Deliverables: a categorical approach to program development in type theory

This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higher-order primitive recursion and higher-order intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The Sigma-types of the calculus enable us to achieve this. There are many similarities with the subset interpretation of Martin-Löf type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical book-keeping, allowing one to concentrate on the essential structure of algorithms. I demonstrate our methodology with a number of small examples, culminating in a machine-checked proof of the Chinese remainder theorem, showing the utility of the deliverables idea. Some drawbacks are also encountered. I consider also semantic aspects of deliverables, examining the definitions in an abstract setting, again firmly based on category theory. The aim is to overcome the clumsiness of the language of categorical combinators, using dependent type theories and their interpretation in fibrations. I elaborate a concrete instance based on the category of sets, which generalises to an arbitrary topos. In the process, I uncover a subsystem of ECC within which one may speak of deliverables defined over the topos. In the presence of enough extra structure, the interpretation extends to the whole of ECC. The wheel turns full circle

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCalpha to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification

Multiple novel prostate cancer susceptibility signals identified by fine-mapping of known risk loci among Europeans

Genome-wide association studies (GWAS) have identified numerous common prostate cancer (PrCa) susceptibility loci. We have fine-mapped 64 GWAS regions known at the conclusion of the iCOGS study using large-scale genotyping and imputation in 25 723 PrCa cases and 26 274 controls of European ancestry. We detected evidence for multiple independent signals at 16 regions, 12 of which contained additional newly identified significant associations. A single signal comprising a spectrum of correlated variation was observed at 39 regions; 35 of which are now described by a novel more significantly associated lead SNP, while the originally reported variant remained as the lead SNP only in 4 regions. We also confirmed two association signals in Europeans that had been previously reported only in East-Asian GWAS. Based on statistical evidence and linkage disequilibrium (LD) structure, we have curated and narrowed down the list of the most likely candidate causal variants for each region. Functional annotation using data from ENCODE filtered for PrCa cell lines and eQTL analysis demonstrated significant enrichment for overlap with bio-features within this set. By incorporating the novel risk variants identified here alongside the refined data for existing association signals, we estimate that these loci now explain ∼38.9% of the familial relative risk of PrCa, an 8.9% improvement over the previously reported GWAS tag SNPs. This suggests that a significant fraction of the heritability of PrCa may have been hidden during the discovery phase of GWAS, in particular due to the presence of multiple independent signals within the same regio

Multiple loci on 8q24 associated with prostate cancer susceptibility

Previous studies have identified multiple loci on 8q24 associated with prostate cancer risk. We performed a comprehensive analysis of SNP associations across 8q24 by genotyping tag SNPs in 5,504 prostate cancer cases and 5,834 controls. We confirmed associations at three previously reported loci and identified additional loci in two other linkage disequilibrium blocks (rs1006908: per-allele OR = 0.87, P = 7.9 x 10(-8); rs620861: OR = 0.90, P = 4.8 x 10(-8)). Eight SNPs in five linkage disequilibrium blocks were independently associated with prostate cancer susceptibility

Common variation near CDKN1A, POLD3 and SHROOM2 influences colorectal cancer risk

We performed a meta-analysis of five genome-wide association studies to identify common variants influencing colorectal cancer (CRC) risk comprising 8,682 cases and 9,649 controls. Replication analysis was performed in case-control sets totaling 21,096 cases and 19,555 controls. We identified three new CRC risk loci at 6p21 (rs1321311, near CDKN1A; P = 1.14 × 10 -10), 11q13.4 (rs3824999, intronic to POLD3; P = 3.65 × 10 -10) and Xp22.2 (rs5934683, near SHROOM2; P = 7.30 × 10 -10) This brings the number of independent loci associated with CRC risk to 20 and provides further insight into the genetic architecture of inherited susceptibility to CRC.</p

Identification of seven new prostate cancer susceptibility loci through a genome-wide association study

Prostate cancer (PrCa) is the most frequently diagnosed male cancer in developed countries. To identify common PrCa susceptibility alleles, we have previously conducted a genome-wide association study in which 541, 129 SNPs were genotyped in 1,854 PrCa cases with clinically detected disease and 1,894 controls. We have now evaluated promising associations in a second stage, in which we genotyped 43,671 SNPs in 3,650 PrCa cases and 3,940 controls, and a third stage, involving an additional 16,229 cases and 14,821 controls from 21 studies. In addition to previously identified loci, we identified a further seven new prostate cancer susceptibility loci on chromosomes 2, 4, 8, 11, and 22 (P=1.6×10−8 to P=2.7×10−33)