Uniform Random Sampling Product Configurations of Feature Models That Have Numerical Features

Analyses of Software Product Lines (SPLs) rely on automated solvers to navigate complex dependencies among features and find legal configurations. Often these analyses do not support numerical features with constraints because propositional formulas use only Boolean variables. Some automated solvers can represent numerical features natively, but are limited in their ability to count and Uniform Random Sample (URS) conigurations, which are key operations to derive unbiased statistics on configuration spaces. Bit-blasting is a technique to encode numerical constraints as propositional formulas. We use bit-blasting to encode Boolean and numerical constraints so that we can exploit existing #SAT solvers to count and URS conigurations. Compared to state-of-art Satisfiability Modulo Theory and Constraint Programming solvers, our approach has two advantages: 1) faster and more scalable coniguration counting and 2) reliable URS of SPL configurations. We also show that our work can be used to extend prior SAT-based SPL analyses to support numerical features and constraints.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Transforming numerical feature models into propositional formulas and the universal variability language

Real-world Software Product Lines (SPLs) need Numerical Feature Models (NFMs) whose features have not only boolean values that satisfy boolean constraints but also have numeric attributes that satisfy arithmetic constraints. An essential operation on NFMs finds near-optimal performing products, which requires counting the number of SPL products. Typical constraint satisfaction solvers perform poorly on counting and sampling. Nemo (Numbers, features, models) is a tool that supports NFMs by bit-blasting, the technique that encodes arithmetic expressions as boolean clauses. The newest version, Nemo2, translates NFMs to propositional formulas and the Universal Variability Language (UVL). By doing so, products can be counted efficiently by #SAT and Binary Decision Tree solvers, enabling finding near-optimal products. This article evaluates Nemo2 with a large set of synthetic and colossal real-world NFMs, including complex arithmetic constraints and counting and sampling experiments. We empirically demonstrate the viability of Nemo2 when counting and sampling large and complex SPLs.Munoz, Pinto and Fuentes work is supported by the European Union’s H2020 research and innovation programme under grant agreement DAEMON 101017109, by the projects co-financed by FEDER, Spain funds LEIA UMA18-FEDERJA-15, IRIS PID2021- 122812OB-I00 (MCI/AEI), and the PRE2019-087496 grant from the Ministerio de Ciencia e Innovación. Funding for open access charge: Universidad de Málaga / CBUA

Internal kinematics and structure of the bulge globular cluster NGC 6569

In the context of a project aimed at characterizing the properties of star clusters in the Galactic bulge, here we present the determination of the internal kinematics and structure of the massive globular cluster NGC 6569. The kinematics has been studied by means of an unprecedented spectroscopic dataset acquired in the context of the ESO-VLT Multi-Instrument Kinematic Survey (MIKiS) of Galactic globular clusters, combining the observations from four different spectrographs. We measured the line-of-sight velocity of a sample of almost 1300 stars distributed between ~0.8" and 770" from the cluster center. From a sub-sample of high-quality measures, we determined the velocity dispersion profile of the system over its entire radial extension (from ~ 5" to ~ 200" from the center), finding the characteristic behavior usually observed in globular clusters, with a constant inner plateau and a declining trend at larger radii. The projected density profile of the cluster has been obtained from resolved star counts, by combining high-resolution photometric data in the center, and the Gaia EDR3 catalog radially extended out to ~20' for a proper sampling of the Galactic field background. The two profiles are properly reproduced by the same King model, from which we estimated updated values of the central velocity dispersion, main structural parameters (such as the King concentration, the core, half-mass, and tidal radii), total mass, and relaxation times. Our analysis also reveals a hint of ordered rotation in an intermediate region of the cluster (40"<r<90", corresponding to $2 r_c<r<4.5 r_c$), but additional data are required to properly assess this possibility.Comment: Accepted for publication in The Astrophysical Journal; 21 pages, 10 figures, 4 table