Algebraic Programming Research Articles

Experimental investigation of the polyhedral structure of BIBDs

We examine the polyhedral structure of Balanced Incomplete Block Designs (BIBDs) in an effort to improve runtimes for finding optimal solutions via constraint or integer program. For example, an optimal BIBD is one that minimizes the range of blocksums. This range, which we refer to as diffsum, has applications in drug trials and Redundant Arrays of Independent Disks (RAID). The polytope we explore is the convex hull of all lexicographically-ordered BIBD solutions satisfying a given set of parameters, ( v , b , r , k , λ ) , with solutions represented as a vector in R kb . The dimension-reducing equations and the facets for some of the smaller polytopes were found by using the Double-Description Method on a algebraic program modelling these solutions. Because of the presence of the lexicographic ordering, none of these polytopes are full dimensional. After describing some of these smaller BIBD polytopes, we generalize some dimension-reducing equations, and evaluate the efficiency of these equations to optimization models.

Multivariate to bivariate reduction for noncommutative polynomial factorization

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely,1.Given an n-variate noncommutative polynomial f∈F〈X〉 over a field F as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial g∈F〈x,y〉; the reduction transforms f into a circuit for g, and given a complete factorization of g, the reduction recovers a complete factorization of f in polynomial time.The reduction works both in the white-box and the black-box setting.2.We show over the field of rationals that bivariate linear matrix factorization problem for 4×4 matrices is at least as hard as factoring square-free integers and for 3×3 matrices it is in polynomial time.

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

The orbit of an n -variate polynomial \(f({\mathbf {x}})\) over a field \(\mathbb {F}\) is the set \(\text {orb}(f) := \lbrace f(A{\mathbf {x}}+{\mathbf {b}}) : A \in \mathrm{GL}(n,\mathbb {F}) \text{ and } {\mathbf {b}}\in \mathbb {F}^n\rbrace\) . The orbit of a polynomial f is a geometrically interesting subset of the set of affine projections of f . Affine projections of polynomials computable by seemingly weak circuit classes can be quite powerful. For example, the polynomial \(\mathsf {IMM}_{3,d}\) —the \((1,1)\) th entry of a product of d generic \(3 \times 3\) matrices—is computable by a constant-width read-once oblivious algebraic branching program (ROABP), yet every polynomial computable by a size- s general arithmetic formula is an affine projection of \(\mathsf {IMM}_{3,\text {poly}(s)}\) as shown by Ben-or and Cleve [ 12 ]. To our knowledge, no efficient hitting set construction was known for \(\text {orb}(\mathsf {IMM}_{3, d})\) before this work. In this article, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: Low-individual-degree polynomials computable by commutative ROABPs . This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials and the orbits of multilinear sparse polynomials . Multilinear polynomials computable by constant-width ROABPs . This implies a quasi-polynomial time hitting set for the orbits of the family \(\lbrace \mathsf {IMM}_{3,d}\rbrace _{d \in \mathbb {N}}\) . Polynomials computable by constant-depth, constant-occur formulas . This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in , and also polynomial-time hitting sets for the orbits of the power symmetric polynomials and the sum-product polynomials . Polynomials computable by occur-once formulas . We say a polynomial has low individual degree if the degree of every variable in the polynomial is at most \(\text {poly}(\log n)\) , where n is the number of variables. The first two results are obtained by building upon and strengthening the rank concentration by translation technique of Agrawal, Saha, and Saxena [ 6 ]; the second result additionally uses the merge-and-reduce idea from Forbes and Shpilka [ 30 ] and Forbes, Shpilka, and Saptharishi [ 27 ]. The proof of the third result adapts the algebraic independence-based technique of Agrawal, Saha, Saptharishi, and Saxena [ 5 ] and Beecken, Mittmann, and Saxena [ 11 ] to reduce to the case of constructing hitting sets for the orbits of sparse polynomials. A similar reduction using the Shpilka-Volkovich (SV) generator-based argument in Shpilka and Volkovich [ 90 ] yields the fourth result. The SV generator plays an important role in all four results.

