Finding Periodic Apartments via Boolean Satisfiability and Orderly Generation

Peer reviewe

Finding Periodic Apartments : A Computational Study of Hyperbolic Buildings

This thesis presents a computational study of a fundamental open conjecture in geometric group theory using an intricate combination of Boolean Satisfiability and orderly generation. In particular, we focus on Gromov’s subgroup conjecture (GSC), which states that “each one-ended hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed surface of genus at least 2”. Several classes of groups have been shown to satisfy GSC, but the status of non-right-angled groups with regard to GSC is presently unknown, and may provide counterexamples to the conjecture. With this in mind Kangaslampi and Vdovina constructed 23 such groups utilizing the theory of hyperbolic buildings [International Journal of Algebra and Computation, vol. 20, no. 4, pp. 591–603, 2010], and ran an exhaustive computational analysis of surface subgroups of genus 2 arising from so-called periodic apartments [Experimental Mathematics, vol. 26, no. 1, pp. 54–61, 2017]. While they were able to rule out 5 of the 23 groups as potential counterexamples to GSC, they reported that their computational approach does not scale to genera higher than 2. We extend the work of Kangaslampi and Vdovina by developing two new approaches to analyzing the subgroups arising from periodic apartments in the 23 groups utilizing different combinations of SAT solving and orderly generation. We develop novel SAT encodings and a specialized orderly algorithm for the approaches, and perform an exhaustive analysis (over the 23 groups) of the genus 3 subgroups arising from periodic apartments. With the aid of massively parallel computation we also exhaust the case of genus 4. As a result we rule out 4 additional groups as counterexamples to GSC leaving 14 of the 23 groups for further inspection. In addition to this our approach provides an independent verification of the genus 2 results reported by Kangaslampi and Vdovina

A SAT Solver and Computer Algebra Attack on the Minimum Kochen-Specker Problem

One of the foundational results in quantum mechanics is the Kochen-Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i.e., a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known to exist, the problem of finding the minimum KS vector system has remained stubbornly open for over 55 years, despite significant attempts by leading scientists and mathematicians. In this paper, we present a new method based on a combination of a SAT solver and a computer algebra system (CAS) to address this problem. Our approach improves the lower bound on the minimum number of vectors in a KS system from 22 to 24, and is about 35,000 times more efficient compared to the previous best computational methods. The increase in efficiency derives from the fact we are able to exploit the powerful combinatorial search-with-learning capabilities of a SAT solver together with the isomorph-free exhaustive generation methods of a CAS. The quest for the minimum KS vector system is motivated by myriad applications such as simplifying experimental tests of contextuality, zero-error classical communication, dimension witnessing, and the security of certain quantum cryptographic protocols. To the best of our knowledge, this is the first application of a novel SAT+CAS system to a problem in the realm of quantum foundations

Proceedings of the 11th Workshop on Nonmonotonic Reasoning

These are the proceedings of the 11th Nonmonotonic Reasoning Workshop. The aim of this series is to bring together active researchers in the broad area of nonmonotonic reasoning, including belief revision, reasoning about actions, planning, logic programming, argumentation, causality, probabilistic and possibilistic approaches to KR, and other related topics. As part of the program of the 11th workshop, we have assessed the status of the field and discussed issues such as: Significant recent achievements in the theory and automation of NMR; Critical short and long term goals for NMR; Emerging new research directions in NMR; Practical applications of NMR; Significance of NMR to knowledge representation and AI in general

Semantic discovery and reuse of business process patterns

Patterns currently play an important role in modern information systems (IS) development and their use has mainly been restricted to the design and implementation phases of the development lifecycle. Given the increasing significance of business modelling in IS development, patterns have the potential of providing a viable solution for promoting reusability of recurrent generalized models in the very early stages of development. As a statement of research-in-progress this paper focuses on business process patterns and proposes an initial methodological framework for the discovery and reuse of business process patterns within the IS development lifecycle. The framework borrows ideas from the domain engineering literature and proposes the use of semantics to drive both the discovery of patterns as well as their reuse