@techreport{ceb-kre-cho-lon-ngu-lud-bar-05-aa-compsec-tr,
  title = {Interval-Type and Affine Arithmetic-Type Techniques for Handling Uncertainty in Expert Systems, with Applications to Geoinformatics and Computer Security},
  author = {Martine C. Ceberio and Vladik Kreinovich and Sanjeev Chopra and Luc {Longpr{\'e}} and Hung T. Nguyen and Bertram Lud{\"a}scher and Chitta Baral},
  year = 2005,
  month = aug,
  institution = {University of Texas at El Paso},
  number = {UTEP-CS-04-37b},
  note = {Short version published in the Proceedings of the 17th World Congress of the International Association for Mathematics and Computers in Simulation IMACS'2005, Paris, France, July 11-15, 2005; full paper~\cite{ceb-kre-cho-lon-ngu-lud-bar-07-aa-expert}. Updated version: UTEP-CS-04-37a, Original file: UTEP-CS-04-37.},
  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 (some probabilities are known to be independent). Examples: 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. Problem: at each step, we ignore the dependence between the intermediate results $F_j$; hence intervals are too wide. Example: the estimate for $P(A\vee \neg A)$ is not 1. Solution: similarly to affine arithmetic, besides $P(F_j)$, we also compute $P(F_j \wedge F_i)$ (or $P(F_j \wedge ... \wedge F_k)$), and on each step, use all combinations of $l$ such probabilities to get new estimates. Results: e.g., $P(A\vee \neg A)$ is estimated as 1.},
  altkeys = {ceb-kre-cho-lon-ngu-lud-bar-05-compsec-tr},
}