Backward Reachability of Array-based Systems by SMT solving: Termination and Invariant Synthesis

The safety of infinite state systems can be checked by a backward reachability procedure. For certain classes of systems, it is possible to prove the termination of the procedure and hence conclude the decidability of the safety problem. Although backward reachability is property-directed, it can unnecessarily explore (large) portions of the state space of a system which are not required to verify the safety property under consideration. To avoid this, invariants can be used to dramatically prune the search space. Indeed, the problem is to guess such appropriate invariants. In this paper, we present a fully declarative and symbolic approach to the mechanization of backward reachability of infinite state systems manipulating arrays by Satisfiability Modulo Theories solving. Theories are used to specify the topology and the data manipulated by the system. We identify sufficient conditions on the theories to ensure the termination of backward reachability and we show the completeness of a method for invariant synthesis (obtained as the dual of backward reachability), again, under suitable hypotheses on the theories. We also present a pragmatic approach to interleave invariant synthesis and backward reachability so that a fix-point for the set of backward reachable states is more easily obtained. Finally, we discuss heuristics that allow us to derive an implementation of the techniques in the model checker MCMT, showing remarkable speed-ups on a significant set of safety problems extracted from a variety of sources.Comment: Accepted for publication in Logical Methods in Computer Scienc

Locality and topology with fat link overlap actions

We study the locality and topological properties of fat link clover overlap (FCO) actions. We find that a small amount of fattening (2-4 steps of APE or 1 step of HYP) already results in greatly improved properties compared to the Wilson overlap (WO). We present a detailed study of the localisation of the FCO and its connection to the density of low modes of $A^\dagger A$. In contrast to the Wilson overlap, on quenched gauge backgrounds we do not find any dependence of the localization of the FCO on the gauge coupling. This suggests that the FCO remains local in the continuum limit. The FCO also faithfully reproduces the zero mode wave functions of typical lattice instantons, not like the Wilson overlap. After a general discussion of different lattice definitions of the topological charge we also show that the FCO together with the Boulder charge are likely to satisfy the index theorem in the continuum limit. Finally, we present a high statistics computation of the quenched topological susceptibility with the FCO action.Comment: 19 pages, LaTe

Topology and chiral symmetry breaking in SU(N) gauge theories

We study the low-lying eigenmodes of the lattice overlap Dirac operator for SU(N) gauge theories with N=2,3,4 and 5 colours. We define a fermionic topological charge from the zero-modes of this operator and show that, as N grows, any disagreement with the topological charge obtained by cooling the fields, becomes rapidly less likely. By examining the fields where there is a disagreement, we are able to show that the Dirac operator does not resolve instantons below a critical size of about rho = 2.5 a, but resolves the larger, more physical instantons. We investigate the local chirality of the near-zero modes and how it changes as we go to larger N. We observe that the local chirality of these modes, which is prominent for SU(2) and SU(3), becomes rapidly weaker for larger N and is consistent with disappearing entirely in the limit of N -> infinity. We find that this is not due to the observed disappearance of small instantons at larger N.Comment: 41 pages, 12 figures, RevTe

A Simplex-Based Extension of Fourier-Motzkin for Solving Linear Integer Arithmetic

International audienceThis paper describes a novel decision procedure for quantifier-free linear integer arithmetic. Standard techniques usually relax the initial problem to the rational domain and then proceed either by projection (e.g. Omega-Test) or by branching/cutting methods (branch-and-bound, branch-and-cut, Gomory cuts). Our approach tries to bridge the gap between the two techniques: it interleaves an exhaustive search for a model with bounds inference. These bounds are computed provided an oracle capable of finding constant positive linear combinations of affine forms. We also show how to design an efficient oracle based on the Simplex procedure. Our algorithm is proved sound, complete, and terminating and is implemented in the Alt-Ergo theorem prover. Experimental results are promising and show that our approach is competitive with state-of-the-art SMT solvers

Effect of vessel wettability on the foamability of "ideal" surfactants and "real-world" beer heads

The ability to tailor the foaming properties of a solution by controlling its chemical composition is highly desirable and has been the subject of extensive research driven by a range of applications. However, the control of foams by varying the wettability of the foaming vessel has been less widely reported. This work investigates the effect of the wettability of the side walls of vessels used for the in situ generation of foam by shaking aqueous solutions of three different types of model surfactant systems (non-ionic, anionic and cationic surfactants) along with four different beers (Guinness Original, Banks’s Bitter, Bass No 1 and Harvest Pale). We found that hydrophilic vials increased the foamability only for the three model systems but increased foam stability for all foams except the model cationic system. We then compared stability of beer foams produced by shaking and pouring and demonstrated weak qualitative agreement between both foam methods. We also showed how wettability of the glass controls bubble nucleation for beers and champagne and used this effect to control exactly where bubbles form using simple wettability patterns

Prospects for Probing the Spacetime of Sgr A* with Pulsars

The discovery of radio pulsars in compact orbits around Sgr A* would allow an unprecedented and detailed investigation of the spacetime of the supermassive black hole. This paper shows that pulsar timing, including that of a single pulsar, has the potential to provide novel tests of general relativity, in particular its cosmic censorship conjecture and no-hair theorem for rotating black holes. These experiments can be performed by timing observations with 100 micro-second precision, achievable with the Square Kilometre Array for a normal pulsar at frequency above 15 GHz. Based on the standard pulsar timing technique, we develop a method that allows the determination of the mass, spin, and quadrupole moment of Sgr A*, and provides a consistent covariance analysis of the measurement errors. Furthermore, we test this method in detailed mock data simulations. It seems likely that only for orbital periods below ~0.3 yr is there the possibility of having negligible external perturbations. For such orbits we expect a ~10^-3 test of the frame dragging and a ~10^-2 test of the no-hair theorem within 5 years, if Sgr A* is spinning rapidly. Our method is also capable of identifying perturbations caused by distributed mass around Sgr A*, thus providing high confidence in these gravity tests. Our analysis is not affected by uncertainties in our knowledge of the distance to the Galactic center, R0. A combination of pulsar timing with the astrometric results of stellar orbits would greatly improve the measurement precision of R0.Comment: 12 pages, 10 Figures, accepted for publication in Ap

Linear Momentum Density in Quasistatic Electromagnetic Systems

We discuss a couple of simple quasistatic electromagnetic systems in which the density of electromagnetic linear momentum can be easily computed. The examples are also used to illustrate how the total electromagnetic linear momentum, which may also be calculated by using the vector potential, can be understood as a consequence of the violation of the action-reaction principle, because a non-null external force is required to maintain constant the mechanical linear momentum. We show how one can avoid the divergence in the interaction linear electromagnetic momentum of a system composed by an idealization often used in textbooks (an infinite straight current) and a point charge.Comment: 22 pages, 5 figures, to appear in Eur. J. Phy

Multiscale modelling of vascular tumour growth in 3D: the roles of domain size & boundary condition

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour