Multi-level Contextual Type Theory

Contextual type theory distinguishes between bound variables and meta-variables to write potentially incomplete terms in the presence of binders. It has found good use as a framework for concise explanations of higher-order unification, characterize holes in proofs, and in developing a foundation for programming with higher-order abstract syntax, as embodied by the programming and reasoning environment Beluga. However, to reason about these applications, we need to introduce meta^2-variables to characterize the dependency on meta-variables and bound variables. In other words, we must go beyond a two-level system granting only bound variables and meta-variables. In this paper we generalize contextual type theory to n levels for arbitrary n, so as to obtain a formal system offering bound variables, meta-variables and so on all the way to meta^n-variables. We obtain a uniform account by collapsing all these different kinds of variables into a single notion of variabe indexed by some level k. We give a decidable bi-directional type system which characterizes beta-eta-normal forms together with a generalized substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

Circulating Sclerostin responses to acute weight and non weight bearing sport activity in pre adolescent males

Mechanical loading, i.e. physical activity and/or exercise, promotes bone formation during growth. Sclerostin, a glycoprotein, mediates osteocytes' response to mechanical loading by inhibiting the Wnt/lf-catenin pathway thereby inhibiting bone formation.Published versio

Prototyping a functional language using higher-order logic programming: a functional pearl on learning the ways of λProlog/Makam

We demonstrate how the framework of higher-order logic programming , as exemplified in the λProlog language design, is a prime vehicle for rapid prototyping of implementations for programming languages with sophisticated type systems. We present the literate development of a type checker for a language with a number of complicated features, culminating in a standard ML-style core with algebraic datatypes and type generalization, extended with staging constructs that are generic over a separately defined language of terms. We add each new feature in sequence, with little to no changes to existing code. Scaling the higher-order logic programming approach to this setting required us to develop approaches to challenges like complex variable binding patterns in object languages and performing generic structural traversals of code, making use of novel constructions in the setting of λProlog, such as GADTs and generic programming. For our development, we make use of Makam, a new implementation of λProlog, which we introduce in tutorial style as part of our (quasi-)literate development. </jats:p