Asymptotic Behavior of Linear Approximations of
Pseudo-Boolean Functions
Paper: jc11-4-2786; 2006/12/1
Asymptotic Behavior of Linear Approximations of
Pseudo-Boolean Functions
Guoli Ding , Robert F. Lax , Peter Chen , and Jianhua Chen
Department
of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
E-mail: ding, lax@math.lsu.edu
Department of Computer Science, Louisiana State University, Baton Rouge, LA 70803, USA
E-mail: chen, jianhua@csc.lsu.edu
[Received March 25, 2006; accepted August 16, 2006]
We study the problem of approximating pseudoBoolean functions by linear pseudo-Boolean functions. Pseudo-Boolean functions generalize ordinary
Boolean functions by allowing the function values to
be real numbers instead of just the 0-1 values. PseudoBoolean functions have been used by AI and theorem proving researchers for efficient constraint satisfaction solving. They can also be applied for modeling uncertainty. We investigate the possibility of efficiently computing a linear approximation of a pseudoBoolean function of arbitrary degree. We show some
example cases in which a simple (efficiently computable) linear approximating function works just as
well as the best linear approximating function, which
may require an exponential amount of computation to
obtain. We conjecture that for any pseudo-Boolean
function of fixed degree k 1 where k is independent
of the number of Boolean variables, the best linear approximating function works better than simply using
the linear part of the target function. We also study
the behavior of the expected best linear approximating function when the target pseudo-Boolean function
to be approximated is random.
Keywords: approximation, uncertainty, pseudo-Boolean
functions
1. Introduction
Pseudo-Boolean functions generalize ordinary Boolean
functions by allowing the function values to be real numbers instead of just the 0-1 values. A pseudo-Boolean
function f is a mapping from the set of all Boolean vectors of length n to the reals. Pseudo-Boolean functions
have been used for solving Constraint Satisfaction Problems by the AI and theorem-proving research communities. For example, Dixon and Ginsberg [4] studied using pseudo-Boolean constraints for solving satisfiability
problems and showed that pseudo-Boolean solvers tend to
be more efficient compared with resolution-based satisfiability solver. Another motivation for studying pseudoBoolean functions from the AI point of view is that these
Vol.11 No.4, 2007
functions provide a flexible framework for modeling uncertainty and yet they are fairly simple compared with
other alternative approaches; e.g., fuzzy logic [14]. Fuzzy
logic allows the variables x i to take values in the interval
0 1 and support quite complex fuzzy inferencing mechanisms using fuzzy rules, whereas pseudo-Boolean functions restrict the variables xi to take only Boolean (0 or 1)
values. Uncertainty modeling by pseudo-Boolean functions is accomplished by allowing function values to be
real numbers.
In this paper, we study the problem of approximating arbitrary pseudo-Boolean functions by linear pseudoBoolean functions. The rationale for using approximations rather than the exact original functions consists in
conceptual simplicity and computational tractability. Linear functions have a rather simple representation and intuitive appeal for human perception and understanding.
Computationally, they are also much easier to handle
compared with most other functions. There are many
works in research on uncertainty in AI that utilize linear functions to model uncertainty. For example, in [11],
linear belief functions are employed to describe the connections between returns of stocks with various factors in
the financial market. A qualitative linear utility formulation is presented in [5] for modeling the usefulness of
epistemic beliefs. Linear discriminant functions are also
widely used in pattern classification sometimes even in
situations in which the underlying patterns are non-linear.
In [9], linear threshold functions are used for the natural
language understanding task.
Let n be a positive integer and B n be the set of all
n-dimensional 0-1 vectors. A pseudo-Boolean function
f x1 x2 xn is a mapping from Bn to the reals. A
pseudo-Boolean function is often closely related to the
probability distribution of Bernoulli random variables.
For a simple example, if X1 X2 , and X3 are independent
identically distributed Bernoulli random variables with
PXi 1 23, then their joint probability distribution is
described by the pseudo-Boolean function f x 1 x2 x3 1271 x 1 x2 x3 x1 x2 x1 x3 x2 x3 x1 x2 x3 in
the sense that PX1 x1 X2 x2 X3 x3 f x1 x2 x3 .
