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

Martine Ceberio, Vladik Kreinovich, Sanjeev Chopra, Luc Longpré, Hung T. Nguyen, Bertram Ludäscher, Chitta Baral

Research output: Contribution to journalArticle

10 Citations (Scopus)

### Abstract

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.

Original language English (US) 403-410 8 Journal of Computational and Applied Mathematics 199 2 https://doi.org/10.1016/j.cam.2005.08.030 Published - Feb 15 2007

### Fingerprint

Expert System
Expert systems
Uncertainty
Interval
Interval Computation
Estimate
NP-complete problem
Heuristics
Query
Oils
Computing
Term
Operator

### Keywords

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

### ASJC Scopus subject areas

• Applied Mathematics
• Computational Mathematics
• Numerical Analysis

### Cite this

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.

In: Journal of Computational and Applied Mathematics, Vol. 199, No. 2, 15.02.2007, p. 403-410.

Research output: Contribution to journalArticle

Ceberio, Martine ; Kreinovich, Vladik ; Chopra, Sanjeev ; Longpré, Luc ; Nguyen, Hung T. ; Ludäscher, Bertram ; Baral, Chitta. / Interval-type and affine arithmetic-type techniques for handling uncertainty in expert systems. In: Journal of Computational and Applied Mathematics. 2007 ; Vol. 199, No. 2. pp. 403-410.
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abstract = "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.",
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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.

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