Towards the Formal Specification and Verification of Maple Programs

In this paper, we present our ongoing work and initial results on the formal specification and verification of MiniMaple (a substantial subset of Maple with slight extensions) programs. The main goal of our work is to find behavioral errors in such programs w.r.t. their specifications by static analysis. This task is more complex for widely used computer algebra languages like Maple as these are fundamentally different from classical languages: they support non-standard types of objects such as symbols, unevaluated expressions and polynomials and require abstract computer algebraic concepts and objects such as rings and orderings etc. As a starting point we have defined and formalized a syntax, semantics, type system and specification language for MiniMaple

Termination of Triangular Integer Loops is Decidable

We consider the problem whether termination of affine integer loops is decidable. Since Tiwari conjectured decidability in 2004, only special cases have been solved. We complement this work by proving decidability for the case that the update matrix is triangular.Comment: Full version (with proofs) of a paper published in the Proceedings of the 31st International Conference on Computer Aided Verification (CAV '19), New York, NY, USA, Lecture Notes in Computer Science, Springer-Verlag, 201

The Origin of Solar Activity in the Tachocline

Solar active regions, produced by the emergence of tubes of strong magnetic field in the photosphere, are restricted to within 35 degrees of the solar equator. The nature of the dynamo processes that create and renew these fields, and are therefore responsible for solar magnetic phenomena, are not well understood. We analyze the magneto-rotational stability of the solar tachocline for general field geometry. This thin region of strong radial and latitudinal differential rotation, between the radiative and convective zones, is unstable at latitudes above 37 degrees, yet is stable closer to the equator. We propose that small-scale magneto-rotational turbulence prevents coherent magnetic dynamo action in the tachocline except in the vicinity of the equator, thus explaining the latitudinal restriction of active regions. Tying the magnetic dynamo to the tachocline elucidates the physical conditions and processes relevant to solar magnetism.Comment: 10 pages, 1 figure, accepted for publication in ApJ

The origin of grand minima in the sunspot cycle

One of the most striking aspects of the 11-year sunspot cycle is that there have been times in the past when some cycles went missing, a most well-known example of this being the Maunder minimum during 1645-1715. Analyses of cosmogenic isotopes (C14 and Be10) indicated that there were about 27 grand minima in the last 11,000 yr, implying that about 2.7% of the solar cycles had conditions appropriate for forcing the Sun into grand minima. We address the question how grand minima are produced and specifically calculate the frequency of occurrence of grand minima from a theoretical dynamo model. We assume that fluctuations in the poloidal field generation mechanism and the meridional circulation produce irregularities of sunspot cycles. Taking these fluctuations to be Gaussian and estimating the values of important parameters from the data of last 28 solar cycles, we show from our flux transport dynamo model that about 1-4% of the sunspot cycles may have conditions suitable for inducing grand minima.Comment: Accepted for publication in Physical Review Letter

Changes in heart rate variability and QT variability during the first trimester of pregnancy

The risk of new-onset arrhythmia during pregnancy is high, presumably relating to changes in both haemodynamic and cardiac autonomic function. The ability to non-invasively assess an individual's risk of developing arrhythmia during pregnancy would therefore be clinically significant. We aimed to quantify electrocardiographic temporal characteristics during the first trimester of pregnancy and to compare these with non-pregnant controls.Ninety-nine pregnant women and sixty-three non-pregnant women underwent non-invasive cardiovascular and haemodynamic assessment during a protocol consisting of various physiological states (postural manoeurvres, light exercise and metronomic breathing). Variables measured included stroke volume, cardiac output, heart rate, heart rate variability, QT and QT variability and QTVI (a measure of the variability of QT relative to that of RR).Heart rate (p < 0.0005, p < 0.0005, p < 0.0005) and cardiac output (p = 0.043, p < 0.0005, p < 0.0005) were greater in pregnant women in all physiological states (respectively for the supine position, light exercise and metronomic breathing state), whilst stroke volume was lower in pregnancy only during the supine position (p < 0.0005). QTe (Q wave onset to T wave end) and QTa (T wave apex) were significantly shortened (p < 0.05) and QTeVI and QTaVI were increased in pregnancy in all physiological states (p < 0.0005). QT variability (p < 0.002) was greater in pregnant women during the supine position, whilst heart rate variability was reduced in pregnancy in all states (p < 0.0005).Early pregnancy is associated with substantial changes in heart rate variability, reflecting a reduction in parasympathetic tone and an increase in sympathetic activity. QTVI shifted to a less favourable value, reflecting a greater than normal amount of QT variability. QTVI appears to be a useful method for quantifying changes in QT variability relative to RR (or heart rate) variability, being sensitive not only to physiological state but also to gestational age. We support the use of non-invasive markers of cardiac electrical variability to evaluate the risk of arrhythmic events in pregnancy, and we recommend the use of multiple physiological states during the assessment protocol

