Algebraic Structure of Combined Traces

Traces and their extension called combined traces (comtraces) are two formal models used in the analysis and verification of concurrent systems. Both models are based on concepts originating in the theory of formal languages, and they are able to capture the notions of causality and simultaneity of atomic actions which take place during the process of a system's operation. The aim of this paper is a transfer to the domain of comtraces and developing of some fundamental notions, which proved to be successful in the theory of traces. In particular, we introduce and then apply the notion of indivisible steps, the lexicographical canonical form of comtraces, as well as the representation of a comtrace utilising its linear projections to binary action subalphabets. We also provide two algorithms related to the new notions. Using them, one can solve, in an efficient way, the problem of step sequence equivalence in the context of comtraces. One may view our results as a first step towards the development of infinite combined traces, as well as recognisable languages of combined traces.Comment: Short variant of this paper, with no proofs, appeared in Proceedings of CONCUR 2012 conferenc

NP-complete Problems and Physical Reality

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and "anthropic computing." The section on soap bubbles even includes some "experimental" results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction

Knapsack Problems in Groups

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure

Oracles and query lower bounds in generalised probabilistic theories

We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying three natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle to be well-defined. The three principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as the marginal of a pure state of a larger system), and strong symmetry existence of non-trivial reversible transformations). Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, then the corresponding lower bound in theories lying at the kth level of Sorkin's hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum oracle needed to solve certain problems are not optimal in the space of all generalised probabilistic theories, although it is not yet known whether the optimal bounds are achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue "Foundational Aspects of Quantum Information" in Foundations of Physic