The internal description of a causal set: What the universe looks like from the inside

We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss.Comment: Version to appear in Comm.Math.Phys. (minor modifications). 37 pages, several eps figure

Locality and Translations in Braided Ribbon Networks

An overview of microlocality in braided ribbon networks is presented. Following this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is followed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.Comment: 12 pages, 12 figure

Quantum causal histories

Quantum causal histories are defined to be causal sets with Hilbert spaces attached to each event and local unitary evolution operators. The reflexivity, antisymmetry, and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators. A quantum causal history in which transitivity holds can be treated as ``directed'' topological quantum field theory. Two examples of such histories are described.Comment: 16 pages, epsfig latex. Some clarifications, minor corrections and references added. Version to appear in Classical and Quantum Gravit

On routing-optimal networks for multiple unicasts

In this paper, we consider the problem of multiple unicast sessions over a directed acyclic graph. It is well known that linear network coding is insufficient for achieving the capacity region, in the general case. However, there exist networks for which routing is sufficient to achieve the whole rate region, and we refer to them as routing-optimal networks. We identify a class of routing-optimal networks, which we refer to as information-distributive networks, defined by three topological features. Due to these features, for each rate vector achieved by network coding, there is always a routing scheme such that it achieves the same rate vector, and the traffic transmitted through the network is exactly the information transmitted over the cut-sets between the sources and the sinks in the corresponding network coding scheme. We present examples of information-distributive networks, including some examples from (1) index coding and (2) from a single unicast session with hard deadline constraint. © 2014 IEEE