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\centerline{\bf PROJECT SUMMARY}
\centerline{ \bf Automated Enumerative Combinatorics} 
 
Doron Zeilberger proposes to continue to develop
an infra-structure for automated combinatorial enumeration
by systematizing and streamlining the many tools that he
has already developed, and creating new ones.
The novelty of his approach is that it goes far beyond just
implementing known algorithms, but rather develops
{\it meta-algorithms} that enable the computer to discover,
all by itself, new algorithms, that either the same, or any other, computer
(possibly running on a numeric, more efficient, computing platform)
can use to compute many terms of important, and often seemingly
intractable, sequences. This way the computer is performing the
kind of research that was formerly the monopoly of humans, but
of course, because of its computing powers, and great patience, can
go much further.

In particular, Zeilberger hopes to introduce a general 
computational theory of
{\it enumeration schemes}, that would include, as very special cases,
his, and other researchers', special-purpose enumeration schemes.
This theory should be general enough that in addition to
its applications to generating new enumeration theorems
for problems that are tractable, would also, hopefully, prove
{\it intractability} in a yet-to-be-found sense that would
be inspired by the computer scientists' notion of \#-P-hardness,
but tailored to {\it symbolic} rather than {\it numeric} computation.

He also hopes to continue, and develop much further, his theory
of symbolic-computational probability, that offers an attractive
alternative to the currently ruling dogma of Kolmogorov-style
measure-theoretic foundation, and since probability and statistics
are so important nowadays, should be of considerable potential for
applications.

Finally, the PI hopes to continue to develop, and extend the scope of,
{\it Wilf-Zeilberger} algorithmic proof theory, that  has already
found many applications in both pure mathematics (topology of knots),
applied mathematics (numerical analysis, approximation theory),
physics (large-scale computations of Feynman integrals), and even
economics.

The {\bf intellectual merit} of this research lies both in
expanding our theoretical knowledge in discrete mathematics,
and enhancing our understanding of the role of computers
in deriving rigorous mathematical results.

Its {\bf broader impact} is in that the {\it methodologies}
developed should be applicable to many other areas of not only
pure mathematics, but also applied mathematics, and hence
it  has potential applications to technological advances in
software development, and elsewhere.

This research is in the field of Combinatorics, whose usefulness
to science and technology is well-known. In particular, computer
science is largely based on combinatorics, as is electronic
communication and the World Wide Web.



 

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