Pseudo-Boolean functions are also the main objects of
study in the theory of cooperative games in economics
(see [6]). Pseudo-Boolean functions also appear in the
Journal of Advanced Computational Intelligence
and Intelligent Informatics
1
Ding, G. et al.
theory of evolutionary computation, where they are called
fitness functions (see [8] and [15]). Jin [8] points out that
evaluation of a fitness function in a real-world application
is sometimes computationally very expensive, thus necessitating the approximation of the function by a low-degree
function. Zhang and Rowe [15] study linear and quadratic
approximations of pseudo-Boolean functions and compare these.
For each variable x i , we define x̄i 1 xi . Then
L x1 x̄1 x2 x̄2 xn x̄n is the set of literals. Clearly,
every pseudo-Boolean function f x 1 x2 xn has a
pseudo-Boolean expression
∑ αZ ∏ z
where
. . . . . . . . . . . . . (1)
zZ
Z L
∏ z is defined to be 1.
z0/
As usual, we assume that
αZ 0 if z z̄ Z for some z L. The degree of expression (1) is defined to be the largest Z such that α Z 0.
Starting with expression (1), if we replace each x̄ i by 1 xi
and then multiply out and collect terms, we obtain a representation of f of the form
f x1 xn ∑
T N
a T ∏ xi
i T
. . . . . (2)
The expression (2) is called a multilinear polynomial. The
following propositions are easy to prove (cf. [2]).
Proposition 1.1 Every pseudo-Boolean function has a
unique multilinear polynomial expression.
Proposition 1.2 If a pseudo-Boolean function has a
pseudo-Boolean expression of degree d, then its multilinear polynomial expression has degree at most d.
Let N 1 2 3 n, x x1 x2 xn , and
f x ∑
T N
a T ∏ xi . . . . . . . . . (3)
iT
Let x be the linear function
aT t 1
∑ 2t ∑
T N
i N
aT
∑ 2t 1
T i
xi
. . . (4)
where t T . It was proved in [6] that x is the best linear approximation of f x, in the sense that it minimizes
∑n
xB
f x l x2 over all linear functions l x.
Even though an explicit expression for x is available,
computing all its coefficients is a time-consuming task
since each coefficient of x is a sum of exponentially
many terms. In this paper, we discuss possibilities of substituting x with an efficiently computable function and
still achieving the same quality of approximation. We will
mainly focus on the asymptotic behavior of different approximations.
Let Lx denote the linear part of f x. We first investigate the possibility of using the function Lx to approximate f x. We show some example functions which can
2
(or cannot) be approximated well by Lx. A conjecture
is presented that suggests the approximation of f x by
Lx will not be effective if f x has degree k 1 for k
independent of n. We then study the problem of approximating by the expectation of x to a random f x with
each coefficient a T being a random variable. We identify
conditions under which such an approximation is (or is
not) reasonable.
2. Relative Quality of the Best Linear Approximation
In this section, we investigate the following natural
question:
Is there a good approximation ˆx of x?
As mentioned earlier, we will take the asymptotic point
of view. That is, we compare ˆx and x when n is
sufficiently large.
Clearly, we want ˆx to be linear. We also want ˆx to
be efficiently computable. There are two natural choices
for ˆx. If all coefficients of x have limits, as n approaches infinity, then this limit is a natural candidate for
ˆx. However, this limit usually does not exist. In such a
case, we propose the following.
Let Lx be the linear function
n
a0/ ∑ ai xi i 1
which is the linear part of f x. If f x is a nonlinear
function, we would like to compute
ρ f ∑xBn f x x2
∑xBn f x Lx2
which would tell us, when comparing with Lx, how
much better the best linear approximation x is. In the
following, we consider a few examples. We will make a
conjecture based on these examples, which says that ρ f is small if f has bounded degree (with the bound being independent of n). This would imply that if f has bounded
degree and n is large, then the linear part of f would not
be nearly as good an approximation to f as the best linear approximation given by Eq. (4) and it would be worth
performing the computations required in Eq. (4).