An analytical approach for the performance calculation of an RCS/RS with several picking stations

Robotic Compact Storage and Retrieval Systems (RCS/RS) offer numerous advantages, including high performance, scalability, and availability, which are essential for modern logistics and warehousing. However, information on the potential performance of RCS/RS is limited, primarily due to the diverse range of configurations available. This paper aims to address this gap by developing an analytical approach to predict the throughput of an RCS/RS with multiple robots serving several picking stations. The approach considers various parameters such as grid size, stack height, number of robots, and filling degree, alongside kinematic data. The cycle time for each robot is calculated assuming a uniform distribution of container stacks. Subsequently, a queueing system with limited capacity is constructed using performance data from a single robot. The analytical approach is validated using a discrete event simulation model of an RCS/RS. Following the validation, an extensive parameter variation and application example are conducted to demonstrate the versatility of the approach. This method offers a straightforward and efficient set of formulas for determining RCS/RS throughput, easily solvable using standard table or algebra programs.

Crop water requirement and irrigation scheduling under climate change scenario, and optimal cropland allocation in lower kulfo catchment

Crop water requirement and irrigation scheduling in Lower Kulfo Catchment of southern Ethiopia have not assessed under climate change scenarios, and the allocation of crop land also not optimal that signifcantly challenges to crop productivity.Therefore, this study was conducted to evaluate the effects of climate change on future crop water requirements, and irrigation scheduling, and to allocate cropland optimally. Bias of projected precipitation and temperature were corrected by utilizing Climate Model data with the hydrologic modeling tool (CMhyd). Alongside, crop water requirements and irrigation scheduling were assessed using Crop Water Assessment Tool. After estimating crop water requirement, crop land were allocated optimally using General Algebraic Modeling System programming with non-negativity constraints (scenario 1), and non-negativity constraints based on farmers adaptation (scenario 2). Average reference evapotranspiration from 2030 to 2050 and 2060 to 2080 was increased by 11.9 %, and 16.2 %, respectively compared with the reference period (2010–2022). The total seasonal crop water requirements were 4,529 mm, 4866.7 mm, and 5272.2 mm under 2010 to 2022, 2030 to 2050, and 2060 to 2080 climate change scenarios, respectively. The meean irrigation interval in 2010–2022, 2030 to 2050, and 2060 to 2080 climate change scenarios were 8 days, 7 days, and 5 days, respectively. This irrigation interval was decreased by 14 % (2030–2050), and 34 % (2060–2080) compared with the reference period. In 2030 to 2050 and 2026 to 2080 climate change scenarios, the required irrigation water at the inlet of main canal increased by 6.8 %, and 18 %, respectively. The optimal allocated area for tomato (60.4 %), maize (20.8 %), and watermelon (18.8 %) in scenario 1 with net benefit of 1.47*108 Ethiopian Birr. The allocated areas in scenario 2 were (48 %) for maize, (31.6 %) for tomato, and (20.4 %) for watermelon with 1.34*108 Ethiopian Birr net benefit it was reduced by 19.1 % compared with the net benefit in scenario 1. Fruit crops alone may not suffice for local food needs and to address this, small farmers should grow maize, tomato, and watermelon. This research aids policymakers in encouraging climate-resilient agriculture and improving small-scale farmers' awareness through conducting workshops and training.

A Survey on Syntactic Pattern Recognition Methods in Bioinformatics

Formal tools and models of syntactic pattern recognition which are used in bioinformatics are introduced and characterized in the paper. They include, among others: stochastic (string) grammars and automata, hidden Markov models, programmed grammars, attributed grammars, stochastic tree grammars, Tree Adjoining Grammars (TAGs), algebraic dynamic programming, NLC- and NCE-type graph grammars, and algebraic graph transformation systems. The survey of applications of these formal tools and models in bioinformatics is presented.

The development of computational methods for Feynman diagrams

Over the last 70 years, Feynman diagrams have played an essential role in the development of many theoretical predictions derived from the standard model Lagrangian. In fact, today they have become an essential and seemingly irreplaceable tool in quantum field theory calculations. In this article, we propose to explore the development of computational methods for Feynman diagrams with a special focus on their automation, drawing insights from both theoretical physics and the history of science. From the latter perspective, the article particularly investigates the emergence of computer algebraic programs, such as the pioneering SCHOONSCHIP, REDUCE, and ASHMEDAI, designed to handle the intricate calculations associated with Feynman diagrams. This sheds light on the many challenges faced by physicists when working at higher orders in perturbation theory and reveal, as exemplified by the test of the validity of quantum electrodynamics at the turn of the 1960s and 1970s, the indispensable necessity of computer-assisted procedures. In the second part of the article, a comprehensive overview of the current state of the algorithmic evaluation of Feynman diagrams is presented from a theoretical point of view. It emphasizes the key algorithmic concepts employed in modern perturbative quantum field theory computations and discusses the achievements, ongoing challenges, and potential limitations encountered in the application of the Feynman diagrammatic method. Accordingly, we attribute the enduring significance of Feynman diagrams in contemporary physics to two main factors: the highly algorithmic framework developed by physicists to tackle these diagrams and the successful advancement of algebraic programs used to process the involved calculations associated with them.

