**Kimberly F. Sellers and Nozer D. Singpurwalla**

The theory of coherent structures is a key element of ``complex stochastic
systems.'' Such systems are germane to the
biomedical, engineering and statistical sciences. With regard to the
biomedical sciences, the theory helps address issues
such as the failure of paired organs. For the engineering sciences, it
helps to address issues pertaining to the
robustness of the electrical power grid and the efficient design of
networks. Regarding statistical sciences, its
underpinnings facilitate a deeper appreciation of the assumptions in
graphical models, neural networks, and ``logic
regression''.

The state of the art in coherent structure theory is driven by two
assertions, whose principles were laid out over forty
years ago. The first is that all units of a system can exist in one of
two possible states: failed or functioning. The
second is that, at any given point in time, each unit can exist in only
one of the above states. Both assertions are
limiting. Units do exist in more than two states - a notion acknowledged
before; however, what
has not been recognized is that it is possible to declare that a unit can
*simultaneously* exist in more than one
state. This feature is a consequence of the view that it is not always
possible to precisely define the subsets of a set
of possible states. Such subsets are known as *vague sets*.

Regarding the first limitation, work has been done under the general label of ``multi-state systems''; however, this work has not capitalized on the mathematics of many-valued propositions developed by logicians. Here, we invoke the truth tables of many-valued logic to define the structure function of multi-state systems and then harness these results in the context of vagueness. In essence, many-valued logic provides a common platform for studying both multi-state and vague systems.

Our development, which has some interplay with philosophy and logic is a contribution to the mathematics of complex stochastic systems whose impact on science, technology and medicine continues to grow.

*Keywords:* consistency profile, fuzzy sets, likelihood function,
membership functions, reliability, probability, maintenance management,
natural language processing, machine
learning, degradation modeling, decision making, utility.

Heidi Sestrich 2003-09-09 Here is the full PDF text for this technical report. It is 751618 bytes long.