Symbolic Variables Research Articles

A compendium of FEM integration formulas for symbolic work

This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work. Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra. Information can be returned in floating‐point (numerical) form, or in exact symbolic form. The latter is useful for computer‐algebra aided FEM work that carries along symbolic variables. A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code. A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations. Some unpublished non‐product rules for pyramid regions were found and included in the collection. For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming. The collection embodies most FEM‐useful formulas of low and moderate order for the seven regions noted above. Some gaps as regard region geometries and omission of non‐product rules are noted in the conclusions. The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification. It can also be used as generator for low‐level floating‐point code modules in Fortran or C. Floating point accuracy can be selected arbitrarily. No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before.

Gröbner Bases in Perturbative Calculations

In this paper we outline the most general and universal algorithmic approach to reduction of loop integrals to basic integrals. The approach is based on computation of Groebner bases for recurrence relations derived from the integration by parts method. In doing so we consider generic recurrence relations when propagators have arbitrary integer powers treated as symbolic variables (indices) for the relations.

Data Dependence Analysis Techniques for Increased Accuracy and Extracted Parallelism

Parallelizing compilers rely on data dependence information in order to produce valid parallel code. Traditional data dependence analysis techniques. such as the Banerjee test and the I-Test, can efficiently compute data dependence information for simple instances of the data dependence problem. However, in more complex cases involving triangular or trapezoidal loop regions, symbolic variables, and multidimensional arrays with coupled subscripts these tests, including the triangular Banerjee test, ignore or simplify many of the constraints and thus introduce approximations, especially when testing for data dependence under direction vector constraints. The Omega test can accurately handle such complex cases, but at a higher computation cost. In this paper we extend the ideas behind the I-Test and present new techniques to handle complex instances of tile dependence problem, which are frequently found in actual source code. In particular, we provide polynomial-time techniques that can prove or disprove data dependences, subject to any direction vector, in loops with triangular or trapezoidal bounds, symbolic variables and multidimensional arrays with coupled subscripts. We also investigate the impact of the proposed data dependence analysis techniques in practice. We perform an extensive experimental evaluation of the data dependence analysis tests, including the I-Test, the Omega test and the proposed new techniques. We compare these tests in terms of data dependence accuracy, compilation efficiency and effectiveness in program parallelization. We run several experiments using the Perfect Club benchmarks and the scientific library Lapack. We analyze the trade-off between accuracy and efficiency and the reasons for any approximation of each data dependence test. We determine the impact of the dependence analysis phase on the total compilation time and we measure the number of loops parallelized by each test. We conclude that we can employ, polynomial-time techniques to improve data dependence accuracy and increase program parallelization to a reasonable computation cost.

Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-Notch protein signalling

Hybrid automata are an eminently suitable modelling framework for biological protein regulatory networks, as the protein concentration dynamics inside each biological cell are modelled using linear differential equations; inputs activate or deactivate these continuous dynamics through discrete switches, which themselves are controlled by protein concentrations reaching given thresholds. This paper proposes an iterative refinement algorithm for computing discrete abstractions of a class of hybrid automata with piecewise affine continuous dynamics and forced discrete transitions, defined completely in terms of symbolic variables and parameters. Furthermore, these discrete abstractions are utilised to compute symbolic parametric backward reachable sets from the equilibria of the hybrid automata, that are guaranteed to be exact or conservative under-approximations. The algorithm is then implemented using MATLAB and QEPCAD, to compute reachable sets for the biologically observed equilibria of the multiple cell Delta-Notch protein signalling automaton with symbolic parameters. The results are analysed to show that novel, non-intuitive, and biologically interesting properties can be deduced from the reachability computation, thus demonstrating the utility of the algorithm.

Abstractions from proofs

The success of model checking for large programs depends crucially on the ability to efficiently construct parsimonious abstractions. A predicate abstraction is parsimonious if at each control location, it specifies only relationships between current values of variables, and only those which are required for proving correctness. Previous methods for automatically refining predicate abstractions until sufficient precision is obtained do not systematically construct parsimonious abstractions: predicates usually contain symbolic variables, and are added heuristically and often uniformly to many or all control locations at once. We use Craig interpolation to efficiently construct, from a given abstract error trace which cannot be concretized, a parsominous abstraction that removes the trace. At each location of the trace, we infer the relevant predicates as an interpolant between the two formulas that define the past and the future segment of the trace. Each interpolant is a relationship between current values of program variables, and is relevant only at that particular program location. It can be found by a linear scan of the proof of infeasibility of the trace.We develop our method for programs with arithmetic and pointer expressions, and call-by-value function calls. For function calls, Craig interpolation offers a systematic way of generating relevant predicates that contain only the local variables of the function and the values of the formal parameters when the function was called. We have extended our model checker Blast with predicate discovery by Craig interpolation, and applied it successfully to C programs with more than 130,000 lines of code, which was not possible with approaches that build less parsimonious abstractions.

RGC: A new conceptual clustering algorithm for mixed incomplete data sets

In this paper, a new conceptual algorithm for the conceptual analysis of mixed incomplete data sets is introduced. This is a logical combinatorial pattern recognition (LCPR) based tool for the conceptual structuralization of spaces. Starting from the limitations of the elaborated conceptual algorithms, our laboratories are working in the application of the methods, the techniques, and in general, the philosophy of the logical combinatorial pattern recognition with the task to improve those limitations. An extension of Michalski's concept of l- complex for any similarity measure, a generalization operator for symbolic variables, and an extension of Michalski's refunion operator are introduced. Finally, the performance of the RGC algorithm is analyzed. A comparison with several known conceptual algorithms is presented.

Multiple instances and symbolic variables in executable sequence charts

We extend live sequence charts (LSCs), a highly expressive variant of sequence diagrams, and provide the extension with an executable semantics. The extension involves support for instances that can bind to multiple objects and symbolic variables that can bind to arbitrary values. The result is a powerful executable language for expressing behavioral requirements on the level of inter-object interaction. The extension is implemented in full in our play-engine tool, with which one can execute the requirements directly without the need to build or synthesize an intra-object system model. It seems that in addition to many advantages in testing and requirements engineering, for some kinds of systems this could lead to the requirements actually serving as the final implementation.

Facilitation of local information processing in the attentional blink as indexed by the shooting line illusion.

Perception of the second of two rapidly sequential targets is impaired if the temporal lag between them is relatively short. This attentional blink (AB) is said to occur because the first target preoccupies attentional resources, leading to a shortage of attention for, and subsequent failure of judgement of, the second target. In the present study, we examined whether the attention which is preoccupied by the first target carries the common characteristic of spatial attention, that is local facilitation of information processing. To index the facilitation, we presented an attention-sensitive probe known as the shooting line illusion in which observers see a progressively drawn line from the attended end to the other. We found that illusory motion was perceived to emanate from the end where the first target was presented. This result demonstrates the inherently spatial nature of attention, even when it is defined by temporal and symbolic variables.

The Effect of Mineralocorticoid Analogues on Na + ,K + -Activated ATPase

The effect of 42 steroids of the 20-ketopregnane series with heterocycles fused in positions 16α and 17α on the activity of Na+,K+-activated ATPase from pig kidney was studied. It was shown that the studied compounds could be divided into two groups. The compounds from the first group stimulate the ATPase at low concentrations (1 × 10–8–1 × 10–7 M) and inhibit it at high concentrations (1 × 10–4 M). The second group of compounds stimulated the sodium pump at either concentration. This is explained by the cooperative action of the ATPase tetramer: after the reaction of its first binding site with the ligand, the tetramer changes the conformation and specificity of its other binding sites. Computer analysis of this series of compounds was carried out and a mathematical model of the dependence of their activities on the structure of their substituents was obtained with a high correlation coefficient and a satisfactory predictive power. This confirmed the structural similarity of the studied compounds with respect to their interaction with the ATPase binding sites. The method of descriptor analysis that was applied in this study is a new variant of approximation; it is based on the use of symbol variables as descriptors.

Effects of Symbol, Referent, and Instructional Variables on the Acquisition of Aided and Unaided Symbols by Individuals with Autism Spectrum Disorders

Individuals with little or no functional speech as a result of autism spectrum disorders frequently rely on aided and unaided augmentative and alternative communication (AAC) symbols and signs to augment or replace natural speech. This has led investigators to identify the variables that are responsible for the acquisition and communicative use of graphic symbols and manual signs. The overall aim of this article is to evaluate the role of symbol, referent, and instructional variables on the acquisition of AAC symbols by individuals with autism and severe speech and language impairments.

