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

AbstractExpert 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

Efficient Geophysical Technique of Vertical Line Elements as a Natural Consequence of General Constraints Techniques

One of the main objectives of geophysics is to find how density d and other physical characteristics depend on a 3-D location (x,y,z). In general, in numerical methods, a way to find the dependence d(x,y,z) is to discretize the space, and to consider, as unknown, e.g., values d(x,y,z) on a 3-D rectangular grid. In this case, the desired density distribution is represented as a combination of point-wise density distributions. In geophysics, it turns out that a more efficient way to find the desired distribution is to represent it as a combination of thin vertical line elements that start at some depth and go indefinitely down. In this paper, we show that the empirical success of such vertical line element techniques can be naturally explained if we recall that, in addition to the equations which relate the observations and the unknown density, we also take into account geophysics-motivated constraints

Model-Order Reduction Using Interval Constraint Solving Techniques

Many natural phenomena can be modeled as ordinary or partial differential equations. A way to find solutions of such equations is to discretize them and to solve the corresponding (possibly) nonlinear large systems of equations. Solving a large nonlinear system of equations is very computationally complex due to several numerical issues, such as high linear-algebra cost and large memory requirements. Model-Order Reduction (MOR) has been proposed as a way to overcome the issues associated with large dimensions, the most used approach for doing so being Proper Orthogonal Decomposition (POD). The key idea of POD is to reduce a large number of interdependent variables (snapshots) of the system to a much smaller number of uncorrelated variables while retaining as much as possible of the variation in the original variables. In this work, we show how intervals and constraint solving techniques can be used to compute all the snapshots at once (I-POD). This new process gives us two advantages over the traditional POD method: 1. handling uncertainty in some parameters or inputs; 2. reducing the snapshots computational cost

Continuous If-Then Statements Are Computable

Abstract. In many practical situations, we must compute the value of an if-then expression f(x) defined as “if c(x) ≥ 0 then f+(x) else f−(x)”, where f+(x), f−(x), and c(x) are computable functions. The value f(x) cannot be computed directly, since in general, it is not possible to check whether a given real number c(x) is non-negative or non-positive. Similarly, it is not possible to compute the value f(x) if the if-then function is discontinuous, i.e., when f+(x0) ̸ = f−(x0) for some x0 for which c(x0) = 0. In this paper, we show that if the if-then expression is continuous, then we can effectively compute f(x). Practical need for if-then statements. In many practical situations, we have different models for describing a phenomenon: – a model f+(x) corresponding to the case when a certain constraint c(x) ≥ 0 is satisfied, and – a model f−(x) corresponding to the case when this constraint is not satisfied