2.1. A Highly Nonlinear Function
Let us consider the case when a T
That is
f x 1, for all T
N.
∑ ∏ xi T N i T
Then its best linear approximation is
x
n 3
2
n1
1
Journal of Advanced Computational Intelligence
and Intelligent Informatics
n
3
3
2
n
∑ xi .
(5)
i 1
Vol.11 No.4, 2007
Asymptotic Behavior of Linear Approximations of
Pseudo-Boolean Functions
Notice that all coefficients of x approach infinity, as n
approaches infinity, thus x does not have a limit function. In such a case, we would like to compare x with
Lx by computing ρ f . From Eq. (5) it is straightforward to verify that
∑
xBn
f x x
2
5
1
9
10
n n
1
9
2.3. A Two-Term Function
Let us consider
f x x1 x2 xn xn1 xn2 x2n (6)
Then its best linear approximation is
and
n
∑n
∑
xB
n
tions of unbounded degree with a small number of terms,
as shown by the example from the next subsection.
f x2 ∑
i 0
Since
xBn
f x
n 2i
2 5n i
x 2 ∑
xB
∑
xBn
Lx2
f x
f x2 it follows from Eqs. (6) and (7) that ρ f 1 o1.
Therefore, we conclude that, asymptotically, Lx performs just as well as x. Since Lx is much easier to
compute, this observation suggests that, in this example,
we should use Lx, instead of x, to approximate this
function f x.
Let us consider the following quadratic function
∑
xi x j nn
1
1i j n
n 1
1 2n
xi 2n1 2n1 i∑1
. . . . . . (9)
∑
f x x
2
∑
f x
ˆ x
2
xB2n
and
x
B2n
2
n1
∑
x
B2n
2
n1
1
f x n1
2n
Lx2
1
1 n
2
which imply that ρ f 1 o1. Therefore, we can
say that ˆx Lx 0 performs just as well as x.
This example suggests that, in general, x can only perform well when the degree of f x is bounded, which can
be considered as supporting evidence for our Conjecture
above.
2.2. A Quadratic Function
f x Clearly, all coefficients of x have limit zero. Thus this
limit function, ˆx, is the same as Lx, and they both are
the zero function. Now it follows from Eq. (9) that
n
x
. . . . . (7)
Then
x
8
n
1
2
n
∑ xi . . . .
(8)
Clearly, x has no limit. Thus we consider ρ f . It
follows from Eq. (8) that
∑n
xB
f x x
f x
Lx2 ∑
2
∑n
xB
n
12n5
nn
and
i 0
n6
2
n 2
i i
i
n2
12 4
3n 2nn
1
which imply that ρ f n22 1 o1. Therefore, x
performs significantly better than Lx.
Fix an integer k 1. Define ρ k n by ρk n sup ρ f ,
where the sup is over all pseudo-Boolean functions f :
Bn R such that deg f k. We make the following conjecture.
Conjecture. If k 1 is a fixed integer, then ρ k n o1.
If this conjecture is true, it provides a justification for
computing x, instead of simply using Lx. We point
out that, if the conjecture is true, it is the best possible in
the sense that the conjecture cannot be extended to funcVol.11 No.4, 2007
3. The Expected Linear Approximation
i 1
We continue our study on the problem of finding a good
approximation to x. In this section, we will consider it
from a different viewpoint.
A random polynomial is a polynomial whose coefficients are random variables. Random polynomials have
applications in engineering and economics – see [1]. In
this section, we consider a random multilinear polynomial; i.e., suppose in Eq. (2) that every a T aT ω is
a random variable on a sample space Ω. A realization
of a random multilinear polynomial is obtained when we
evaluate the random variable coefficients at an element ω
of the sample space Ω. Then we ask:
(*) Instead of computing the best linear approximation
x for each realization of f x, is it possible to find a
single linear pseudo-Boolean function that depends only
on the expected values of each a T and that serves as a
good approximation for every realization of f x?
3.1. A Negative Answer
In this subsection, we exhibit a scenario under which
the answer to question (*) is negative.
