AC-KBO Revisited

Equational theories that contain axioms expressing associativity and commutativity (AC) of certain operators are ubiquitous. Theorem proving methods in such theories rely on well-founded orders that are compatible with the AC axioms. In this paper we consider various definitions of AC-compatible Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are revisited. The former is enhanced to a more powerful version, and we modify the latter to amend its lack of monotonicity on non-ground terms. We further present new complexity results. An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given. The various orders are compared on problems in termination and completion.Comment: 31 pages, To appear in Theory and Practice of Logic Programming (TPLP) special issue for the 12th International Symposium on Functional and Logic Programming (FLOPS 2014

New results on rewrite-based satisfiability procedures

Program analysis and verification require decision procedures to reason on theories of data structures. Many problems can be reduced to the satisfiability of sets of ground literals in theory T. If a sound and complete inference system for first-order logic is guaranteed to terminate on T-satisfiability problems, any theorem-proving strategy with that system and a fair search plan is a T-satisfiability procedure. We prove termination of a rewrite-based first-order engine on the theories of records, integer offsets, integer offsets modulo and lists. We give a modularity theorem stating sufficient conditions for termination on a combinations of theories, given termination on each. The above theories, as well as others, satisfy these conditions. We introduce several sets of benchmarks on these theories and their combinations, including both parametric synthetic benchmarks to test scalability, and real-world problems to test performances on huge sets of literals. We compare the rewrite-based theorem prover E with the validity checkers CVC and CVC Lite. Contrary to the folklore that a general-purpose prover cannot compete with reasoners with built-in theories, the experiments are overall favorable to the theorem prover, showing that not only the rewriting approach is elegant and conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page

The Planet Nine Hypothesis

Over the course of the past two decades, observational surveys have unveiled the intricate orbital structure of the Kuiper Belt, a field of icy bodies orbiting the Sun beyond Neptune. In addition to a host of readily-predictable orbital behavior, the emerging census of trans-Neptunian objects displays dynamical phenomena that cannot be accounted for by interactions with the known eight-planet solar system alone. Specifically, explanations for the observed physical clustering of orbits with semi-major axes in excess of $\sim250\,$AU, the detachment of perihelia of select Kuiper belt objects from Neptune, as well as the dynamical origin of highly inclined/retrograde long-period orbits remain elusive within the context of the classical view of the solar system. This newly outlined dynamical architecture of the distant solar system points to the existence of a new planet with mass of $m_9\sim 5-10\,M_{\oplus}$, residing on a moderately inclined orbit ($i_9\sim15-25\deg$) with semi-major axis $a_9\sim 400 - 800\,$AU and eccentricity between $e_9 \sim 0.2 - 0.5$. This paper reviews the observational motivation, dynamical constraints, and prospects for detection of this proposed object known as Planet Nine.Comment: 92 pages, 28 figures, published in Physics Report

Inversion of surficial sediment thickness from under-ice acoustic transmission measurement

Author Posting. © Acoustical Society of America, 2021. This article is posted here by permission of Acoustical Society of America for personal use, not for redistribution. The definitive version was published in Journal of the Acoustical Society of America 149(1), (2021): 371, https://doi.org/10.1121/10.0003328.The under-ice acoustic transmission experiment of 2013, conducted under ice cover in the Fram Strait, was analyzed for bottom interactions for the purpose of developing a model of the seabed. Using the acoustic signals, as well as data from other sources, including cores, gravimetric, refraction, and seismic surveys, it was deduced that the seabed may be modeled as a thin surficial layer overlaid on a deeper sediment. The modeling was based on the Biot–Stoll model for acoustic propagation in porous sediments, aided by more recent developments that improve parameter estimation and depth dependence due to consolidation. At every stage, elastic and fluid approximations were explored to simplify the model and improve computational efficiency. It was found the surficial layer could be approximated as a fluid, but the deeper sediment required an elastic model. The full Biot–Stoll model, while instrumental in guiding the model construction, was not needed for the final computation. The model could be made to agree with the measurements by adjusting the surficial layer thickness.The fieldwork was performed under funding from the Research Council of Norway through the UNDER-ICE (Grant No. 226373) project and ENGIE E&P Norway providing additional support. This analysis was supported by the United States Office of Naval Research, Ocean Acoustics Program.2021-07-1

