Towards Efficient Verification of Population Protocols

Population protocols are a well established model of computation by anonymous, identical finite state agents. A protocol is well-specified if from every initial configuration, all fair executions reach a common consensus. The central verification question for population protocols is the well-specification problem: deciding if a given protocol is well-specified. Esparza et al. have recently shown that this problem is decidable, but with very high complexity: it is at least as hard as the Petri net reachability problem, which is EXPSPACE-hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class WS3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that WS3 has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership problem for WS3 reduces to solving boolean combinations of linear constraints over N. This allowed us to develop the first software able to automatically prove well-specification for all of the infinitely many possible inputs.Comment: 29 pages, 1 figur

The effects of short-lived radionuclides and porosity on the early thermo-mechanical evolution of planetesimals

The thermal history and internal structure of chondritic planetesimals, assembled before the giant impact phase of chaotic growth, potentially yield important implications for the final composition and evolution of terrestrial planets. These parameters critically depend on the internal balance of heating versus cooling, which is mostly determined by the presence of short-lived radionuclides (SLRs), such as aluminum-26 and iron-60, as well as the heat conductivity of the material. The heating by SLRs depends on their initial abundances, the formation time of the planetesimal and its size. It has been argued that the cooling history is determined by the porosity of the granular material, which undergoes dramatic changes via compaction processes and tends to decrease with time. In this study we assess the influence of these parameters on the thermo-mechanical evolution of young planetesimals with both 2D and 3D simulations. Using the code family I2ELVIS/I3ELVIS we have run numerous 2D and 3D numerical finite-difference fluid dynamic models with varying planetesimal radius, formation time and initial porosity. Our results indicate that powdery materials lowered the threshold for melting and convection in planetesimals, depending on the amount of SLRs present. A subset of planetesimals retained a powdery surface layer which lowered the thermal conductivity and hindered cooling. The effect of initial porosity was small, however, compared to those of planetesimal size and formation time, which dominated the thermo-mechanical evolution and were the primary factors for the onset of melting and differentiation. We comment on the implications of this work concerning the structure and evolution of these planetesimals, as well as their behavior as possible building blocks of terrestrial planets.Comment: 19 pages, 11 figures, 5 tables; accepted for publication in Icarus; for associated video files, see http://timlichtenberg.net/2015_porosity.html or http://dx.doi.org/10.1016/j.icarus.2016.03.00

An Improbable Solution to the Underluminosity of 2M1207B: A Hot Protoplanet Collision Afterglow

We introduce an alternative hypothesis to explain the very low luminosity of the cool (L-type) companion to the ~25 M_Jup ~8 Myr-old brown dwarf 2M1207A. Recently, Mohanty et al. (2007) found that effective temperature estimates for 2M1207B (1600 +- 100 K) are grossly inconsistent with its lying on the same isochrone as the primary, being a factor of ~10 underluminous at all bands between I (0.8 um) and L' (3.6 um). Mohanty et al. explain this discrepency by suggesting that 2M1207B is an 8 M_Jup object surrounded by an edge-on disk comprised of large dust grains producing 2.5^m of achromatic extinction. We offer an alternative explanation: the apparent flux reflects the actual source luminosity. Given the temperature, we infer a small radius (~49,000 km), and for a range of plausible densities, we estimate a mass < M_Jup. We suggest that 2M1207B is a hot protoplanet collision afterglow and show that the radiative timescale for such an object is >~1% the age of the system. If our hypothesis is correct, the surface gravity of 2M1207B should be an order of magnitude lower than predicted by Mohanty et al. (2007).Comment: ApJ Letters, in press (11 pages

Supernova enrichment and dynamical histories of solar-type stars in clusters

We use N-body simulations of star cluster evolution to explore the hypothesis that short-lived radioactive isotopes found in meteorites, such as 26-Al, were delivered to the Sun's protoplanetary disc from a supernova at the epoch of Solar System formation. We cover a range of star cluster formation parameter space and model both clusters with primordial substructure, and those with smooth profiles. We also adopt different initial virial ratios - from cool, collapsing clusters to warm, expanding associations. In each cluster we place the same stellar population; the clusters each have 2100 stars, and contain one massive 25M_Sun star which is expected to explode as a supernova at about 6.6Myr. We determine the number of Solar (G)-type stars that are within 0.1 - 0.3pc of the 25M_Sun star at the time of the supernova, which is the distance required to enrich the protoplanetary disc with the 26-Al abundances found in meteorites. We then determine how many of these G-dwarfs are unperturbed `singletons'; stars which are never in close binaries, nor suffer sub-100au encounters, and which also do not suffer strong dynamical perturbations. The evolution of a suite of twenty initially identical clusters is highly stochastic, with the supernova enriching over 10 G-dwarfs in some clusters, and none at all in others. Typically only ~25 per cent of clusters contain enriched, unperturbed singletons, and usually only 1 - 2 per cluster (from a total of 96 G-dwarfs in each cluster). The initial conditions for star formation do not strongly affect the results, although a higher fraction of supervirial (expanding) clusters would contain enriched G-dwarfs if the supernova occurred earlier than 6.6Myr. If we sum together simulations with identical initial conditions, then ~1 per cent of all G-dwarfs in our simulations are enriched, unperturbed singletons.Comment: 14 pages, 5 figures, accepted for publication in MNRA