped the change would increase transit’s attractiveness and usefulness to the community. One year after the service restructuring, overall performance results are similar to those experienced in other cities that have implemented major service changes. Overall ridership and productivity are lower than before the service restructuring, due to the short time frame for rider adjustments and longer-than-anticipated headways, but new ridership has appeared in previously un-served or under-served corridors and neighborhoods. The service restructuring resulted in longer walks to bus stops, due to the removal of stops from many neighborhoods and their relocation to major roads, but overall transit travel times are shorter due to more direct routing. No particular neighborhoods or community groups disproportionately benefited from or were harmed by the change. The service restructuring was supported by some segments of the community who viewed the older system as ill-suited to the increasingly decentralized community, while it was opposed by other community stakeholders who worried about the loss of service in some neighborhoods and issues of access and safety, particularly affecting elderly and disabled riders, at new stop locations. StarMetro’s extensive public outreach efforts and ongoing service adjustments have reduced the intensity of the opposition to the service restructuring over time, although some segments of the community continue to voice their concerns about the effects of the change on transit-dependent, disabled, and elderly riders

Time and Location Aware Mobile Data Pricing

Mobile users' correlated mobility and data consumption patterns often lead to severe cellular network congestion in peak hours and hot spots. This paper presents an optimal design of time and location aware mobile data pricing, which incentivizes users to smooth traffic and reduce network congestion. We derive the optimal pricing scheme through analyzing a two-stage decision process, where the operator determines the time and location aware prices by minimizing his total cost in Stage I, and each mobile user schedules his mobile traffic by maximizing his payoff (i.e., utility minus payment) in Stage II. We formulate the two-stage decision problem as a bilevel optimization problem, and propose a derivative-free algorithm to solve the problem for any increasing concave user utility functions. We further develop low complexity algorithms for the commonly used logarithmic and linear utility functions. The optimal pricing scheme ensures a win-win situation for the operator and users. Simulations show that the operator can reduce the cost by up to 97.52% in the logarithmic utility case and 98.70% in the linear utility case, and users can increase their payoff by up to 79.69% and 106.10% for the two types of utilities, respectively, comparing with a time and location independent pricing benchmark. Our study suggests that the operator should provide price discounts at less crowded time slots and locations, and the discounts need to be significant when the operator's cost of provisioning excessive traffic is high or users' willingness to delay traffic is low.Comment: This manuscript serves as the online technical report of the article accepted by IEEE Transactions on Mobile Computin