cs.SC

arXiv:2501.00364v1 Announce Type: new Abstract: Reward machines (RMs) are an effective approach for addressing non-Markovian rewards in reinforcement learning (RL) through finite-state machines. Traditional RMs, which label edges with propositional logic formulae, inherit the limited expressivity of propositional logic. This limitation hinders the learnability and transferability of RMs since complex tasks will require numerous states and edges. To overcome these challenges, we propose First-Order Reward Machines ($\texttt{FORM}$s), which use first-order logic to label edges, resulting in more compact and transferable RMs. We introduce a novel method for $\textbf{learning}$ $\texttt{FORM}$s and a multi-agent formulation for $\textbf{exploiting}$ them and facilitate their transferability, where multiple agents collaboratively learn policies for a shared $\texttt{FORM}$. Our experimental results demonstrate the scalability of $\texttt{FORM}$s with respect to traditional RMs. Specifically, we show that $\texttt{FORM}$s can be effectively learnt for tasks where traditional RM learning approaches fail. We also show significant improvements in learning speed and task transferability thanks to the multi-agent learning framework and the abstraction provided by the first-order language.

arXiv:2501.00563v1 Announce Type: cross Abstract: We present a new Python package called "motives", a symbolic manipulation package based on SymPy capable of handling and simplifying motivic expressions in the Grothendieck ring of Chow motives and other types of $\lambda$-rings. The package is able to manipulate and compare arbitrary expressions in $\lambda$-rings and, in particular, it contains explicit tools for manipulating motives of several types of commonly used moduli schemes and moduli stacks of decorated bundles on curves. We have applied this new tool to advance in the verification of Mozgovoy's conjectural formula for the motive of the moduli space of twisted Higgs bundles, proving that it holds in rank 2 and 3 for any curve of genus up to 18 and any twisting bundle of small degree.

arXiv:2412.18294v1 Announce Type: new Abstract: This paper presents an advanced method for addressing the inverse kinematics and optimal path planning challenges in robot manipulators. The inverse kinematics problem involves determining the joint angles for a given position and orientation of the end-effector. Furthermore, the path planning problem seeks a trajectory between two points. Traditional approaches in computer algebra have utilized Gr\"obner basis computations to solve these problems, offering a global solution but at a high computational cost. To overcome the issue, the present authors have proposed a novel approach that employs the Comprehensive Gr\"obner System (CGS) and CGS-based quantifier elimination (CGS-QE) methods to efficiently solve the inverse kinematics problem and certify the existence of solutions for trajectory planning. This paper extends these methods by incorporating smooth curves via cubic spline interpolation for path planning and optimizing joint configurations using shortest path algorithms to minimize the sum of joint configurations along a trajectory. This approach significantly enhances the manipulator's ability to navigate complex paths and optimize movement sequences.

arXiv:2412.18143v1 Announce Type: new Abstract: Graphs are the most suitable structures for modeling objects and interactions in applications where component inter-connectivity is a key feature. There has been increased interest in graphs to represent domains such as social networks, web site link structures, and biology. Graph stores recently rose to prominence along the NoSQL movement. In this work we will focus on NOSQL graph databases, describing their peculiarities that sets them apart from other data storage and management solutions, and how they differ among themselves. We will also analyze in-depth two different graph database management systems - AllegroGraph and Neo4j that uses the most popular graph models used by NoSQL stores in practice: the resource description framework (RDF) and the labeled property graph (LPG), respectively.

arXiv:2412.17099v1 Announce Type: cross Abstract: A finite group with an integral representation has two induced canonical actions, one on polynomials and one on Laurent polynomials. Knowledge about the invariants is in either case applied in many computations by means of symmetry reduction techniques, for example in algebraic systems solving or optimization. In this article, we realize the two actions as the additive action on the symmetric algebra and the multiplicative action on the group algebra of a lattice with Weyl group symmetry. By constructing explicit equivariant isomorphisms, we draw algorithmic relations between the two, which allow the transfer and preservation of representation- and invariant-theoretic properties. Our focus lies on the multiplicative coinvariant space, which is identified with the regular representation and harmonic polynomials.