Suppose f x is a linear function. That is, a T 0, for
all T N with T 1. In this case, it is clear that x Journal of Advanced Computational Intelligence
and Intelligent Informatics
3
Ding, G. et al.
f x. Moreover, the linear approximation obtained from
the expected values of each a T is
n
ˆ x
E a0/ ∑ E ai xi Let
D
D i 1
Obviously,
∑n
xB
x
f x 2
0
Let us assume that the random variables a T , for all
T 1, are independent. Let b0 a0/ E a0/ and
bi ai E ai , for i 1 2 n. Then E b i 0
(i 0 1 2 n), E b20 V a0/ , and E b2i V ai (i 1 2 n). For each vector x B n , let us define
X i N : xi 1. It follows that
ˆ x
2
2
b0 ∑ bi
iX
2
b0 ∑ b2i
i X
2 ∑ bi b j E bi b j E bi E b j 0
n
2nV a0/ 2n1 ∑ V ai i 1
0 and β is finite, then
The implication of this theorem is the following. If the
variance of every a T is positive and finite, then we can
say that, on average, to approximate f x, its best linear
approximation x is not too much better than x, the
best linear approximation of f x. For example, if all the
aT are independent random variables with expected value
1 and positive variance, then the linear pseudo-Boolean
function from Eq. (5) would be a good approximation for
every realization of f x if n is large.
We prove the theorem by proving a sequence of three
lemmas, using notation as above.
∑ ∑ λX2T V
E D is bounded below by a positive number.
This example shows that the answer to (*) is negative
in general. That is, we have to compute x for each
f x because the difference between ∑ f x x2
xB
ˆ x
f x
x
2
where
f x
i 1
f x ∑
xBn
if T
X;
if T
X Proof. It is routine to verify that
n
aT 2t
V a0/ ∑ V ai ∑n
T NX N
which can be arbitrarily large, provided
and E t 1
1 2t
λX T t 1 2T X Thus
x 2 f x Lemma 3.1
where the last sum is taken over all indices i j in X 0. Notice that
δ
xB
E D E D 5 n n 5
O1
E D
6
5
xBn
f x
∑n
x 2
f x
Theorem. If α
But what can we say about
δ E ∑ f x ˆx2 ?
∑
xBn
∑
aT t 1
aT ∑
2t
T N
∑
aT ∑
aT λX T T X
x
Bn
2
x
T X
, under any measure, can be ar-
T N
bitrarily large.
∑
iX
aT t 1
2t
N
∑
T
aT
∑ 2t 1
T i
aT
T X t 1
N2
∑
T
Therefore,
3.2. A Positive Answer
In this subsection, we assume that
E f x
V aT β for all T N. Again, we assume that a S and aT are independent if S T . Let aT E aT , for all T N. Let
f x
∑ aT ∏ xi α
T N
4
a t 1
∑ T 2t ∑
T N
iN
a
∑ 2t T 1
T i
∑ λX2T E
∑ λX2T V
aT ∑
∑
S T N
2
λX S λX T aS aT
∑ λX2T
T N
T N
xi a2T 2
T N
Then its best linear approximation is
iT
2
T N
x
x
∑
S T N
λX2T V aT f x
aT 2
λX S λX T aS aT
x 2 which proves the lemma.
Lemma 3.2 E D α 3n 1
Journal of Advanced Computational Intelligence
and Intelligent Informatics
o1.