Relative Entropy Application to Study the Elastoplastic Behavior of S235JR Structural Steel.

The main issue in this work is to study the limit functions necessary for the reliability assessment of structural steel with the use of the relative entropy apparatus. This will be done using a few different mathematical theories relevant to this relative entropy, namely those proposed by Bhattacharyya, Kullback-Leibler, Jeffreys, and Hellinger. Probabilistic analysis in the presence of uncertainty in material characteristics will be delivered using three different numerical strategies-Monte Carlo simulation, the stochastic perturbation method, as well as the semi-analytical approach. All of these methods are based on the weighted least squares method approximations of the structural response functions versus the given uncertainty source, and they allow efficient determination of the first two probabilistic moments of the structural responses including stresses, displacements, and strains. The entire computational implementation will be delivered using the finite element method system ABAQUS and computer algebra program MAPLE, where relative entropies, as well as polynomial response functions, will be determined. This study demonstrates that the relative entropies may be efficiently used in reliability assessment close to the widely engaged first-order reliability method (FORM). The relative entropy concept enables us to study the probabilistic distance of any two distributions, so that structural resistance and extreme effort in elastoplastic behavior need not be restricted to Gaussian distributions.

Second quantization-based symmetry-adapted perturbation theory: Generalizing exchange beyond single electron pair approximation.

This paper presents a general second-quantized form of a permutation operator interchanging n pairs of electrons between interacting subsystems in the framework of the symmetry-adapted perturbation theory (SAPT). We detail the procedure for constructing this operator through the consecutive multiplication of single-pair permutation operators. This generalized form of the permutation operator has enabled the derivation of universal formulas for S2n approximations of the exchange energies in the first and second order of the interaction operator. We present expressions for corrections of S4 approximations and assess its efficacy on a selection of systems anticipated to exhibit a slowly converging overlap expansion. Additionally, we outline a method to sum the overlap expansion series to infinity in second-quantization, up to the second order in V. This new approach offers an alternative to the existing formalism based on density-matrix formulations. When combined with a symbolic algebra program for automated derivations, it paves the way for advancements in SAPT theory, particularly for intricate wavefunction theories.

Semirings for probabilistic and neuro-symbolic logic programming

The field of probabilistic logic programming (PLP) focuses on integrating probabilistic models into programming languages based on logic. Over the past 30 years, numerous languages and frameworks have been developed for modeling, inference and learning in probabilistic logic programs. While originally PLP focused on discrete probability, more recent approaches have incorporated continuous distributions as well as neural networks, effectively yielding neuro-symbolic methods. We provide an overview and synthesis of this domain, thereby contributing a unified algebraic perspective on the different flavors of PLP, showing that many if not most of the extensions of PLP can be cast within a common algebraic logic programming framework, in which facts are labeled with elements of a semiring and disjunction and conjunction are replaced by addition and multiplication. This does not only hold for the PLP variations itself but also for the underlying execution mechanism that is based on (algebraic) model counting. In order to showcase and explain this unified perspective, we focus on the ProbLog language and its extensions.

The Method Of Logical Networks For Modeling Adaptive Knowledge Testing Systems

The subject of the research is the development of mathematical and algorithmic support of the intellectual toolkit for the analysis of sets of test tasks and the modeling of the process of interpreting the quality of sets of test tasks, which allows for objective and comprehensive continuous control of the knowledge of subjects of training, subject to the implementation of the concept of virtual distributed training (retraining).
 The purpose of the research is to improve the effectiveness of monitoring the knowledge of subjects of education in the distance form of education through the use of adaptive computer testing methods based on models of logical networks and the algebra of finite predicates.
 The following tasks are solved in the article: the formation of a testing model in a distributed virtual learning environment and a model of validity assessment based on the content of sets of test tasks.
 The following methods are used: methods of logical networks and algebraic programming based on the algebra of finite predicates and predicate operations, intellectual analysis of information.
 The following results were obtained: the principles of intellectual analysis were formulated in the development of a model of a universal logical network and its application to actual tasks of artificial intelligence in the field of informal information processing, namely, in the construction of knowledge testing systems for distributed virtual learning
 Conclusions. Algorithms for optimal multi-stage adaptive testing of knowledge as part of distributed virtual learning models and methods for analyzing the success of training subjects have been improved. The use of conjunctive decomposition with binary predicates achieves the goal of the research, because in this way any multi-place predicate can be represented by a logical network simulating the process of knowledge testing, the model of the subject of learning is described.