Proof Theory at Work: Complexity Analysis of Term Rewrite Systems

This thesis is concerned with investigations into the "complexity of term rewriting systems". Moreover the majority of the presented work deals with the "automation" of such a complexity analysis. The aim of this introduction is to present the main ideas in an easily accessible fashion to make the result presented accessible to the general public. Necessarily some technical points are stated in an over-simplified way.Comment: Cumulative Habilitation Thesis, submitted to the University of Innsbruc

Hi-inflected verbal *CóC-stems in Cuneiform and Hieroglyphic Luwian

In Luwian, as in Hittite, the *CóC-stem formation is the counterpart of PIE. perfect *C1e-C1óC2-. In Proto-Anatolian the PIE. perfect shows hardly any traces of a reduplication syllable (although there are examples); principally, it shows only the o-ablaut. Structurally, the Hittite hi-verbs are best compared to the isolated PIE. verb *u̯eid- 'to know', which was unreduplicated but inflected as a perfect. While this situation has been well studied in Hittite, a study of this kind focusing specifically on Luwian is still lacking. This article aims to explore this issue in Cuneiform and Hieroglyphic Luwian

AC Dependency Pairs Revisited

Rewriting modulo AC, i.e., associativity and/or commutativity of certain symbols, is among the most frequently used extensions of term rewriting by equational theories. In this paper we present a generalization of the dependency pair framework for termination analysis to rewriting modulo AC. It subsumes existing variants of AC dependency pairs, admits standard dependency graph analyses, and in particular enjoys the minimality property in the standard sense. As a direct benefit, important termination techniques are easily extended; we describe usable rules and the subterm criterion for AC termination, which properly generalize the non-AC versions. We also perform these extensions within IsaFoR - the Isabelle formalization of rewriting - and thereby provide the first formalization of AC dependency pairs. Consequently, our certifier CeTA now supports checking proofs of AC termination

Lightcurves of 20--100 kilometer Kuiper Belt Objects using the Hubble Space Telescope

We report high precision photometry of three small and one larger Kuiper Belt Objects (KBOs) obtained with the Advanced Camera for Surveys onboard the Hubble Space Telescope (ACS/HST). The three small bodies are the smallest KBOs for which lightcurve measurements are available. 2003 BF91 has a diameter of 20 kilometers (assuming 10% albedo) and a 1.09 magnitude, 9.1-hour lightcurve that is feasibly explained by the rotation of an elongated, coherent body that is supported by material strength and best imagined as an icy outer Solar System analog to asteroid (243) Ida. Two other small KBOs, 2003 BG91 and 2003 BH91 (diameters 31 and 18 km, with albedo 10%), exhibit an unremarkable lightcurve and no detectable photometric variation, respectively. For the larger KBO 2000 FV53 (116 km diameter, assuming 10% albedo) we strongly detect a non-sinusoidal periodic (7.5 hours) brightness variation with a very small amplitude (0.07 mag). This KBO may be nearly spherical, a result that might not be unusual in the Kuiper Belt but would be remarkable among outer Solar System satellites of similar size. We carry out a study of possible physical states and bulk densities under the assumptions of both fluid equilibrium and finite, non-zero internal friction. The densities for the these KBOs are likely to be in the range 1--2 g/cm3, and a plausible solution for 2000 FV53 is a rubble pile of this density that is held slightly out of the minimum-energy shape by internal friction among constituent blocks that are relatively small. Our interpretation of 2000 FV53 as a pulverized but essentially primordial object and 2003 BF91 as a collisional fragment is consistent with models of collisional timescales in the outer Solar System. We compile all published KBO lightcurve data and compare our results to the larger population. [abridged]Comment: AJ, in press. Tables 1-4 will be electronic only in published version but appear here in full. Figures 1,3,5 in colo