Vol.11 No.4, 2007
Asymptotic Behavior of Linear Approximations of
Pseudo-Boolean Functions
Proof. By Lemma 3.1,
E D ∑ ∑
It is straightforward to verify that
λX2T V aT T NX N
α ∑ ∑
T NX N
λX2T
∑ λX2T
X T
t 1
2t
1
2nt 1
t 1
2t
2nt 1
t 1
2t
2
2
2
t 1
2t
1
2nt
∑
∑
0/ R N T S T
2nt
∑
1
t
1 2s
2t
t
t
1 ∑
t
s
s 0
2
2nt
t
1 2s
2t
n
t
∑
t 0
3n
2t 1 t 2nt
Lemma 3.3
α
2β n5
5
n 5
2
n
n3
3
2t 1 t n
3
2
f x
81
T
∑ µX T aT
2
5
2
x
t
x
∑
T T
¼
E d x 2
µX S s
1
β
n
2T X . Then
2t
∑ aS ∑ µX T aT
T N
∑ aS ∑ µX T aT S X
T N
T
∑ µX2 T
aT 2
T N
µX T µX T aT aT
¼
N
¼
x
2
f x
x 2 aS ∑ µX2 T V
∑ µX SV
T N
2s2s 1
n
∑
s
n
x
x 0
n
5
2
x
∑
a T 12s
x s1
s 2s
s 0
n5
5
On the other hand,
∑ ∑ µX2 T
∑ ∑ µX2 T
T NX N
n
∑
n
t
5
2
t
∑
i 0
t
i
t
1 2i
2t
2
2nt
n5
5
The result follows from the last three equations immediately.
D 2α S X
It follows that
∑ µX S aS aS
µX T aS a For each S X, we have
n
E D
2
S X
t 0
n5
5
Then
S X
Vol.11 No.4, 2007
2
¼
Notice that f x x2 can be expressed in a similar
way (with each aT replaced by a T ). Let
4
and
f x
µX T a
X NT N
n
n2 26n 81
9
Proof. Let µX T
S X T N S T
2n3t 1 t 2
o1
3n 1
∑
2n2t 1 t which proves the lemma.
∑
T N
aS aS
S S¼ X
X NS X
∑
2
∑ ∑ µX S
2
1t 12t
λX2T
2
2
Therefore,
∑ ∑
d x f x
2n2t 1 t 2n3t 1 t 2
T NX N
n ∑ a2S
2
1 2s
2t
S T
2
2nt
∑ aS
2
S X
S X
nt
∑ aS
T N
∑ λX2T
2
2
T . Then, for
X N
nt
2
S X
∑ λX2T
X T
For any T X N, let S X T and R X
any fixed T N,
x
f x
5
2
n5
5
Proof of the Theorem. The theorem follows from the last
two lemmas immediately.
4. Discussion
Pseudo-Boolean functions have been studied extensively by the Operations Research community for optimizations. In recent years the AI research community, in
particular, the theorem-proving researchers have investigated the use of pseudo-Boolean functions and pseudo-
Journal of Advanced Computational Intelligence
and Intelligent Informatics
5
Ding, G. et al.
Boolean optimizations for various AI applications. In
[13], pseudo-Boolean functions are used for modeling a
job-shop scheduling problem. Lower bounding method
has been developed in [12] for pseudo-Boolean optimization problems. The existing AI literature indicates the
usefulness of pseudo-Boolean functions for AI-related
optimization and constraint solving.
We take the perspective that pseudo-Boolean functions
can be used for modeling uncertainty, in addition to being
a model for constraints. The function value f x for a
pseudo-Boolean function f on a point x can be seen as
modeling the degree of certainty (positive or negative) of
x belonging to the concept described by the function f .
Consider the following terrorist profiling problem.
Suppose we know (from existing records) that there are
five basic attributes relevant for recognizing instances of
terrorists/non-terrorists:
x: having traveled to certain sensitive regions;
y: being from some specific ethnic groups;
z: being an old person;
u: having attended a pilot school;
v: having past criminal record.
Assume that we also know that the most relevant composite attributes for terrorists profiling are: x y z xu xv.
Assume that we have also seen three specific known instances of terrorist together with their ratings (how dangerous they are) given by some source (say, CIA). We
model the terrorists profiles by a pseudo-Boolean function of the form
f x y z u v ax by cz dxu exv
Here V = x y z u v is the set of Boolean variables and
each of the coefficients a b c d e takes value in 1 1.