Automated design of dynamic programming schemes for RNA folding with pseudoknots

Although RNA secondary structure prediction is a textbook application of dynamic programming (DP) and routine task in RNA structure analysis, it remains challenging whenever pseudoknots come into play. Since the prediction of pseudoknotted structures by minimizing (realistically modelled) energy is NP-hard, specialized algorithms have been proposed for restricted conformation classes that capture the most frequently observed configurations. To achieve good performance, these methods rely on specific and carefully hand-crafted DP schemes. In contrast, we generalize and fully automatize the design of DP pseudoknot prediction algorithms. For this purpose, we formalize the problem of designing DP algorithms for an (infinite) class of conformations, modeled by (a finite number of) fatgraphs, and automatically build DP schemes minimizing their algorithmic complexity. We propose an algorithm for the problem, based on the tree-decomposition of a well-chosen representative structure, which we simplify and reinterpret as a DP scheme. The algorithm is fixed-parameter tractable for the treewidth tw of the fatgraph, and its output represents a O(ntw+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(n^{tw+1})$$\\end{document} algorithm (and even possibly O(ntw)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(n^{tw})$$\\end{document} in simple energy models) for predicting the MFE folding of an RNA of length n. We demonstrate, for the most common pseudoknot classes, that our automatically generated algorithms achieve the same complexities as reported in the literature for hand-crafted schemes. Our framework supports general energy models, partition function computations, recursive substructures and partial folding, and could pave the way for algebraic dynamic programming beyond the context-free case.

Conversations reflecting boundary-objects-related details of a teacher’s local practices with spreadsheet algebra programs on variables

The ways teachers converse about their work in relation to information and communications technologies (ICTs) are worth studying. We analyse how a teacher converses about her local practices in relation to two spreadsheet algebra programs (SAPs) on variables. During the conversations we noticed that the teacher keeps different policy documents – boundary objects – firmly in view, in relation to the design of the two other boundary objects, namely the two SAPs. The policy documents provide details on the operative curricula which entail the intended, implemented and examined curricula. Of these curricula, the teacher regarded the examined curriculum and associated examinations as most important. Also, she conversed about how she intends to align the design features of the two SAPs with particular policy documents, especially in the context of the South African high-stakes National Senior Certificate examinations and the attendant examination pressure. Our results confirm current professional development (PD) literature suggestions that emphasise fostering coherence, for example between policy boundary objects details and what university-based PD providers do when they interact with school teachers.Contribution: The results provide guidelines for university-based PD providers to integrate SAPs or other ICTs related to algebra and variables by keeping teachers’ local practices in view. These providers should note that different policy-related boundary objects shape the ways teachers understand and converse about their local practices, namely their work at the classroom level.

Exploiting the Sparseness of Control-Flow and Call Graphs for Efficient and On-Demand Algebraic Program Analysis

Algebraic Program Analysis (APA) is a ubiquitous framework that has been employed as a unifying model for various problems in data-flow analysis, termination analysis, invariant generation, predicate abstraction and a wide variety of other standard static analysis tasks. APA models program summaries as elements of a regular algebra . Suppose that a summary in A is assigned to every transition of the program and that we aim to compute the effect of running the program starting at line s and ending at line t . APA first computes a regular expression capturing all program paths of interest. In case of intraprocedural analysis, models all paths from s to t , whereas in the interprocedural case it models all interprocedurally-valid paths, i.e. ‍paths that go back to the right caller function when a callee returns. This regular expression is then interpreted over the algebra to obtain the desired result. Suppose the program has n lines of code and each evaluation of an operation in the regular algebra takes O ( k ) time. It is well-known that a single APA query, or a set of queries with the same starting point s , can be answered in O ( n · α( n ) · k ), where α is the inverse Ackermann function. In this work, we consider an on-demand setting for APA: the program is given in the input and can be preprocessed. The analysis has to then answer a large number of on-line queries, each providing a pair ( s , t ) of program lines which are the start and end point of the query, respectively. The goal is to avoid the significant cost of running a fresh APA instance for each query. Our main contribution is a series of algorithms that, after a lightweight preprocessing of O ( n · lg n · k ), answer each query in O ( k ) time. In other words, our preprocessing has almost the same asymptotic complexity as a single APA query, except for a sub-logarithmic factor, and then every future query is answered instantly, i.e. ‍by a constant number of operations in the algebra. We achieve this remarkable speedup by relying on certain structural sparsity properties of control-flow and call graphs (CFGs and CGs). Specifically, we exploit the fact that control-flow graphs of real-world programs have a tree-like structure and bounded treewidth and nesting depth and that their call graphs have small treedepth in comparison to the size of the program. Finally, we provide experimental results demonstrating the effectiveness and efficiency of our approach and showing that it beats the runtime of classical APA by several orders of magnitude.

