We explore the features of a user interface where formal proofs can be built through gestural actions. In particular, we show how proof construction steps can be associated to drag-and-drop actions. We argue that this can provide quick and intuitive proof construction steps. This work builds on theoretical tools coming from deep inference. It also resumes and integrates some ideas of the former proof-by-pointing project.

Quantification of the effect of spatial tumour sampling on the patterns of mutations detected in next-generation sequencing data is largely lacking. Here we use a spatial stochastic cellular automaton model of tumour growth that accounts for somatic mutations, selection, drift and spatial constraints, to simulate multi-region sequencing data derived from spatial sampling of a neoplasm. We show that the spatial structure of a solid cancer has a major impact on the detection of clonal selection and genetic drift from both bulk and single-cell sequencing data. Our results indicate that spatial constrains can introduce significant sampling biases when performing multi-region bulk sampling and that such bias becomes a major confounding factor for the measurement of the evolutionary dynamics of human tumours. We also propose a statistical inference framework that incorporates spatial effects within a growing tumour and so represents a further step forwards in the inference of evolutionary dynamics from genomic data. Our analysis shows that measuring cancer evolution using next-generation sequencing while accounting for the numerous confounding factors remains challenging. However, mechanistic model-based approaches have the potential to capture the sources of noise and better interpret the data.

The immortal strand hypothesis poses that stem cells could produce differentiated progeny while conserving the original template strand, thus avoiding accumulating somatic mutations. However, quantitating the extent of non-random DNA strand segregation in human stem cells remains difficult in vivo. Here we show that the change of the mean and variance of the mutational burden with age in healthy human tissues allows estimating strand segregation probabilities and somatic mutation rates. We analysed deep sequencing data from healthy human colon, small intestine, liver, skin and brain. We found highly effective non-random DNA strand segregation in all adult tissues (mean strand segregation probability: 0.98, standard error bounds (0.97,0.99)). In contrast, non-random strand segregation efficiency is reduced to 0.87 (0.78,0.88) in neural tissue during early development, suggesting stem cell pool expansions due to symmetric self-renewal. Healthy somatic mutation rates differed across tissue types, ranging from 3.5 × 10-9/bp/division in small intestine to 1.6 × 10-7/bp/division in skin.

Most tissues in multicellular organisms are maintained by continuous cell renewal processes. However, high turnover of many cells implies a large number of error-prone cell divisions. Hierarchical organized tissue structures with stem cell driven cell differentiation provide one way to prevent the accumulation of mutations, because only few stem cells are long lived. We investigate the deterministic dynamics of cells in such a hierarchical multi compartment model, where each compartment represents a certain stage of cell differentiation. The dynamics of the interacting system is described by ordinary differential equations coupled across compartments. We present analytical solutions for these equations, calculate the corresponding extinction times and compare our results to individual based stochastic simulations. Our general compartment structure can be applied to different tissues, as for example hematopoiesis, the epidermis, or colonic crypts. The solutions provide a description of the average time development of stem cell and non stem cell driven mutants and can be used to illustrate general and specific features of the dynamics of mutant cells in such hierarchically structured populations. We illustrate one possible application of this approach by discussing the origin and dynamics of PIG-A mutant clones that are found in the bloodstream of virtually every healthy adult human. From this it is apparent, that not only the occurrence of a mutant but also the compartment of origin is of importance.

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.

Abstract We define a generic notion of cut that applies to many first-order theories. We prove a generic cut elimination theorem showing that the cut elimination property holds for all theories having a so-called pre-model. As a corollary, we retrieve cut elimination for several axiomatic theories, including Church's simple type theory.

We consider two alternative definitions for the conversion rule in pure type systems. We study the consequences of this choice for the metatheory and point out the related implementation issues. We relate two open problems by showing that if a PTS allows the construction of a fixed point combinator, then Church-Rosser for /spl betaspl eta/-reduction fails. We present a new formalization of Russell's paradox in a slight extension of Martin-Lof's inconsistent theory with Type:Type and show that the resulting term leads to a fix-point construction. The main consequence is that the corresponding system is non-confluent. This example shows that in some typed /spl lambda/-calculi, the Church-Rosser proof for the /spl betaspl eta/-reduction is not purely combinatorial anymore, as in pure /spl lambda/-calculus, but relies on the normalization and thus the logical consistency of the system.>