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Project 3.1. Study the articles [ 15 , 37 ] and construct the state spaces and wiring
diagrams of the models.
Project 3.2. Study the article [ 16 ] and, without computing the entire state space,
find the cycle structure of the following Boolean monomial dynamical system: F
:
2 → Z
2 , where
F
Z
x 2 ).
Project 3.3. In [ 17 ], the authors describe an equivalence among three discrete
modeling classes, namely logical models, bounded Petri nets , and PDSs. What is
the significance of such an equivalence? In addressing this question, consider the
following:
1. In their article, the authors describe logical models and bounded Petri nets. How
easy is it to describe or construct these types of models?
2. What algebraic structure do they have?
3. How do their description and algebraic structure compare to those of PDSs?
4. Do you think it is easier to develop an algorithm for PDSs than for logical models
or Petri nets?
(
x 1 ,
x 2 ,
x 3 ,
x 4 ,
x 5 ,
x 6 ,
x 7 ) = (
x 7 ,
x 1 x 3 ,
x 4 ,
x 5 ,
x 6 ,
x 7 ,
3.3 COMPUTATIONAL ALGEBRA PRELIMINARIES
In this section we introduce some basic concepts from computational algebra that we
will need for the rest of the chapter. For a comprehensive introductory treatment of
the subject some excellent sources are [ 19 - 21 ].
In linear algebra we learn how to solve systems of linear equations using Gaus-
sian elimination. However, the algorithm cannot be easily adjusted for application to
systems of polynomials of degree larger than one. Simultaneously solving systems of
polynomial equations is needed in many areas but a systematic computational method
was not developed until the 1960s when the so-called Gröbner bases were developed
[ 22 , 23 ]. The role of a Gröbner basis is similar to that of a basis for a vector space and
the algorithm for finding it can be seen as a multivariate, nonlinear generalization of
Gaussian elimination for linear systems.
An issue that arises when using multivariable polynomials is that polynomial divi-
sion is generally not well defined. Polynomial division in one variable is always well
defined in the sense that there is a unique remainder upon division of polynomials.
This is because there is a unique monomial ordering (see Definition 3.4 ) and the divi-
sion of polynomial f by polynomial g proceeds by dividing the highest power of the
variable in g into the highest power in f . In other words, the one-variable monomials
are ordered using degree ordering:
···
x m + 1
x m
x 2
.
With multivariate polynomials, however, there is more than one way of ordering their
monomials (terms) and thus to carry out long division. So the remainder of long
division is not uniquely determined in general.
···
x
1
 
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