@mastersthesis{cho-05-aa-thesis,
  title = {Affine Arithmetic-Type Techniques for Handling Uncertainty in Expert Systems},
  author = {Chopra, Sanjeev},
  school = {University of Texas at El Paso},
  year = 2005,
  month = dec,
  advisor = {Vladik Kreinovich},
  comment = {See~\cite{ceb-kre-cho-lon-ngu-lud-bar-07-aa-expert}},
  abstract = {Expert knowledge consists of statements $S_j$: facts and rules. The expert's degree of confidence in each statement $S_j$ can be described as a (subjective) probability. For example, if we are interested in oil; we should look at seismic data (confidence 90{\%}); a bank $A$ trusts a client $B$, so if we trust $A$, we should trust $B$ too (confidence 99{\%}). If a query $Q$ is deducible from facts and rules, what is our confidence $P(Q)$ in $Q$?  We can describe $Q$ as a propositional formula $F$ in terms of $S_j$; computing $P(Q)$ exactly is NP-hard, so heuristics are needed. Traditionally, expert systems use technique similar to straightforward interval computations: we parse $F$ and replace each computation step with corresponding probability operation. The problem with this approach is that at each step, we ignore the dependence between the intermediate results $F_j$; hence intervals are too wide. For example, the estimate for $P(A\vee \neg A)$ is not 1.  In this thesis, we propose a new solution to this problem; similarly to affine arithmetic, besides $P(F_j)$, we also compute $P(F_j \wedge F_i)$ (or $P(F_{j1} \wedge\cdots\wedge F_{jk})$), and on each step, use all combinations of $\ell$ such probabilities to get new estimates. As a result, for the above stated e.g., $P(A\vee \neg A)$ is estimated as 1.}
}