Compliance as Dissidence: Young Catholics and Sexual Ethics

This paper examines the sexual and reproductive attitudes of post-Vatican II Catholics in the United States. This study finds that even though most young Catholics possess liberal attitudes regarding homosexuality, abortion, premarital sex, and the use of contraceptives, about 10 percent of young Catholics accept the official teachings of the Catholic church regarding sexual issues. These young traditionalists represent a nonconformist minority residing within a more permissive religious culture. This article uses classic symbolic interactionist variables to determine how young Catholics differ on matters of sexual ethics. Results suggest that having a strong religious self-concept, a significant other who is traditional, and a religiously active mother in childhood encourages the acceptance of traditional attitudes. A path model is proposed and tested using these variables. Ultimately, a strong religious self-concept shows the greatest correlation with the possession of traditional attitudes among young Catholics.

Codes on graphs: normal realizations

A generalized state realization of the Wiberg (1996) type is called normal if symbol variables have degree 1 and state variables have degree 2. A natural graphical model of such a realization has leaf edges representing symbols, ordinary edges representing states, and vertices representing local constraints. Such a graph can be decoded by any version of the sum-product algorithm. Any state realization of a code can be put into normal form without essential change in the corresponding graph or in its decoding complexity. Group or linear codes are generated by group or linear state realizations. On a cycle-free graph, there exists a well-defined minimal canonical realization, and the sum-product algorithm is exact. However, the cut-set bound shows that graphs with cycles may have a superior performance-complexity tradeoff, although the sum-product algorithm is then inexact and iterative, and minimal realizations are not well-defined. Efficient cyclic and cycle-free realizations of Reed-Muller (RM) codes are given as examples. The dual of a normal group realization, appropriately defined, generates the dual group code. The dual realization has the same graph topology as the primal realization, replaces symbol and state variables by their character groups, and replaces primal local constraints by their duals. This fundamental result has many applications, including to dual state spaces, dual minimal trellises, duals to Tanner (1981) graphs, dual input/output (I/O) systems, and dual kernel and image representations. Finally a group code may be decoded using the dual graph, with appropriate Fourier transforms of the inputs and outputs; this can simplify decoding of high-rate codes.

COPAS: A New Algorithm for the Partial Input Encoding Problem

Frequently, the logic designer deals with functions with symbolic input variables. The binary encoding of such symbols should be chosen to optimize the final implementation. Conventionally, this input encoding (IE) problem has been solved in a two‐step process. First step generates constraints on the relationship between codes for different symbols, called group constraints. In a following step, symbols are encoded such that constraints are satisfied. This paper addresses the partial input encoding problem (PIE), a variation of the IE problem which generates codes of minimum length. The role of group constraints within the framework of the PIE problem has been questioned. This paper describes an algorithm that unlike conventional approaches, which try to maximize the number of satisfied constraints, targets the economical implementation of each input constraint. The proposed approach is based on a powerful heuristic that produces high quality results in shorter time compared to previous algorithm.

Symbolic timing analysis of asynchronous systems

We extend the time separations of events (TSE) timing analysis algorithm into the symbolic domain, that is, we allow symbolic variables to be used to specify unknown parameters of the model (essentially, unknown delays) and verification algorithms which are capable of identifying not just failure or success, but also the constraints on these symbolic variables which will ensure successful verification. The two contributions are (1) an iterative algorithm which continuously narrows down the domain of interest and (2) a practical method for reducing the representation of symbolic expressions containing minimizations and maximizations defined for a given domain. The algorithm applies to asynchronous circuits without conditional behavior. Although this may seem a severe restriction, this is a large and useful subclass which includes, e,g,, pipeline structures. We report experimental results for several asynchronous circuits to demonstrate that symbolic analysis is feasible and that the output provided is what a designer (or perhaps a synthesis tool) would often want to know.

A constraint-based genetic algorithm for concurrent engineering

Concurrent engineering is a complex problem in that a large number of considerations have to be brought to bear during the design stage. Compounding this complexity is that concurrent engineering problems are also likely to encompass both numeric and symbolic variables, and constraints that range from simple algebraic constraints, through conditional constraints, to potentially very complex database constraints. Previous approaches have concentrated on finding good solutions to simplified problems or, alternatively, to finding a feasible solution which may not be close to optimal. This paper presents an approach to concurrent engineering problems that extends the conventional genetic algorithm approach to handle the complex variety of variables and constraints that are inherent in a typical concurrent engineering problem. This poses some severe research challenges and this paper describes three genetic operators (the constraint-based initialization, crossover, and mutation operators) that preserve feasibility during initialization, crossover, and mutation. Preserving feasibility is achieved by incorporating local constraint propagation, boundary propagation, and set propagation. This overall approach is called the constraint-based genetic algorithm (CBGA) and aims to determine a near-optimal solution to concurrent engineering problems. The paper describes the background to constraint modelling, and genetic algorithms. The constraint-based initialization, crossover and mutation operators are then detailed, together with a description of the implementation of the CBGA in a system called Comet. Finally an example application is given.