arXiv:2411.00031v2 Announce Type: replace Abstract: Solving algebra problems (APs) continues to attract significant research interest as evidenced by the large number of algorithms and theories proposed over the past decade. Despite these important research contributions, however, the body of work remains incomplete in terms of theoretical justification and scope. The current contribution intends to fill the gap by developing a review framework that aims to lay a theoretical base, create an evaluation scheme, and extend the scope of the investigation. This paper first develops the State Transform Theory (STT), which emphasizes that the problem-solving algorithms are structured according to states and transforms unlike the understanding that underlies traditional surveys which merely emphasize the progress of transforms. The STT, thus, lays the theoretical basis for a new framework for reviewing algorithms. This new construct accommodates the relation-centric algorithms for solving both word and diagrammatic algebra problems. The latter not only highlights the necessity of introducing new states but also allows revelation of contributions of individual algorithms obscured in prior reviews without this approach.

arXiv:2412.16161v1 Announce Type: new Abstract: In this short article I introduce the evitaicossa package which provides functionality for antiassociative algebras in the R programming language; it is available on CRAN at https://CRAN.R-project.org/package=evitaicossa.

arXiv:2412.17280v1 Announce Type: new Abstract: This study presents a comprehensive mathematical framework for modeling the flight dynamics of a six-degree-of-freedom fixed-wing aircraft as a rigid body with three control surfaces: rudder, elevators, and ailerons. The framework consists of 35 differential-algebraic equations (DAEs) and requires 30 constants to be specified. It supports both direct and inverse flight dynamics analyses. In direct dynamics, the historical profiles of control inputs (deflection angles and engine thrust) are specified, and the resulting flight trajectory is predicted. In inverse dynamics, the desired flight trajectory and an additional constraint are specified to determine the required control inputs. The framework employs wind axes for linear-momentum equations and body axes for angular-momentum equations, incorporates two flight path angles, and provides formulas for aerodynamic force and moment coefficients. Key advantages include improved computational efficiency, elimination of Euler angle singularities, and independence from symmetry assumptions with regard to the aircraft's moments of inertia. The model also accounts for nonlinear air density variations with altitude, up to 20 km above mean sea level, making it suitable for accurate and efficient flight dynamics simulations.

arXiv:2412.17336v1 Announce Type: new Abstract: Knowledge graphs (KGs), which store an extensive number of relational facts, serve various applications. Recently, personalized knowledge graphs (PKGs) have emerged as a solution to optimize storage costs by customizing their content to align with users' specific interests within particular domains. In the real world, on one hand, user queries and their underlying interests are inherently evolving, requiring PKGs to adapt continuously; on the other hand, the summarization is constantly expected to be as small as possible in terms of storage cost. However, the existing PKG summarization methods implicitly assume that the user's interests are constant and do not shift. Furthermore, when the size constraint of PKG is extremely small, the existing methods cannot distinguish which facts are more of immediate interest and guarantee the utility of the summarized PKG. To address these limitations, we propose APEX$^2$, a highly scalable PKG summarization framework designed with robust theoretical guarantees to excel in adaptive summarization tasks with extremely small size constraints. To be specific, after constructing an initial PKG, APEX$^2$ continuously tracks the interest shift and adjusts the previous summary. We evaluate APEX$^2$ under an evolving query setting on benchmark KGs containing up to 12 million triples, summarizing with compression ratios $\leq 0.1\%$. The experiments show that APEX outperforms state-of-the-art baselines in terms of both query-answering accuracy and efficiency.

arXiv:2412.15829v1 Announce Type: new Abstract: Large knowledge graphs capture information of a large number of entities and their relations. Among the many relations they capture, class subsumption assertions are usually present and expressed using the \texttt{rdfs:subClassOf} construct. From our examination, publicly available knowledge graphs contain many potentially erroneous cyclic subclass relations, a problem that can be exacerbated when different knowledge graphs are integrated as Linked Open Data. In this paper, we present an automatic approach for resolving such cycles at scale using automated reasoning by encoding the problem of cycle-resolving to a MAXSAT solver. The approach is tested on the LOD-a-lot dataset, and compared against a semi-automatic version of our algorithm. We show how the number of removed triples is a trade-off against the efficiency of the algorithm.