### Abstract

Expert knowledge consists of statements S_{j} (facts and rules). The facts and rules are often only true with some probability. For example, if we are interested in oil, we should look at seismic data. If in 90% of the cases, the seismic data were indeed helpful in locating oil, then we can say that if we are interested in oil, then with probability 90% it is helpful to look at the seismic data. In more formal terms, we can say that the implication "if oil then seismic" holds with probability 90%. Another example: a bank A trusts a client B, so if we trust the bank A, we should trust B too; if statistically this trust was justified in 99% of the cases, we can conclude that the corresponding implication holds with probability 99%. If a query Q is deducible from facts and rules, what is the resulting probability p (Q) in Q? We can describe the truth of Q as a propositional formula F in terms of S_{j}, i.e., as a combination of statements S_{j} linked by operators like &, ∨, and ¬; 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 ∨ ¬ A) is not 1. Solution: similar to affine arithmetic, besides P (F_{j}), we also compute P (F_{j} & F_{i}) (or P (F_{j1} & & F_{jd})), and on each step, use all combinations of l such probabilities to get new estimates. Results: e.g., P (A ∨ ¬ A) is estimated as 1.

Original language | English (US) |
---|---|

Pages (from-to) | 403-410 |

Number of pages | 8 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 199 |

Issue number | 2 |

DOIs | |

State | Published - Feb 15 2007 |

### Fingerprint

### Keywords

- Affine arithmetic
- Computer security
- Expert systems
- Fuzzy logic
- Geoinformatics
- Interval computations

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Numerical Analysis

### Cite this

*Journal of Computational and Applied Mathematics*,

*199*(2), 403-410. https://doi.org/10.1016/j.cam.2005.08.030

**Interval-type and affine arithmetic-type techniques for handling uncertainty in expert systems.** / Ceberio, Martine; Kreinovich, Vladik; Chopra, Sanjeev; Longpré, Luc; Nguyen, Hung T.; Ludäscher, Bertram; Baral, Chitta.

Research output: Contribution to journal › Article

*Journal of Computational and Applied Mathematics*, vol. 199, no. 2, pp. 403-410. https://doi.org/10.1016/j.cam.2005.08.030

}

TY - JOUR

T1 - Interval-type and affine arithmetic-type techniques for handling uncertainty in expert systems

AU - Ceberio, Martine

AU - Kreinovich, Vladik

AU - Chopra, Sanjeev

AU - Longpré, Luc

AU - Nguyen, Hung T.

AU - Ludäscher, Bertram

AU - Baral, Chitta

PY - 2007/2/15

Y1 - 2007/2/15

N2 - Expert knowledge consists of statements Sj (facts and rules). The facts and rules are often only true with some probability. For example, if we are interested in oil, we should look at seismic data. If in 90% of the cases, the seismic data were indeed helpful in locating oil, then we can say that if we are interested in oil, then with probability 90% it is helpful to look at the seismic data. In more formal terms, we can say that the implication "if oil then seismic" holds with probability 90%. Another example: a bank A trusts a client B, so if we trust the bank A, we should trust B too; if statistically this trust was justified in 99% of the cases, we can conclude that the corresponding implication holds with probability 99%. If a query Q is deducible from facts and rules, what is the resulting probability p (Q) in Q? We can describe the truth of Q as a propositional formula F in terms of Sj, i.e., as a combination of statements Sj linked by operators like &, ∨, and ¬; 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 Fj; hence intervals are too wide. Example: the estimate for P (A ∨ ¬ A) is not 1. Solution: similar to affine arithmetic, besides P (Fj), we also compute P (Fj & Fi) (or P (Fj1 & & Fjd)), and on each step, use all combinations of l such probabilities to get new estimates. Results: e.g., P (A ∨ ¬ A) is estimated as 1.

AB - Expert knowledge consists of statements Sj (facts and rules). The facts and rules are often only true with some probability. For example, if we are interested in oil, we should look at seismic data. If in 90% of the cases, the seismic data were indeed helpful in locating oil, then we can say that if we are interested in oil, then with probability 90% it is helpful to look at the seismic data. In more formal terms, we can say that the implication "if oil then seismic" holds with probability 90%. Another example: a bank A trusts a client B, so if we trust the bank A, we should trust B too; if statistically this trust was justified in 99% of the cases, we can conclude that the corresponding implication holds with probability 99%. If a query Q is deducible from facts and rules, what is the resulting probability p (Q) in Q? We can describe the truth of Q as a propositional formula F in terms of Sj, i.e., as a combination of statements Sj linked by operators like &, ∨, and ¬; 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 Fj; hence intervals are too wide. Example: the estimate for P (A ∨ ¬ A) is not 1. Solution: similar to affine arithmetic, besides P (Fj), we also compute P (Fj & Fi) (or P (Fj1 & & Fjd)), and on each step, use all combinations of l such probabilities to get new estimates. Results: e.g., P (A ∨ ¬ A) is estimated as 1.

KW - Affine arithmetic

KW - Computer security

KW - Expert systems

KW - Fuzzy logic

KW - Geoinformatics

KW - Interval computations

UR - http://www.scopus.com/inward/record.url?scp=33750198108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750198108&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2005.08.030

DO - 10.1016/j.cam.2005.08.030

M3 - Article

AN - SCOPUS:33750198108

VL - 199

SP - 403

EP - 410

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 2

ER -