Towards A Mean-Field Formulation Of The Babcock-Leighton Type Solar Dynamo. I. Alpha Coefficient Versus Durney's Double Ring Approach

We develop a model of the solar dynamo in which, on the one hand, we follow the Babcock-Leighton approach to include surface processes like the production of poloidal field from the decay of active regions, and, on the other hand, we attempt to develop a mean field theory that can be studied in quantitative detail. One of the main challenges in developing such models is to treat the buoyant rise of toroidal field and the production of poloidal field from it near the surface. We build up a dynamo model with two contrasting methods of treating buoyancy. In one method, we incorporate the generation of the poloidal field near the solar surface by Durney's procedure of double ring eruption. In the second method, the poloidal field generation is treated by a positive alpha-effect concentrated near the solar surface, coupled with an algorithm for handling buoyancy. The two methods are found to give qualitatively similar results.Comment: 32 pages, 27 figures, uses aastex.cls and epsfig.st

Lifting CDCL to template-based abstract domains for program verification

The success of Conflict Driven Clause Learning (CDCL) for Boolean satisfiability has inspired adoption in other domains. We present a novel lifting of CDCL to program analysis called Abstract Conflict Driven Learning for Programs (ACDLP). ACDLP alternates between model search, which performs over-approximate deduction with constraint propagation, and conflict analysis, which performs under-approximate abduction with heuristic choice. We instantiate the model search and conflict analysis algorithms with an abstract domain of template polyhedra, strictly generalizing CDCL from the Boolean lattice to a richer lattice structure. Our template polyhedra can express intervals, octagons and restricted polyhedral constraints over program variables. We have implemented ACDLP for automatic bounded safety verification of C programs. We evaluate the performance of our analyser by comparing with CBMC, which uses Boolean CDCL, and Astrée, a commercial abstract interpretation tool. We observe two orders of magnitude reduction in the number of decisions, propagations, and conflicts as well as a 1.5x speedup in runtime compared to CBMC. Compared to Astrée, ACDLP solves twice as many benchmarks and has much higher precision. This is the first instantiation of CDCL with a template polyhedra abstract domain

The Three-dimensional Evolution of Rising, Twisted Magnetic Flux Tubes in a Gravitationally Stratified Model Convection Zone

We present three-dimensional numerical simulations of the rise and fragmentation of twisted, initially horizontal magnetic flux tubes which evolve into emerging Omega-loops. The flux tubes rise buoyantly through an adiabatically stratified plasma that represents the solar convection zone. The MHD equations are solved in the anelastic approximation, and the results are compared with studies of flux tube fragmentation in two dimensions. We find that if the initial amount of field line twist is below a critical value, the degree of fragmentation at the apex of a rising Omega-loop depends on its three-dimensional geometry: the greater the apex curvature of a given Omega-loop, the lesser the degree of fragmentation of the loop as it approaches the photosphere. Thus, the amount of initial twist necessary for the loop to retain its cohesion can be reduced substantially from the two-dimensional limit. The simulations also suggest that as a fragmented flux tube emerges through a relatively quiet portion of the solar disk, extended crescent-shaped magnetic features of opposite polarity should form and steadily recede from one another. These features eventually coalesce after the fragmented portion of the Omega-loop emerges through the photosphere.Comment: 17 pages, 17 figures, uses AAS LaTeX macros v5.0. ApJ, in pres

Proving Safety with Trace Automata and Bounded Model Checking

Loop under-approximation is a technique that enriches C programs with additional branches that represent the effect of a (limited) range of loop iterations. While this technique can speed up the detection of bugs significantly, it introduces redundant execution traces which may complicate the verification of the program. This holds particularly true for verification tools based on Bounded Model Checking, which incorporate simplistic heuristics to determine whether all feasible iterations of a loop have been considered. We present a technique that uses \emph{trace automata} to eliminate redundant executions after performing loop acceleration. The method reduces the diameter of the program under analysis, which is in certain cases sufficient to allow a safety proof using Bounded Model Checking. Our transformation is precise---it does not introduce false positives, nor does it mask any errors. We have implemented the analysis as a source-to-source transformation, and present experimental results showing the applicability of the technique