An Algebraic Framework for Compositional Program Analysis

The purpose of a program analysis is to compute an abstract meaning for a program which approximates its dynamic behaviour. A compositional program analysis accomplishes this task with a divide-and-conquer strategy: the meaning of a program is computed by dividing it into sub-programs, computing their meaning, and then combining the results. Compositional program analyses are desirable because they can yield scalable (and easily parallelizable) program analyses. This paper presents algebraic framework for designing, implementing, and proving the correctness of compositional program analyses. A program analysis in our framework defined by an algebraic structure equipped with sequencing, choice, and iteration operations. From the analysis design perspective, a particularly interesting consequence of this is that the meaning of a loop is computed by applying the iteration operator to the loop body. This style of compositional loop analysis can yield interesting ways of computing loop invariants that cannot be defined iteratively. We identify a class of algorithms, the so-called path-expression algorithms [Tarjan1981,Scholz2007], which can be used to efficiently implement analyses in our framework. Lastly, we develop a theory for proving the correctness of an analysis by establishing an approximation relationship between an algebra defining a concrete semantics and an algebra defining an analysis.Comment: 15 page

The Turbulent Story of X-ray Bursts: Effects of Shear Mixing on Accreting Neutron Stars

During accretion, a neutron star (NS) is spun up as angular momentum is transported through its liquid surface layers. We study the resulting differentially rotating profile, focusing on the impact this has for type I X-ray bursts. The viscous heating is found to be negligible, but turbulent mixing can be activated. Mixing has the greatest impact when the buoyancy at the compositional discontinuity between accreted matter and ashes is overcome. This occurs preferentially at high accretion rates or low spin frequencies and may depend on the ash composition from the previous burst. We then find two new regimes of burning. The first is ignition in a layer containing a mixture of heavier elements with recurrence times as short as ~5-30 minutes, similar to short recurrence time bursts. When mixing is sufficiently strong, a second regime is found where accreted helium mixes deep enough to burn stably, quenching X-ray bursts altogether. The carbon-rich material produced by stable helium burning would be important for triggering and fueling superbursts.Comment: 3 pages, 3 figures. To appear in the proceedings of "Forty Years of Pulsars: Millisecond Pulsars, Magnetars and More" held in Montreal, Canada, August 12-17, 200

Turbulent Mixing in the Surface Layers of Accreting Neutron Stars

During accretion a neutron star (NS) is spun up as angular momentum is transported through its surface layers. We study the resulting differentially rotating profile, focusing on the impact this has for type I X-ray bursts. The predominant viscosity is likely provided by the Tayler-Spruit dynamo. The radial and azimuthal magnetic field components have strengths of ~10^5 G and ~10^10 G, respectively. This leads to nearly uniform rotation at the depths of interest for X-ray bursts. A remaining small shear transmits the accreted angular momentum inward to the NS interior. Though this shear gives little viscous heating, it can trigger turbulent mixing. Detailed simulations will be required to fully understand the consequences of mixing, but our models illustrate some general features. Mixing has the greatest impact when the buoyancy at the compositional discontinuity between accreted matter and ashes is overcome. This occurs at high accretion rates, at low spin frequencies, or may depend on the ashes from the previous burst. We then find two new regimes of burning. The first is ignition in a layer containing a mixture of heavier elements from the ashes. If ignition occurs at the base of the mixed layer, recurrence times as short as ~5-30 minutes are possible. This may explain the short recurrence time of some bursts, but incomplete burning is still needed to explain these bursts' energetics. When mixing is sufficiently strong, a second regime is found where accreted helium mixes deep enough to burn stably, quenching X-ray bursts. We speculate that the observed change in X-ray burst properties near one-tenth the Eddington accretion rate is from this mechanism. The carbon-rich material produced by stable helium burning would be important for triggering and fueling superbursts. (abridged)Comment: Accepted for publication in The Astrophysical Journal, 16 pages, 15 figure

Thermonuclear burst observations for model comparisons: a reference sample

We present a sample of observations of thermonuclear (type-I) X-ray bursts, selected for comparison with numerical models. Provided are examples of four distinct cases of thermonuclear ignition: He-ignition in mixed H/He fuel (case 1 of Fujimoto et al. 1981); He-ignition in pure He fuel, following exhaustion of accreted H by steady burning (case 2); ignition in (almost) pure He accumulated from an evolved donor in an ultracompact system; and an example of a superburst, thought to arise from ignition of a layer of carbon fuel produced as a by-product of more frequent bursts. For regular bursts, we measured the recurrence time and calculated averaged burst profiles from RXTE observations. We have also estimated the recurrence time for pairs of bursts, including those observed during a transient outburst modelled using a numerical ignition code. For each pair of bursts we list the burst properties including recurrence time, fluence and peak flux, the persistent flux level (and inferred accretion rate) as well as the ratio of persistent flux to fluence. In the accompanying material we provide a bolometric lightcurve for each burst, determined from time-resolved spectral analysis. Along with the inferred or adopted parameters for each burst system, including distance, surface gravity, and redshift, these data are suggested as a suitable test cases for ignition models.Comment: 14 pages, 3 figures, 2 tables and associated data provided on-line; accepted by PAS

