This paper deals with the problem of safety verification of non-linear hybrid systems. We start from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid. We put this method into an abstraction refinement framework and improve it by developing an additional refinement step that employs interval constraint propagation to add information to the abstraction without introducing new grid elements. Moreover, the resulting method allows switching conditions, initial states and unsafe states to be described by complex constraints instead of sets that correspond to grid elements. Nevertheless, the method can be easily implemented since it is based on a well-defined set of constraints, on which one can run any constraint propagation based solver. Tests of such an implementation are promising.

In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance V = 1 n · n∑ (xi − E) i=1

We present the design of the Boost interval arithmetic library, a C++ library designed to efficiently handle mathematical intervals in a generic way. Interval computations are an essential tool for reliable computing. Increasingly a number of mathematical proofs have relied on global optimization problems solved using branch-andbound algorithms with interval computations; it is therefore extremely important to have a mathematically correct implementation of interval arithmetic. Various implementations exist with diverse semantics. Our design is unique in that it uses policies to specify three independent variable behaviors: rounding, checking, comparisons. As a result, with the proper policies, our interval library is able to emulate almost any of the specialized libraries available for interval arithmetic, without any loss of performance nor sacrificing the ease of use. This library is openly available at www.boost.org.

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1

We show how quantum computing can speed up computations related to processing probabilistic, interval, and fuzzy uncertainty.

Abstract. In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn). Measurements are never 100 % accurate; hence, the measured values �xi are different from xi, and the resulting estimate �y = f(�x1,..., �xn) is different from the desired value y = f(x1,..., xn). How different? Traditional engineering to error estimation in data processing assumes that we know the probabilities of different def measurement error ∆xi = �xi − xi. In many practical situations, we only know the upper bound ∆i for this error; hence, after the measurement, the only information that we have about xi is that it belongs def to the interval xi = [�xi − ∆i, �xi + ∆i]. In this case, it is important to find the range y of all possible values of y = f(x1,..., xn) when xi ∈ xi. We start with a brief overview of the corresponding interval computation problems. We then discuss what to do when, in addition to the upper bounds ∆i, we have some partial information about the probabilities of different values of ∆xi.

In the traditional interval computations approach to handling uncertainty, we assume that we know the intervals xi of possible values of different parameters xi, and we assume that an arbitrary combination of these values is possible. In geometric terms, in the traditional interval computations approach, the set of possible combinations x = (x1,..., xn) is a box x = x1 ×... × xn. In many real-life situations, in addition to knowing the intervals xi of possible values of each variable xi, we also know additional restrictions on the possible combinations of xi; in this case, the set x of possible values of x is a subset of the original box. For example, in addition to knowing the bounds on x1 and x2, we may also know that the difference between x1 and x2 cannot exceed a certain amount. Informally speaking, the parameters xi are no longer independent – in the sense that the set of possible values of xi may depend on the values of other parameters. In interval computations, we start with independent inputs; as we follow computations, we get dependent intermediate results: e.g., for x1 − x 2 1, the values of x1

In many application areas, it is important to detect outliers. The traditional engineering approach to outlier detection is that we start with some “normal ” values x1,..., xn, compute the sample average E, the sample standard deviation σ, and then mark a value x as an outlier if x is outside the k0sigma interval [E − k0 · σ, E + k0 · σ] (for some pre-selected parameter k0). In real life, we often have only interval ranges [xi, xi] for the normal values x1,..., xn. In this case, we only have intervals of possible values for the bounds L def = E − k0 · σ and U def = E + k0 · σ. We can therefore identify outliers as values that are outside all k0-sigma intervals, i.e., values which are outside the interval [L, U]. In general, the problem of computing L and U is NP-hard; a polynomial-time algorithm is known for the case when the measurements are sufficiently accurate, i.e., � when “narrowed ” intervals � 1 + α2 1 + α2

In statistical analysis of measurement results it is often necessary to compute the range [V, V] of the population variance V = 1 n · n∑ (xi − E) 2 where E = 1 n · n∑ xi when we only know the intervals i=1 [˜xi − ∆i, ˜xi + ∆i] of possible values of the xi. While V can be computed efficiently, the problem of computing V is, in general, NP-hard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm ” (Reliable Computing, 2006, Vol. 12, No. 4, pp. 273–280) we showed that in

In chip design, one of the main objectives is to decrease its clock cycle. On the design stage, this time is usually estimated by using worst-case (interval) techniques, in which we only use the bounds on the parameters that lead to delays. This analysis does not take into account that the probability of the worst-case values is usually very small; thus, the resulting estimates are over-conservative, leading to unnecessary over-design and under-performance of circuits. If we knew the exact probability distributions of the corresponding parameters, then we could use Monte-Carlo simulations (or the corresponding analytical techniques) to get the desired estimates. In practice, however, we only have partial information about the corresponding distributions, and we want to produce estimates that are valid for all distributions which are consistent with this information.