Computations Stiefel-Whitney classes of real representations for non-prime power groups

In the current study, the researchers conduct computation of the Stiefel- Whitney classes of real representations of non-prime power groups such as Mathieu groups, symmetric groups, alternating groups and Janko groups. Stiefel- Whitney classes are conducted by using polytope convex hull with vector in n-dimention our computation using the HAP (Homological Algebra Programming.

Examples of Expansions in Fractional Powers, and Applications

We approximate the solution of a generalized form of the Bagley–Torvik equation using Taylor’s expansions in fractional powers. Then, we study the fractional Laguerre-type logistic equation by considering the fractional exponential function and its Laguerre-type form. To verify our findings, we conduct numerical tests using the computer algebra program Mathematica©.

Automatic Generation of Spatial Accelerator for Tensor Algebra

Tensor algebra finds applications in various domains including machine learning applications, data analytics and others. Spatial hardware accelerators are widely used to boost the performance of tensor algebra applications. It has a complex hardware architecture and rich design space. Prior approaches based on manual implementation lead to low programming productivity, making it hard to explore the large design space. In this paper, we propose Tensorlib, a framework for generating spatial hardware accelerators for tensor algebra applications. Tensorlib is motivated by the observation that, tensor dataflows can be expressed with linear transformations, and they share common hardware modules which can be reused across different designs. Tensorlib first uses Space-Time Transformation to explore different dataflows, which can compactly represent the hardware dataflow using a transformation matrix. Next, we identify the common structures of different dataflows and build parameterized hardware module templates. Our generation framework can select the needed hardware modules for each dataflow, connect the modules using a specified interconnection pattern, and automatically generate the complete hardware accelerator design. Tensorlib remarkably improves the productivity for the development and optimization of spatial hardware architecture, providing a rich design space with trade-offs in performance, area, and power. Experiments show that Tensorlib can automatically generate hardware designs with different dataflows for a variety of tensor algebra programs. Tensorlib can achieve 318 MHz frequency and 786 GFLOP/s throughput for matrix multiplication kernel on Xilinx VU9P FPGA, which outperforms the state-of-the-art generators.

Teaching Materials for Algebra and Linear Programs Based on Pancasila Student Profiles

The aims of this study were: 1) to develop textbooks based on Pancasila student profiles for Algebra and Linear Programming courses; and 2) developing instruments to measure the achievement of the dimensions of the Pancasila student profile. The quality of the products developed is assessed based on: 1) validity aspects, 2) practical aspects, and 3) effectiveness aspects of students' mathematical creative and critical thinking abilities. The research method used is the ADDIE Research and Development (R D) model. The research subjects were 19 students in the first semester of the Mathematics Education Study Program. The instruments used to obtain the quality of the textbooks developed were: 1) textbook assessment sheets to measure validity, 2) response questionnaires and observation sheets to measure practicality, 3) critical thinking skills and mathematical creative questions to measure effectiveness. The validity quality of textbooks fulfilling the valid criteria is shown by the average score of 3.53 out of a maximum score of 4 which means that it is very valid for practicality quality to get an average score of 3.32 out of a maximum score of 4 which means practical, quality of effectiveness in terms of critical thinking skills and creative mathematical students who meet the affective criteria are shown, namely 58% which means sufficient and the average score for critical and creative thinking skills is 67.27 which means quite effective. Textbooks and RPS are suitable for use in lectures

Sextactic points on the Fermat cubic curve and arrangements of conics

The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of 6-division points on the Fermat cubic F and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction in the search of conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to F, studying higher degree curves passing through higher order division points on F, studying curves passing through intersection points of already constructed curves, taking the duals etc.) and we hope some younger colleagues might find pleasure in following proposed paths as well as finding their own.

F. S. Macaulay: From plane curves to Gorenstein rings

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.