A Parallel Implementation of Stickel\u27s AC Unification Algorithm in a Message-Passing Environment

Unification algorithms are an essential component of automated reasoning and term rewriting systems. Unification finds a set of substitutions or unifiers that, when applied to variables in two or more terms, make those terms identical or equivalent. Most systems use Robinson\u27s unification algorithm or some variant of it. However, terms containing functions exhibiting properties such as associativity and commutativity may be made equivalent without appearing identical. Systems employing Robinson\u27s unification algorithm must use some mechanism separate from the unification algorithm to reason with such functions. Often this is done by incorporating the properties into a rule base and generating equivalent terms which can be unified by Robinson\u27s algorithm. However, rewriting the terms in this manner can generate large numbers of useless terms in the problem space of the system. If the properties of the functions are incorporated into the unification algorithm itself, there is no need to rewrite the terms such that they appear identical. Stickel developed an algorithm to unify two terms containing associative and commutative functions. The unifiers (there may be more than one) are found by creating a homogeneous linear Diophantine equation with integer coefficients from the terms being unified. The unifiers can be constructed from solutions to this equation. The unifiers generated from one solution of the Diophantine equation are independent of any other solution to the equation. Therefore, once the Diophantine equation has been solved, the unifiers can be calculated from the solutions in parallel. We have implemented Stickel\u27s AC unification algorithm to run in parallel on a local area network of Sun 4/110 workstations in an effort to improve the speed of AC unification

Parallel machine architecture and compiler design facilities

The objective is to provide an integrated simulation environment for studying and evaluating various issues in designing parallel systems, including machine architectures, parallelizing compiler techniques, and parallel algorithms. The status of Delta project (which objective is to provide a facility to allow rapid prototyping of parallelized compilers that can target toward different machine architectures) is summarized. Included are the surveys of the program manipulation tools developed, the environmental software supporting Delta, and the compiler research projects in which Delta has played a role

Exact Tests via Complete Enumeration: A Distributed Computing Approach

The analysis of categorical data often leads to the analysis of a contingency table. For large samples, asymptotic approximations are sufficient when calculating p-values, but for small samples the tests can be unreliable. In these situations an exact test should be considered. This bases the test on the exact distribution of the test statistic. Sampling techniques can be used to estimate the distribution. Alternatively, the distribution can be found by complete enumeration. A new algorithm is developed that enables a model to be defined by a model matrix, and all tables that satisfy the model are found. This provides a more efficient enumeration mechanism for complex models and extends the range of models that can be tested. The technique can lead to large calculations and a distributed version of the algorithm is developed that enables a number of machines to work efficiently on the same problem

Department of Applied Mathematics Academic Program Review, Self Study / June 2010

The Department of Applied Mathematics has a multi-faceted mission to provide an exceptional mathematical education focused on the unique needs of NPS students, to conduct relevant research, and to provide service to the broader community. A strong and vibrant Department of Applied Mathematics is essential to the university's goal of becoming a premiere research university. Because research in mathematics often impacts science and engineering in surprising ways, the department encourages mathematical explorations in a broad range of areas in applied mathematics with specific thrust areas that support the mission of the school

Strategic directions in constraint programming

An abstract is not available

Implementation and Evaluation of Algorithmic Skeletons: Parallelisation of Computer Algebra Algorithms

This thesis presents design and implementation approaches for the parallel algorithms of computer algebra. We use algorithmic skeletons and also further approaches, like data parallel arithmetic and actors. We have implemented skeletons for divide and conquer algorithms and some special parallel loops, that we call ‘repeated computation with a possibility of premature termination’. We introduce in this thesis a rational data parallel arithmetic. We focus on parallel symbolic computation algorithms, for these algorithms our arithmetic provides a generic parallelisation approach. The implementation is carried out in Eden, a parallel functional programming language based on Haskell. This choice enables us to encode both the skeletons and the programs in the same language. Moreover, it allows us to refrain from using two different languages—one for the implementation and one for the interface—for our implementation of computer algebra algorithms. Further, this thesis presents methods for evaluation and estimation of parallel execution times. We partition the parallel execution time into two components. One of them accounts for the quality of the parallelisation, we call it the ‘parallel penalty’. The other is the sequential execution time. For the estimation, we predict both components separately, using statistical methods. This enables very confident estimations, although using drastically less measurement points than other methods. We have applied both our evaluation and estimation approaches to the parallel programs presented in this thesis. We haven also used existing estimation methods. We developed divide and conquer skeletons for the implementation of fast parallel multiplication. We have implemented the Karatsuba algorithm, Strassen’s matrix multiplication algorithm and the fast Fourier transform. The latter was used to implement polynomial convolution that leads to a further fast multiplication algorithm. Specially for our implementation of Strassen algorithm we have designed and implemented a divide and conquer skeleton basing on actors. We have implemented the parallel fast Fourier transform, and not only did we use new divide and conquer skeletons, but also developed a map-and-transpose skeleton. It enables good parallelisation of the Fourier transform. The parallelisation of Karatsuba multiplication shows a very good performance. We have analysed the parallel penalty of our programs and compared it to the serial fraction—an approach, known from literature. We also performed execution time estimations of our divide and conquer programs. This thesis presents a parallel map+reduce skeleton scheme. It allows us to combine the usual parallel map skeletons, like parMap, farm, workpool, with a premature termination property. We use this to implement the so-called ‘parallel repeated computation’, a special form of a speculative parallel loop. We have implemented two probabilistic primality tests: the Rabin–Miller test and the Jacobi sum test. We parallelised both with our approach. We analysed the task distribution and stated the fitting configurations of the Jacobi sum test. We have shown formally that the Jacobi sum test can be implemented in parallel. Subsequently, we parallelised it, analysed the load balancing issues, and produced an optimisation. The latter enabled a good implementation, as verified using the parallel penalty. We have also estimated the performance of the tests for further input sizes and numbers of processing elements. Parallelisation of the Jacobi sum test and our generic parallelisation scheme for the repeated computation is our original contribution. The data parallel arithmetic was defined not only for integers, which is already known, but also for rationals. We handled the common factors of the numerator or denominator of the fraction with the modulus in a novel manner. This is required to obtain a true multiple-residue arithmetic, a novel result of our research. Using these mathematical advances, we have parallelised the determinant computation using the Gauß elimination. As always, we have performed task distribution analysis and estimation of the parallel execution time of our implementation. A similar computation in Maple emphasised the potential of our approach. Data parallel arithmetic enables parallelisation of entire classes of computer algebra algorithms. Summarising, this thesis presents and thoroughly evaluates new and existing design decisions for high-level parallelisations of computer algebra algorithms

Multiple-Criteria Decision Making

Decision-making on real-world problems, including individual process decisions, requires an appropriate and reliable decision support system. Fuzzy set theory, rough set theory, and neutrosophic set theory, which are MCDM techniques, are useful for modeling complex decision-making problems with imprecise, ambiguous, or vague data.This Special Issue, “Multiple Criteria Decision Making”, aims to incorporate recent developments in the area of the multi-criteria decision-making field. Topics include, but are not limited to:- MCDM optimization in engineering;- Environmental sustainability in engineering processes;- Multi-criteria production and logistics process planning;- New trends in multi-criteria evaluation of sustainable processes;- Multi-criteria decision making in strategic management based on sustainable criteria