- A Problem Solving Environment for Numerical Partial
Differential Equations
R.L. Akers,
P. Baffes,
E. Kant,
C. Randall,
S. Steinberg,
R.L Young
Abstract:
SciNapse is a problem solving environment for numerically solving
partial differential equations that is currently under commercial
development. The main interface to the system is the "problem
specification language". This language allows an initial-boundary
value problem for a system of partial differential equations to be
specified in terms of invariant differential operators or in a
particular coordinate system. The region in which the differential
equations hold is specified in terms of a union of rectangles in some
coordinate system. This allow rather general regions apropriate for
discretization using logically-rectangular grids. The methods for
discretizing the differential equations and the boundary conditions
can be specified by giving keywords for common methods or by giving
detailed mathematical replacement rules, or some combination of
both. In the case of finite-difference discretizations, simple ways
exist for specifying grid staggering and the placement of the boundaries
in the staggered grid.
The SciNapse system includes a template language for specifying
algorithms. One typical use for templates is to specify the over-all
time evolution in terms of a general method for taking individual time
steps. Then the particular method for taking a time step can be
chosen from a library of templates, e.g. Runge-Kutta-Fehlberg or
Dormand-Prince integrators, or a new template may be written. If the
method is implicit, then a solver may also be chosen from a library of
solvers, e.g. preconditioned conjugate gradient or quasi-minimal
residual, or a new solver template can be written. SciNapse has
heuristics for choosing solvers, and many other features of the
solution algorithm. The SciNapse system is implemented in Mathematica
and the templates are executable Mathematica code, so they can easily
be tested for correctness in Mathematica.
SciNapse is a knowledge-based program synthesis system implemented
with objects, transformation rules, and a reasoning system. A great
deal of knowledge engineering has gone into providing the system with
extensive information about how to solve initial boundary value problems
in an intelligent manner. The system starts with an abstract notion
of what is needed to write a program recorded in its objects and
starts filling in the details based on rules associated with the
objects. Information may be given to SciNapse in files written by the
user, from a simple terminal interface, or in a graphical user
interface. When SciNapse realizes that it doesn't have a piece of
information that it needs to specify a problem, it will ask the user
for that information, typically by giving a menu of alternatives.
Users can have the system use its own knowledge whenever possible or
can request that they be asked to confirm system choices. If some
information has already been given to SciNapse, then the reasoning
system uses this to eliminate alternatives. Using constaints (hard and
fast rules about legal options and combinations), heuristic rules of
thumb for making decisions, and defaults, SciNapse can take much of
the burden of writing a program off the user.
- Computer Algebra and Artificial Intelligence
Jacques Calmet
Abstract:
The relationship between CA and AI is stressed in the first part of the talk.
It is emphasized that this relationship leads to cross fertilization of the
two domains.
The second part of the talk is devoted to two specific applications. The
first one shows how to better prove theorems using computer algebra systems.
Examples using Maple and Magma will be presented. The second application is
for a problem in vision and more specifically in traffic scene
reconstruction. It is shown how Mathematica can be used to improve an
approach based upon multi-agents.
If time permits, a possible non-standard application outside AI will be
suggested: the use of Computer Algebra in Econometrics.
- Thermodyanmics with Maple
Ross Taylor
Abstract:
Computer algebra systems (of which Maple is but one example)
have enormous potential, not just in thermodynamics, but in all
areas of science and engineering.
However, to be useful in thermodynamics a computer algebra
system needs:
- The ability to work with total differentials of undefined
functions, for example dS where S=S(U,V).
- To be able to work with the subscripted (indexed) partial
derivatives of thermodynamic functions found in many
thermodynamic formulae.
- The ability to differentiate undefined or arbitrary sums (a
summation where the upper limit is a symbol rather than a
number).
As far as the author is aware, none of the commercially
available computer algebra is able to do any of the above right
out of the box. In this paper we describe extensions to Maple
so that it can be used to rapidly develop the expressions needed
to compute thermodynamic properties of mixtures of any number of
components.
This paper also presents examples of thermodynamic computations
carried out using Maple with an emphasis on graphical
visualization of both symbolic and numerical results.
One of the advantages of using a computer algebra system is the
greatly increased accuracy of derivations of thermodynamic
properties and numerical computations. That is, students are
much more likely to get it right. On the debit side is a
possible decrease in the students ability to carry out such
derivations and computations by hand. Other advantages and
disadvantages of using a computer algebra package in
undergraduate education will also be discussed.
- A Topology-Independent Model for Railway Interlocking
Systems
Eugenio Roanes-Lozano,
Luis M. Laita
Abstract:
Two decision models for interlocking systems are presented. The first for the
case that the turnouts have spring switches and the second for the case that
the turnouts should not be trailed through a switch set against. In both of
them, the algorithm is independent from the topology of the station and is
based on the calculation of accessibility in an oriented graph.
Surprisingly brief implementations in Maple V of both models have been
developed. The package analyzes any change in the position of the switches
and the clearances given by the semaphores and supervises if it is safe to
authorize the change. In the second model, which turnouts could be trailed
through switches set against is also studied. To finish with, an example is
given.
- CA and Electricity Distribution
Antonio Montes
Abstract:
The Load Flow Problem is very important in Electrical
Engineering. It provides systems of multivariate polynomial
equations. Instead of ordinary numerical solutions it would be very
convenient to obtain algebraic solutions. Combining different
Computer Algebra techniques, namely Groebner basis, dialytic
elimination, resultants and comprehensive Groebner basis, we provide
a method to solve algebraically simple networks, allowing us also to
study the numerical stability of the problem and of the algorithm.
- Regular Expressions Simplification.
Alejandro A.R. Trejo,
Guillermo Fernandez Anaya
Abstract:
Although Kleene's regular expressions are known since middle 50's and that
play a central role in many areas of computer science, such as in the design
of sequential circuits and, very specially, in the compiler theory,
at the present time an authomatic method for their simplification is
not available. Such a method becomes desirable whenever
you face involved language descriptions which require long and tedius
sequences of manipulations.
This talk is devoted to present a technique to simplify regular expressions.
Several ideas of Computer Algebra and symbolic computing
are applied to the regular expression symbolic domain.
An implementation of the described technique has been developed.
The talk is divided in the following sections:
In section 1, the problem is stated. We present a wide view of the
question proposed for solution.
In section 2, the technique for simplifying
is outlined. In section 3, the method of representation and manipulation of
expressions is explained. Finally, in section 4, the implemented regular
expression simplifier system is presented, together with examples of
its usage.
- Computer Algebra application to the distribution
of sample correlation coefficient.
Shigekazu Nakagawa,
Naoto Niki,
Hiroki Hashiguchi
Abstract:
Let $r$ be the correlation coefficient based on a
sample of size $n$ from a population $F$.
The value of $r$ remains unchanged however
observations may be exchanged or permuted.
The statistic $r$ is one of symmetric statistics.
Our goal is divided into two parts:
- obtaining the asymptotic cumulants of $r$,
- as a consequence of 1., deriving the asymptotic expansions
for probability integrals and percentiles of $r$,
where $F$ has population cumulants of requisite order.
In accomplishing 1, the computational difficulty arises there.
If we want the more precise approximation, the order
of requisite approximate cumulants of $r$ increases.
Raising the order may cost a large amount of algebraic
computation often beyond human power.
Here computer algebra induces an increasing interest.
The authors have taken a new look at symmetric statistics
and developed the symbolic and algebraic
algorithm for changing of bases of symmetric polynomials.
Both the asymptotic cumulants and the approximate distributions are given.