Suppose the known instances of terrorists are given by:
f 11101 05 f 11010 18 and f 10011 20. It
is intuitive to see from the examples that attributes xu, x,
xv are quite important to make an instance’s rating highly
positive (more likely to be a terrorist) - the instance vector
10011 with x xu xv 1 and y z 0 has the highest
rating among the three instances. On the other hand, the
attribute z seems to reduce an instance’s rating: the vector 11101 (with z 1) has much smaller rating than the
other two instances (with z 0. In [3], we described a
method for learning the “best” pseudo-Boolean function
that is compatible with a limited set of training data. This
method involves considering a polyhedron whose points
correspond to vectors of coefficients of pseudo-Boolean
functions that are compatible with the data. The best
function then corresponds to the centroid of this polyhedron. (This is similar to the Bayes point used in the
theory of Support Vector Machines [7].) Using this learning method here, we obtain the pseudo-Boolean function
f 077x 015y 077z 088xu 035xv.
Note that this pseudo-Boolean function is not a linear one. The best approximating linear function is given
6
by 088 0354 077 088 0352x 015y 077z 0882u 0352v. That is,
03075 1385x 015y
077z 044u 0175v
Applying the resulting to the 3 known instance vectors, we obtain 11101 06325, 11010 16675,
and 10011 16925. On the other hand, if we apply
the function Lx, the linear part of f , L 077x 015y
077z, the results are much worse: L11101 015,
L11010 092, and L10011 077. This example illustrates that the best linear approximation of a quadratic
function is generally better than using the linear part of
the original function, which is related to our Conjecture
in section 2.
5. Conclusions
Pseudo-Boolean functions are useful for AI applications such as constraint solving and uncertainty modeling. Approximating a pseudo-Boolean function of arbitrary degree by a linear pseudo-Boolean function is often
desired for simplicity and computational efficiency. Using the linear part of the target function as an approximation would produce similar asymptotic effect as the best
linear approximation function does for some cases and
save computations, but in general this approach may not
be valid. We also considered random multilinear polynomials and found conditions that tell when the best linear
approximation to the multilinear polynomial given by the
expected values of the coefficient random variables may
be used to approximate every realization of the random
multilinear polynomial.
Acknowledgements
This research was partially supported by NSF grant ITR0326387 and AFOSR grants FA9550-05-1-0454, F49620-03-10238, F49620-03-1-0239 and F49620-03-1-0241. We thank the
referees for helpful comments.
References:
[1] A. T. Bharucha-Reid and M. Sambandham, “Random Polynomials,” Academic Press, 1986.
[2] E. Boros and P. L. Hammer, “Pseudo-Boolean Optimization,” Discrete Appl. Math., 123 (1-3), pp. 155-225, 2002.
[3] G. Ding, J. Chen, R. Lax, and P. Chen, “Efficient learning of
pseudo-boolean functions from limited training data,” Foundations
of Intelligent Systems: 15th International Symposium, ISMIS 2005,
Lect. Notes in Comp. Sci. 3488 (Springer, 2005), pp. 323-331.
[4] H. E. Dixon and M. L. Ginsberg, “Inference methods for a pseudoBoolean satisfiability solver,” Proc. of American Association of AI
Conference, 2002.
[5] P. H. Giang and P. P. Shenoy, “A Qualitative Linear Utility Theory for Spohn’s Theory of Epistemic Beliefs,” Proceedings of Int.
Conference on Uncertainty in AI, San Francisco, CA, pp. 220-229,
2000.
[6] P. L. Hammer and R. Holzman, “Approximations of PseudoBoolean Functions; Applications to Game Theory,” ZOR – Methods
and Models of Operations Research, 36, pp. 3-21, 1992.
[7] R. Herbrich and T. Graepel, “Bayes Point Machines,” Journal of
Machine Learning Research, 1, pp. 245-279, 2001.
[8] Y. Jin, “A Comprehensive Survey of Fitness Approximation in Evolutionary Computation,” Soft Computing, 9, pp. 3-12, 2005.
Journal of Advanced Computational Intelligence
and Intelligent Informatics
Vol.11 No.4, 2007
Asymptotic Behavior of Linear Approximations of
Pseudo-Boolean Functions
[9] R. Khardon, D. Roth, and L. G. Valiant, “Relational Learning for
NLP using Linear Threshold Elements,” Proceedings of Int. Joint
Conference on AI (IJCAI’99), Aug., 1999.