On the Parsimony of the Multi-Layer Perceptrons when Processing Encoded Symbolic Variables

This article addresses the issue of symbolic processing with Multi-Layer Perceptrons through encoding. Given an encoding, we propose a lower bound of the number of parameters for an MLP to perform a random mapping of its input symbolic space to its output symbolic space. In the case of what we call binary encoding, the needed number of parameters may be theoretically computed. Given these two results, we show that the most efficient encodings are the ones which use one input unit per value.

UNIK-PMA: A unifier of optimization model with rule-based systems by the post-model analysis

Post-Model Analysis (PMA) is a framework that unifies an optimization model with a rule-based system and enables the multi-objective decision modeling that considers both numeric and symbolic objectives and decision variables. In this research, we have developed a system UNIK-PMA that implements the PMA procedure on the knowledge-assisted optimization modeler UNIK-OPT and a backward chaining rule-based system UNIK-BWD. This paper particularly elaborates the process of generating constraints of the linear programming model from the rule-based goals, which is a crucial step of PMA. UNIK-PMA is illustrated with an example of aggregate production planning.

Performance Prediction in Symbolic Scheduling of Partitioned Programs with Weight Variation

In this paper we consider the symbolic scheduling of partitioned loop programs which are modeled as iterative task graphs (ITGs). Each task in such a graph is coarse grained and contains a large chunk of computations. The weights of computation and communication vary from one iteration to another depending on the index value of the loop. The goal of scheduling such graphs is to incorporate the symbolic variables in weight functions and loop bounds and provide an asymptotically optimal schedule and predict its performance accurately. We provide a lower bound for optimal scheduling when weights of iterative task graphs change monotonically in the course of iterations and there is a sufficient number of processors. We present a technique that devises a valid symbolic schedule without searching all task instances and examine the asymptotic performance of this schedule compared to an optimal solution. Finally, we present case studies and experimental results on nCUBE-2 to verify our solutions.

Exploitation of symbolic information in interprocedural dependence analysis

The requirement for a very accurate dependence analysis to underpin software tools to aid the generation of efficient parallel implementations of scalar code is argued. The current status of dependence analysis is shown to be inadequate for the generation of efficient parallel code, causing too many conservative assumptions to be made. This paper summarises the limitations of conventional dependence analysis techniques, and then describes a series of extensions which enable the production of a much more accurate dependence graph. The extensions include analysis of symbolic variables, the development of a symbolic inequality disproof algorithm and its exploitation in a symbolic Banerjee inequality test; the use of inference engine proofs; the exploitation of exact dependence and dependence pre-domination attributes; interprocedural array analysis; conditional variable definition tracing; integer array tracing and division calculations. Analysis case studies on typical numerical code is shown to reduce the total dependencies estimated from conventional analysis by up to 50%. The techniques described in this paper have been embedded within a suite of tools, CAPTools, which combines analysis with user knowledge to produce efficient parallel implementations of numerical mesh based codes.

MultiPolynomial Resultant Algorithms

Computational methods for manipulating sets of polynomial equations are becoming of greater importance due to the use of polynomial equations in various applications. In some cases we need to eliminate variables from a given system of polynomial equations to obtain a "symbolically smaller" system, while in others we desire to compute the numerical solutions of non-linear polynomial equations. Recently, the techniques of Gröbner bases and polynomial continuation have received much attention as algorithmic methods for these symbolic and numeric applications. When it comes to practice, these methods are slow and not effective for a variety of reasons. In this paper we present efficient techniques for computing multipolynomial resultant algorithms and show their effectiveness for manipulating system of polynomial equations. In particular, we present efficient algorithms for computing the resultant of a system of polynomial equations (whose coefficients may be symbolic variables). The algorithm can also be used for interpolating polynomials from their values and expanding symbolic determinants. Furthermore, it is possible to come up with tight bounds on the running time and storage requirements of the algorithm. Finally, we use the symbolic elimination algorithm to compute the real or complex solutions of non-linear polynomial equations. It reduces the problem to finding roots of univariate polynomials. We also discuss the implementation of these algorithms and discuss their performance on some applications.