Computability, G\"odel's Incompleteness Theorem, and an inherent limit on the predictability of evolution

The process of evolutionary diversification unfolds in a vast genotypic space of potential outcomes. During the past century there have been remarkable advances in the development of theory for this diversification, and the theory's success rests, in part, on the scope of its applicability. A great deal of this theory focuses on a relatively small subset of the space of potential genotypes, chosen largely based on historical or contemporary patterns, and then predicts the evolutionary dynamics within this pre-defined set. To what extent can such an approach be pushed to a broader perspective that accounts for the potential open-endedness of evolutionary diversification? There have been a number of significant theoretical developments along these lines but the question of how far such theory can be pushed has not been addressed. Here a theorem is proven demonstrating that, because of the digital nature of inheritance, there are inherent limits on the kinds of questions that can be answered using such an approach. In particular, even in extremely simple evolutionary systems a complete theory accounting for the potential open-endedness of evolution is unattainable unless evolution is progressive. The theorem is closely related to G\"odel's Incompleteness Theorem and to the Halting Problem from computability theory.Comment: Journal of the Royal Society, Interface 201

When logic meets engineering: introduction to logical issues in the history and philosophy of computer science

Introduction to a Journal Special issue on Logical Issues in the History and Philosophy of Computer Scienc

Identification and in vitro Analysis of the GatD/MurT Enzyme-Complex Catalyzing Lipid II Amidation in Staphylococcus aureus

The peptidoglycan of Staphylococcus aureus is characterized by a high degree of crosslinking and almost completely lacks free carboxyl groups, due to amidation of the D-glutamic acid in the stem peptide. Amidation of peptidoglycan has been proposed to play a decisive role in polymerization of cell wall building blocks, correlating with the crosslinking of neighboring peptidoglycan stem peptides. Mutants with a reduced degree of amidation are less viable and show increased susceptibility to methicillin. We identified the enzymes catalyzing the formation of D-glutamine in position 2 of the stem peptide. We provide biochemical evidence that the reaction is catalyzed by a glutamine amidotransferase-like protein and a Mur ligase homologue, encoded by SA1707 and SA1708, respectively. Both proteins, for which we propose the designation GatD and MurT, are required for amidation and appear to form a physically stable bi-enzyme complex. To investigate the reaction in vitro we purified recombinant GatD and MurT His-tag fusion proteins and their potential substrates, i.e. UDP-MurNAc-pentapeptide, as well as the membrane-bound cell wall precursors lipid I, lipid II and lipid II-Gly5. In vitro amidation occurred with all bactoprenol-bound intermediates, suggesting that in vivo lipid II and/or lipid II-Gly5 may be substrates for GatD/MurT. Inactivation of the GatD active site abolished lipid II amidation. Both, murT and gatD are organized in an operon and are essential genes of S. aureus. BLAST analysis revealed the presence of homologous transcriptional units in a number of gram-positive pathogens, e.g. Mycobacterium tuberculosis, Streptococcus pneumonia and Clostridium perfringens, all known to have a D-iso-glutamine containing PG. A less negatively charged PG reduces susceptibility towards defensins and may play a general role in innate immune signaling

A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of K\"ahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an $n$-dimensional cone a rescaling of the canonical potential is an $n$-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde

Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets

We investigate a class of nominal algebraic Henkin-style models for the simply typed lambda-calculus in which variables map to names in the denotation and lambda-abstraction maps to a (non-functional) name-abstraction operation. The resulting denotations are smaller and better-behaved, in ways we make precise, than functional valuation-based models. Using these new models, we then develop a generalisation of \lambda-term syntax enriching them with existential meta-variables, thus yielding a theory of incomplete functions. This incompleteness is orthogonal to the usual notion of incompleteness given by function abstraction and application, and corresponds to holes and incomplete objects.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

The good, the bad and the ugly

This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form of the Bad Company objection and introduce an epistemic issue connected to this general problem that will be the focus of the rest of the paper. In the third section, we present an alternative approach within philosophy of mathematics, a view that emerges from Hilbert's Grundlagen der Geometrie (1899, Leipzig: Teubner; Foundations of geometry (trans.: Townsend, E.). La Salle, Illinois: Open Court, 1959.). We will argue that Bad Company-style worries, and our concomitant epistemic issue, also affects this conception and other foundationalist approaches. In the following sections, we then offer various ways to address our epistemic concern, arguing, in the end, that none resolves the issue. The final section offers our own resolution which, however, runs against the foundationalist spirit of the Scottish neo-logicist program

Simplivariate Models: Uncovering the Underlying Biology in Functional Genomics Data

One of the first steps in analyzing high-dimensional functional genomics data is an exploratory analysis of such data. Cluster Analysis and Principal Component Analysis are then usually the method of choice. Despite their versatility they also have a severe drawback: they do not always generate simple and interpretable solutions. On the basis of the observation that functional genomics data often contain both informative and non-informative variation, we propose a method that finds sets of variables containing informative variation. This informative variation is subsequently expressed in easily interpretable simplivariate components