Interactive Simplifier Tracing and Debugging in Isabelle

The Isabelle proof assistant comes equipped with a very powerful tactic for term simplification. While tremendously useful, the results of simplifying a term do not always match the user's expectation: sometimes, the resulting term is not in the form the user expected, or the simplifier fails to apply a rule. We describe a new, interactive tracing facility which offers insight into the hierarchical structure of the simplification with user-defined filtering, memoization and search. The new simplifier trace is integrated into the Isabelle/jEdit Prover IDE.Comment: Conferences on Intelligent Computer Mathematics, 201

Isabelle/PIDE as Platform for Educational Tools

The Isabelle/PIDE platform addresses the question whether proof assistants of the LCF family are suitable as technological basis for educational tools. The traditionally strong logical foundations of systems like HOL, Coq, or Isabelle have so far been counter-balanced by somewhat inaccessible interaction via the TTY (or minor variations like the well-known Proof General / Emacs interface). Thus the fundamental question of math education tools with fully-formal background theories has often been answered negatively due to accidental weaknesses of existing proof engines. The idea of "PIDE" (which means "Prover IDE") is to integrate existing provers like Isabelle into a larger environment, that facilitates access by end-users and other tools. We use Scala to expose the proof engine in ML to the JVM world, where many user-interfaces, editor frameworks, and educational tools already exist. This shall ultimately lead to combined mathematical assistants, where the logical engine is in the background, without obstructing the view on applications of formal methods, formalized mathematics, and math education in particular.Comment: In Proceedings THedu'11, arXiv:1202.453

Capturing Hiproofs in HOL Light

Hierarchical proof trees (hiproofs for short) add structure to ordinary proof trees, by allowing portions of trees to be hierarchically nested. The additional structure can be used to abstract away from details, or to label particular portions to explain their purpose. In this paper we present two complementary methods for capturing hiproofs in HOL Light, along with a tool to produce web-based visualisations. The first method uses tactic recording, by modifying tactics to record their arguments and construct a hierarchical tree; this allows a tactic proof script to be modified. The second method uses proof recording, which extends the HOL Light kernel to record hierachical proof trees alongside theorems. This method is less invasive, but requires care to manage the size of the recorded objects. We have implemented both methods, resulting in two systems: Tactician and HipCam

Numerical investigation of the stability of stationary solutions in the theory of cathode spots in arcs in vacuum and ambient gas

The stability of stationary spots on cathodes of arcs in vacuum and ambient gas is investigated by means of the simulation of the temporal evolution of perturbations imposed over steady-state solutions. Two cases of loading conditions are considered, namely, spots operating at a fixed current (the case typical of small-scale experiments) and spots operating at a fixed voltage (the case typical of high-power circuit breakers). Results are reported on spots on large copper cathodes of vacuum arcs and on spots on tungsten cathodes of high-pressure argon arcs. It is shown, in particular, that if the ballast resistance in small-scale laboratory experiments with a high-current arc is insufficient, the potential consequence may be a thermal explosion of a spot, if the arc burns in vacuum, and massive melting of the cathode surface, if the arc burns in ambient gas. This conclusion conforms to trends observed in the experiment.info:eu-repo/semantics/publishedVersio

Pronounced genetic structure and low genetic diversity in European red-billed chough (Pyrrhocorax pyrrhocorax) populations

Conservation Genetics August 2015, Volume 16, Issue 4, pp 1011–1012 Erratum to: Pronounced genetic structure and low genetic diversity in European red-billed chough (Pyrrhocorax pyrrhocorax) populations Erratum to: Conserv Genet (2012) 13:1213–1230 DOI 10.1007/s10592-012-0366-6 In the original publication, Tables 3 and 6 were published with incorrect estimates of population heterozygosities. All other diversity statistics were correct as originally presented. Updated versions of Tables 3 and 6 with corrected heterozygosity estimates confirmed using Arlequin 3.5 (Excoffier and Lischer 2010) as in Dávila et al. (2014) are provided in this erratum. Discrepancies were minor for populations on the British Isles. The correct estimates for Spain are slightly larger than those reported for La Palma by Dávila et al. (2014), but this does not necessarily affect their interpretation that choughs on La Palma may have originated from multiple migration events. The original conclusion that chough populations on the British Isles have low genetic diversity compared to continental European populations remains and is now, in fact, strengthened.Peer reviewedPostprin

Simulation of thermal instability in non-uniformities on the surface of cathodes of vacuum arcs

Instability stemming from the strong dependence of electron emission current on the local surface temperature plays an important role in current transfer to hot cathodes of arc discharges. In the case of vacuum arcs, this instability may lead to micro explosions on cathode surface even if the surface is planar. This work is concerned with numerical simulation of effect produced by surface non-uniformities. It is found that the effect is non-trivial: the presence of surface non-uniformities can not only accelerate the development of the instability, which is what one would expect intuitively, but also slow it down and even suppress.info:eu-repo/semantics/publishedVersio

Phenomenological approach to simulation of propagation of spots over cathodes of high-power vacuum circuit breakers

A phenomenological description of an ensemble of a large number of spots on negative contacts of high-power vacuum circuit breakers is developed by means of generalization of the concept of random walk of a single cathode spot in low-current vacuum arcs. The model is formulated in terms of a convection-diffusion equation governing the evolution of the number of spots per unit area, taking into account the variation of the number of spots with the arc current and the “retrograde repulsion” between spots. The approach is applied to description of the distribution of cathode spots during the initial expansion process after arc ignition in conditions of two independent experiments simulating high-power switches. A reasonably good agreement between the theory and the experiment is found. The developed model can be used as a module of global numerical models of the interruption process in high-power vacuum circuit breakers.info:eu-repo/semantics/publishedVersio

Monte Carlo study of the critical properties of the three-dimensional 120-degree model

We report on large scale finite-temperature Monte Carlo simulations of the classical $120^\circ$ or $e_g$ orbital-only model on the simple cubic lattice in three dimensions with a focus towards its critical properties. This model displays a continuous phase transition to an orbitally ordered phase. While the correlation length exponent $\nu\approx0.665$ is close to the 3D XY value, the exponent $\eta \approx 0.15$ differs substantially from O(N) values. We also introduce a discrete variant of the $e_g$ model, called $e_g$-clock model, which is found to display the same set of exponents. Further, an emergent U(1) symmetry is found at the critical point $T_c$, which persists for $T<T_c$ below a crossover length scaling as $\Lambda \sim \xi^a$, with an unusually small $a\approx1.3$.Comment: 13 pages, 6 figure