On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers

For finite sums of non-negative powers of arithmetic progressions the generating functions (ordinary and exponential ones) for given powers are computed. This leads to a two parameter generalization of Stirling and Eulerian numbers. A direct generalization of Bernoulli numbers and their polynomials follows. On the way to find the Faulhaber formula for these sums of powers in terms of generalized Bernoulli polynomials one is led to a one parameter generalization of Bernoulli numbers and their polynomials. Generalized Lah numbers are also considered.Comment: 28 page

Thermonuclear X-ray Bursts: Theory vs. Observations

I review our theoretical understanding of thermonuclear flashes on accreting neutron stars, concentrating on comparisons to observations. Sequences of regular Type I X-ray bursts from GS 1826-24 and 4U 1820-30 are very well described by the theory. I discuss recent work which attempts to use the observed burst properties in these sources to constrain the composition of the accreted material. For GS 1826-24, variations in the burst energetics with accretion rate indicate that the accreted material has solar metallicity; for 4U 1820-30, future observations should constrain the hydrogen fraction, testing evolutionary models. I briefly discuss the global bursting behavior of burst sources, which continues to be a major puzzle. Finally, I turn to superbursts, which naturally fit into the picture as unstable carbon ignition in a thick layer of heavy elements. I present new time-dependent models of the cooling tails of superbursts, and discuss the various interactions between superbursts and normal Type I bursts, and what can be learned from them.Comment: 11 pages, 6 figures; to appear in Proc. of the 2nd BeppoSAX Meeting: "The Restless High-Energy Universe" (Amsterdam, May 5-8, 2003), E.P.J. van den Heuvel, J.J.M. in 't Zand, and R.A.M.J. Wijers (Eds

Deep Neural Networks as Gaussian Processes

It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. Recently, kernel functions which mimic multi-layer random neural networks have been developed, but only outside of a Bayesian framework. As such, previous work has not identified that these kernels can be used as covariance functions for GPs and allow fully Bayesian prediction with a deep neural network. In this work, we derive the exact equivalence between infinitely wide deep networks and GPs. We further develop a computationally efficient pipeline to compute the covariance function for these GPs. We then use the resulting GPs to perform Bayesian inference for wide deep neural networks on MNIST and CIFAR-10. We observe that trained neural network accuracy approaches that of the corresponding GP with increasing layer width, and that the GP uncertainty is strongly correlated with trained network prediction error. We further find that test performance increases as finite-width trained networks are made wider and more similar to a GP, and thus that GP predictions typically outperform those of finite-width networks. Finally we connect the performance of these GPs to the recent theory of signal propagation in random neural networks.Comment: Published version in ICLR 2018. 10 pages + appendi

Alignment-based compositional semantics for instruction following

This paper describes an alignment-based model for interpreting natural language instructions in context. We approach instruction following as a search over plans, scoring sequences of actions conditioned on structured observations of text and the environment. By explicitly modeling both the low-level compositional structure of individual actions and the high-level structure of full plans, we are able to learn both grounded representations of sentence meaning and pragmatic constraints on interpretation. To demonstrate the model's flexibility, we apply it to a diverse set of benchmark tasks. On every task, we outperform strong task-specific baselines, and achieve several new state-of-the-art results.Comment: in proceedings of EMNLP 201

Bounded Expectations: Resource Analysis for Probabilistic Programs

This paper presents a new static analysis for deriving upper bounds on the expected resource consumption of probabilistic programs. The analysis is fully automatic and derives symbolic bounds that are multivariate polynomials of the inputs. The new technique combines manual state-of-the-art reasoning techniques for probabilistic programs with an effective method for automatic resource-bound analysis of deterministic programs. It can be seen as both, an extension of automatic amortized resource analysis (AARA) to probabilistic programs and an automation of manual reasoning for probabilistic programs that is based on weakest preconditions. As a result, bound inference can be reduced to off-the-shelf LP solving in many cases and automatically-derived bounds can be interactively extended with standard program logics if the automation fails. Building on existing work, the soundness of the analysis is proved with respect to an operational semantics that is based on Markov decision processes. The effectiveness of the technique is demonstrated with a prototype implementation that is used to automatically analyze 39 challenging probabilistic programs and randomized algorithms. Experimental results indicate that the derived constant factors in the bounds are very precise and even optimal for many programs

Recent contributions to the calculus of finite differences: a survey

We retrace the recent history of the Umbral Calculus. After studying the classic results concerning polynomial sequences of binomial type, we generalize to a certain type of logarithmic series. Finally, we demonstrate numerous typical examples of our theory. Nous passons en revue ici les resultats recents du calcul ombral. Nous nous interessons tout d'abord aux resultats classique appliqu\'es aux suites de polyn\^omes de type binomial, pius elargions le champ d'\'etude aux series logarithmiques. Enfin nous donnons de nombreaux exemples types d'application de cette th\'eorie