From Gödel to Einstein: Computability between logic and physics at CiE 2006

Abstract

Computability in Europe (CiE) is an informal network of European scientists working on computability theory, including its foundations, technical development, and applications. Among the aims of the network is advancing our theoretical understanding of what can and cannot be computed, by any means of computation. Its scientific vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs, and machines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory or relativity. Computations may be very general, depending upon the foundations of set theory; or very specific, using the combinatorics of finite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations. The sources of new ideas and methods include practical developments in areas such as neural networks, quantum computation, natural computation, molecular computation, computational learning. Applications are everywhere, especially in algebra, analysis and geometry, or data types and programming. The conferences CiE 2005 in Amsterdam and CiE 2006 in Swansea are at the start of a new conference series CiE-CS that will reconvene in 2007 in Siena, 2008 in Athens, 2009 in Heidelberg and 2010 in Ponta Delgada (Acores). CiE 2006 focused on the variety of logical approaches to all kinds of computational barriers, like the practical and feasible ones, centered around the P vs. NP problem; the computable ones connected to models of computers and programming languages; and the hypercomputable ones. These three kinds of barriers are ubiquitous in computability theory: They occur in classical computability theory on discrete data; in computations with continuous data like higher types, real numbers or topological spaces; and in the physical sciences where we find for example quantum computers, analogue computers, and other computing systems based on classical, relativistic and quantum mechanics. The conference provided an interdisciplinary venue for researchers from computer science and mathematics to exchange ideas, approaches and techniques in their respective work, thereby generating a wider community for work