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

Abstract

We introduce the concept of regular expression characteristics as a unified way to concisely express bounds on time-series constraints. This allows us not only to define time-series constraints in a compositional way, but also to deal with their combinatorial aspect in a compositional way, without developing ad-hoc bounds for each time-series constraint separately. IntroductionA time series is here a sequence of integers, corresponding to measurements taken over a time interval. Time series are common in many application areas, such as the output of electric power stations over multiple days [9], or the manpower required in a call-centre [5], or the daily capacity of a hospital clinic over a period of years. Time series are constrained by physical or organisational limits, which restrict the evolution of the series.We showed in [6] that many constraints γ( X 1 , X 2 , . . . , Xn , N ) on an unknown time series X = X 1 , X 2 , . . . , Xn can be specified in a compositional way by a triple σ, f, g , where σ is a regular expression over the alphabet Σ = {'<', '=', '>'} (we assume the reader is familiar with regular expressions [13]), while f ∈ {max, min, one, surf, width} is called a feature function, and g ∈ {Max, Min, Sum} is called an aggregator function. Volume II of the global constraint catalogue [4] contains 266 such functional time-series constraints. 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. The word '>>' is a word of length 2 on Σ, which does not belong to L Decreasing .• Inflexion = '<(<|=)*> | >(>|=)*<' is a regular expression on Σ. The word v = '>=<' is a word of length 3 on Σ that belongs to L Inflexion . The word υ has multiple proper factors, e.g., '>', '<'. The word '>=<<' does not belong to L Inflexion .Definition 2 A regular expression r is a non-fixed length regular expression if not all words of Lr have the same length.Example 2 We give two examples of regular expressions, a first one with a fixed length and a second one with a non-fixed length.• The Decreasing = '>' regular expression has a fixed length since L Decreasing contains a single word.• The Inflexion = '<(<|=)*> | >(>|=)*<' regular expression does not have a fixed length since L Inflexion contains words of different length.Definition 3 A regular expression over an alphabet A is disjunction-capsuled if it is in the form of 'r 1 r 2 . . . rp', where every r i (with i ∈ [1, p]) is, either a letter of the alphabet A, or a regular expression whose regular language contains the ...

Synthesising a Database of Parameterised Linear and Non-Linear Invariants for Time-Series Constraints

“…For a regular expression σ, its language [19] is denoted by Lσ. The size [5] of a regular expression σ, denoted by ωσ, is the number of letters in the shortest word of Lσ.…”

“…where c is a natural number, and up Ri (n) is the maximum value of R i among all time series of length n [5]. Such invariants are required when the set of feasible combinations of R 1 and R 2 is non-convex and therefore linear invariants are not enough for fully describing it.…”

“…. , Xn with every X i ranging over [0, 3], four among [9] constraints that restrict the variables R 0 , R 1 , R 2 , R 3 to be the number of occurrences of values 0, 1, 2, 3, respectively, in X, as well as the four corresponding stretch [31] constraints restricting the stretch length in X to be respectively in [1,4], [2,5], [3,5], and [1,2]. In addition assume that value 2 (resp.…”

“…-(iv) and (v) are related to the fact that infeasible combinations of R 1 and R 2 can be located on a ray or an interval. -(vi) and (vii) are respectively linked to the fact that quite often infeasible combinations of R 1 and R 2 within the convex hull are very close to the minimum or the maximum values [5] of R k (with k ∈ [1, 2]), i.e. c is a very small constant, typically 0 or 1.…”

“…, Xn , R) timeseries constraint and a gap atomic relation C defined by R = up R (n). We showed in [5] that the maximum value of R for a given time-series length n is max 0, n−1…”