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axioms weaker than probability axioms, thus belief function can be viewed as a
superset of existing probability functions. With belief function, even without
precise probabilities, constrains on probability distributions can be set based on
prior domain knowledge. The theory has well established theoretical foundations,
yet its definition and computation is comparatively complex. In recent years, this
theory of evidence has gained more and more research focuses, and many
research achievements and application systems have been developed. For
example, Lotfi Zadeh illustrated how the Dempster-Shafer theory can be viewed
as an instance of inference from second-order relations, and applied in a
relational database.
In 1965, Lotfi Zadeh proposed the Fuzzy Set, based on which a series of
research have been made, including fuzzy logic, fuzzy decision making,
probability theory, etc. For reasoning with natural language, Zadeh introduced
fuzzy quantization to represent fuzzy propositions in natural language, defined
concepts of linguistic variable, linguistic value and probability distribution,
developed possibility theory and approximate reasoning. His work has attracted
much research focuses. Fuzzy mathematics has been widely applied to expert
systems and intelligent controllers, as well as for the research of fuzzy computer.
Chinese researchers have done a lot in theoretical research and practical
applications, drawing much attention from the international academics. However,
many theoretical problems still remain to be solved in this domain. There are also
some different views and disputes, such as, what is the basis for fuzzy logic?
What about the problem of consistency and completeness of fuzzy logic? In the
future, research focuses of uncertain reasoning may be centralized on the
following three aspects: first, to solve existing problems of current uncertainty
theories; second, to study the efficient and effective discrimination capabilities
and judgment mechanisms of human beings for new theories and new
methodologies to deal with uncertainties; and third, to explore methods and
technologies to synthetically process varieties of uncertainties.
Theorem proving is a kind of specific intelligent behavior of human, which
not only relies on logic deductions based on premises, but also requires certain
intuitive skills. Automated theorem proving adopts a suit of symbol systems to
formalize the process of human theorem proving into symbol calculation that can
be automatically implemented by computers, i.e., to mechanize the intelligence
process of reasoning and deduction. The mechanical theorem proving in
elementary geometry and differential geometry proposed by Professor Wenjun
Wu of Chinese Academy of Sciences is highly valued all over the world.
