Systematic Derivation of Bounds and Glue Constraints for Time-Series Constraints

Abstract

Abstract. Integer time series are often subject to constraints on the aggregation of the integer features of all occurrences of some pattern within the series. For example, the number of inflexions may be constrained, or the sum of the peak maxima, or the minimum of the peak widths. It is currently unknown how to maintain domain consistency efficiently on such constraints. We propose parametric ways of systematically deriving glue constraints, which are a particular kind of implied constraints, as well as aggregation bounds that can be added to the decomposition of time-series constraints [5]. We evaluate the beneficial propagation impact of the derived implied constraints and bounds, both alone and together.

Deriving generic bounds for time-series constraints based on regular expressions characteristics

“…, Xn , N ) time-series constraints with every X i being over the same integer interval domain [ , u]. Table 2 provides a synthesis of the bounds on N obtained in Sections 4, 5 and [3], when g, f is in { Max, min , Max, width , Min, width , Sum, one , Sum, width }. The theorems and the propositions mentioned in Table 2 were applied for computing sharp bounds on N for 93 time-series constraints of Volume II of the global constraint catalogue [4].…”

“…When σ has the nb-simple property for [ , u], there exists a time series of length n over [ , u] whose signature contains no σ-patterns, and thus such a time series yields the default value of Max (respectively Min), which is −∞ (respectively +∞). [3]. Table 3 provides for each of the regular expressions in Table 1 the corresponding value of each regular expression characteristics.…”

“…This is an extended version of parts of the CP 2016 paper [3] which involves a subset of the original authors. This paper introduces 6 new characteristics of regular expressions to express generic bounds on time-series constraints, which were not discussed in the original paper [3].1 are regular expressions that respectively describe the regular languages Lr 1 ∪ Lr 2 , Lr 1 ∩ Lr 2 , and L * r1 .Example 1 Consider the alphabet Σ = {'<', '=', '>'}.• Decreasing = '>' is a regular expression on Σ. The word v = '>' is a word of length 1 on Σ that belongs to L Decreasing , and it does not have any proper factors.…”

“…Since we have too many time-series constraints deriving such bounds needs to be done in a systematic way. Motivated by this, we sketched in [3] a methodology to obtain such bounds and illustrated it only for time-series constraints when g = Max and f = min.…”

“…The contribution of this paper, which makes explicit the approach sketched in [3], is to introduce the notion of regular expression characteristics that provides a unified way to concisely express bounds on the result variable N of a time-series constraint. Six regular expression characteristics are introduced, which allows coming up in a compositional way with bounds when g, f ∈ { Sum, one , Max, width , Min, width , Sum, width }: five main theorems (see Theorems 1, 2, 3, 4, and 5) allow obtaining 95 bounds implemented in Volume II of the global constraint catalogue [4].…”