[10] R. Lax, G. Ding, P. Chen, and J. Chen, “Approximating PseudoBoolean Functions on Non-uniform Domains,” Proceedings of
IJCAI-05, pp. 1754-1755, 2005.
[11] L. Liu, C. Shenoy, and P. P. Shenoy, “A Linear Belief Function Approach to Portfolio Evaluation,” Proceedings of Int. Conference on
Uncertainty in AI, San Francisco, CA, pp. 370-377, 2003.
[12] V. M. Manquinho and J. Marques-Silva, “Integration of Lower
Bound Estimates in Pseudo-Boolean Optimization,” Proc. of IEEE
Int. Conference on Tools with AI, 2004.
[13] S. Prestwich and C. Quirke, “Boolean and Pseudo-Boolean Models
for Scheduling,” Proc. of Second International Workshop on Modeling and Reformulating Constraint Satisfaction Problems, 2003.
[14] L. A. Zadeh and J. Kacprzyk (Eds.), “Fuzzy Logic for the Management of Uncertainty,” John Wiley & Sons, 1992.
[15] H. Zhang and J. E. Rowe, “Best Approximations of Fitness Functions of Binary Strings,” Natural Computing, 3, pp. 113-124, 2004.
Name:
Peter P. Chen
Affiliation:
Department of Computer Science, Louisiana
State University
Address:
Baton Rouge, Louisiana, 70803, USA
Brief Biographical History:
Dr. Peter Chen, a Distinguished Chair Professor at LSU (Baton Rouge),
has taught at MIT, UCLA, and Harvard.
Main Works:
¯ Prof. Chen is internationally known for his work on the
Entity-Relationship (ER) model. He has received many awards including
ACM/AAAI Allen Newell Award, IEEE Harry Goode Award, Pan
Wen-Yuen Outstanding Research Award, DAMA International
Achievement Award, and Stevens Software Method Innovation Award. He
has served as a keynote speaker at approximately 30 international
conferences.
Name:
Guoli Ding
Membership in Learned Societies:
Affiliation:
Department of Mathematics, Louisiana State
University
¯ Fellow of IEEE, ACM, AAAS
¯ Member of European Academy of Sciences
¯ Invited Expert in XML Schema and XLink WGs of W3C
¯ Member of U.S. NSF/CISE advisory committee and U.S. Air Force
Scientific Advisory Board
¯ Listed in Who’s Who in America and Who’s Who in the World
Address:
Baton Rouge, Louisiana, 70803, USA
Brief Biographical History:
1991 Received Ph.D. from Rutgers University
1991- Assistant Professor, Mathematics Department at Louisiana State
University
????- Associate Professor, Mathematics Department at Louisiana State
University
????- Professor, Mathematics Department at Louisiana State University
Name:
Jianhua Chen
Affiliation:
Main Works:
Department of Computer Science, Louisiana
State University
¯ Prof. Ding has published many research articles in the fields of graph
theory and matroid theory.
Address:
Baton Rouge, Louisiana, 70803, USA
Name:
Brief Biographical History:
Robert F. Lax
1988 Received Ph.D. in Computer Science from Jilin University, Chang
Chun, China
1988- Visiting Assistant Professor, the Computer Science Department of
Louisiana State University
????- Assistant Professor, the Computer Science Department of Louisiana
State University
currently Associate Professor, the Computer Science Department of
Louisiana State University
Affiliation:
Department of Mathematics, Louisiana State
University
Main Works:
¯ Dr. Chen’s research interests include machine learning and data mining,
Web mining, fuzzy Logic and Fuzzy Clustering, Knowledge
Representation and Reasoning.
Address:
Baton Rouge, Louisiana, 70803, USA
Brief Biographical History:
1973 Received Ph.D. from MIT
1973- Faculty member, Department of Mathematics, Louisiana State
University
Main Works:
¯ Prof. Lax has published research articles in the fields of algebraic curves
and algebraic coding theory.
Vol.11 No.4, 2007
Journal of Advanced Computational Intelligence
and Intelligent Informatics
7