Experience Implementing a Performant Category-Theory Library in Coq

We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.Comment: The final publication will be available at link.springer.com. This version includes a full bibliography which does not fit in the Springer version; other than the more complete references, this is the version submitted as a final copy to ITP 201

Homotopy Type Theory in Lean

We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201

Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters

The Expectation Monad in Quantum Foundations

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and also the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.Comment: In Proceedings QPL 2011, arXiv:1210.029

Modal dependent type theory and dependent right adjoints

In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory, and spatial and cohesive type theory. In this paper we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K-axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any finite limit category with an adjunction of endofunctors gives rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal lambda-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes

The Principle of General Tovariance

We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining these ideas is topos theory, a framework earlier proposed for physics by Isham and collaborators.Our principle of general tovariance states that any mathematical structure appearing in the laws of physics must be definable in an arbitrary topos (with natural numbers object) and must be preserved under so‐called geometric morphisms. This principle identifies geometric logic as the mathematical language of physics and restricts the constructions and theorems to those valid in intuitionism: neither Aristotle's principle of the excluded third nor Zermelo's Axiom of Choice may be invoked. Subsequently, our equivalence principle states that any algebra of observables (initially defined in the topos Sets ) is empirically equivalent to a commutative one in some other topos

An optimality-based model of the dynamic feedbacks between natural vegetation and the water balance

The hypothesis that vegetation adapts optimally to its environment gives rise to a novel framework for modeling the interactions between vegetation dynamics and the catchment water balance that does not rely on prior knowledge about the vegetation at a particular site. We present a new model based on this framework that includes a multilayered physically based catchment water balance model and an ecophysiological gas exchange and photosynthesis model. The model uses optimization algorithms to find those static and dynamic vegetation properties that would maximize the net carbon profit under given environmental conditions. The model was tested at a savanna site near Howard Springs (Northern Territory, Australia) by comparing the modeled fluxes and vegetation properties with long-term observations at the site. The results suggest that optimality may be a useful way of approaching the prediction and estimation of vegetation cover, rooting depth, and fluxes such as transpiration and CO2 assimilation in ungauged basins without model calibration

CGIAR modeling approaches for resource-constrained scenarios: I. Accelerating crop breeding for a changing climate.

Crop improvement efforts aiming at increasing crop production (quantity, quality) and adapting to climate change have been subject of active research over the past years. But, the question remains 'to what extent can breeding gains be achieved under a changing climate, at a pace sufficient to usefully contribute to climate adaptation, mitigation and food security?'. Here, we address this question by critically reviewing how model-based approaches can be used to assist breeding activities, with particular focus on all CGIAR (formerly the Consultative Group on International Agricultural Research but now known simply as CGIAR) breeding programs. Crop modeling can underpin breeding efforts in many different ways, including assessing genotypic adaptability and stability, characterizing and identifying target breeding environments, identifying tradeoffs among traits for such environments, and making predictions of the likely breeding value of the genotypes. Crop modeling science within the CGIAR has contributed to all of these. However, much progress remains to be done if modeling is to effectively contribute to more targeted and impactful breeding programs under changing climates. In a period in which CGIAR breeding programs are undergoing a major modernization process, crop modelers will need to be part of crop improvement teams, with a common understanding of breeding pipelines and model capabilities and limitations, and common data standards and protocols, to ensure they follow and deliver according to clearly defined breeding products. This will, in turn, enable more rapid and better-targeted crop modeling activities, thus directly contributing to accelerated and more impactful breeding efforts.Online Version of Record before inclusion in an issue