Measuring the Likelihood of Numerical Constraints

Abstract

Our goal is to measure the likelihood of the satisfaction of numerical constraints in the absence of prior information. We study expressive constraints, involving arithmetic and complex numerical functions, and even quantification over numbers. Such problems arise in processing incomplete data, or analyzing conditions in programs without a priori bounds on variables. We show that for constraints on n variables, the proper way to define such a measure is as the limit of the part of the n-dimensional ball that consists of points satisfying the constraints, when the radius increases. We prove that the existence of such a limit is closely related to the notion of o-minimality from model theory. Thus, for constraints definable with the usual arithmetic and exponentiation, the likelihood is well defined, but adding trigonometric functions is problematic. We look at computing and approximating such likelihoods for order and linear constraints, and prove an impossibility result for approximating with multiplicative error. However, as the likelihood is a number between 0 and 1, an approximation scheme with additive error is acceptable, and we give it for arbitrary linear constraints.

Queries with Arithmetic on Incomplete Databases

“…That is, the ratio Vol (R ) ∩ /Vol( ) shows the proportion of the -dimensional ball of radius occupied by vectors satisfying , and ( ) defines the asymptotic behavior of this ratio. □ Next we use a known fact, namely that ( ) exists for every formula over R. It was shown in [11], using results of [20,22], how to definably approximate volumes of sets definable in R. Combining it with previous results, we obtain the main result of this section. We finish this section by returning to the example from the introduction and computing ( , , ( )), where refers to the only market segment present in the database.…”

“…•¯is the scalar product,¯∈ R and ′ ∈ R. For such a formula , let˜denote its homogenized version, i.e., the formula obtained from by replacing each atomic formula of numerical type¯•¯< ′ by¯•¯< 0. Then we know (see [11]) that…”

“…As in Section 7, we start with (¯,¯), a database with numerical nulls, and tuples¯,¯and use Theorem 5.4 to produce a formula (¯) with free variables over R so that ( , , (¯,¯)) = ( ). If only uses linear constraints, and is in disjunctive normal form, the existence of an AFPRAS follows from results of [11]. Now, however, the query is in FO(+, •, <) and thus could be a formula over R of arbitrary shape that also uses multiplication, which complicates the search for an AFPRAS considerably.…”

Reasoning about Measures of Unmeasurable Sets

“…Our goal is to define such measures and study their structural and computational properties. This problem was first studied (Console, Hofer, and Libkin 2019) using the following approach. It first defined the measure of a set X restricted to the ball of radius r as m r (X) = Vol(X ∩ B n r )/Vol(B n r ) ∈ [0, 1], and then set m(X) = lim r→∞ m r (X).…”

“…The proof of the existence of m(X) cited above relied on a result from (Karpinski and Macintyre 1997) on approximability of volumes by firstorder formulae. The proof of (Console, Hofer, and Libkin 2019) used the result as stated in that paper, but the actual proof of (Karpinski and Macintyre 1997) added an extra assumption. It restricted sets whose volumes are approximated to subsets of [0, 1] n , rather than [−r, r] n for arbitrary r > 0, as the construction of (Console, Hofer, and Libkin 2019) crucially required.…”