Approximate Comparison of Functions Computed by Distance Automata

Colcombet, Thomas; Daviaud, Laure

doi:10.1007/s00224-015-9643-3

Cited by 8 publications

(18 citation statements)

References 11 publications

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“…Each path can be described through a word w, its cost being the sum, in conventional algebra, of the transition costs. The cost of the path is the minimal cost of all paths described with the same word w. A (min, +) automaton can also be referred in the literature as a distance automaton (Colcombet and Daviaud 2013). A (max, +) automaton is the dual representation with transition weights related to a gain.…”

Section: (Min +) and (Max +) Automatamentioning

confidence: 99%

Towards a tropical automaton product minimizing global completion times

et al. 2020

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The main contribution of this paper is oriented towards system modeling with resource conflicts, in which a set of tasks must be completed at the earliest. Mutual exclusion between tasks with underlying resource conflicts is modeled as subsystems through local tropical automata. Parallel task execution is explored through a new framework for a synchronous product of tropical automata directly minimizing the global completion time of a set of tasks. A dater-based analysis is proposed to determine an optimal qualitative schedule and relies on proposed definitions of global, private, and synchronizing daters. This approach drastically narrows the solution spectrum, and allows finding the optimal task schedule. The approach is applied to a flow network as a case study, but can be applied to systems of different nature.

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Section: (Min +) and (Max +) Automatamentioning

confidence: 99%

Towards a tropical automaton product minimizing global completion times

et al. 2020

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“…We consider an example of intervals without restart for the σ = DecreasingTerrace regular expression. For the time series X = 4, 3, 3, 2, 2, 1, 4, 2, 2, 1 , the intervals [1,6] and [7,10] are intervals without restart corresponding to the subseries t 1 = 4, 3, 3, 2, 2, 1 and t 2 = 4, 2, 2, 1 , because: * Each X i (with i ∈ [1,6] or i ∈ [7,10]) belongs to at least one extended σ-pattern (Condition (1) of Definition 17). * The subseries t 1 (respectively t 2 ) contains 2 (respectively 1) extended σ-patterns of shortest length 4 [Proof of (i)] We construct a time series t that we prove to have the maximum number of σ-patterns among all ground time series of length n without considering any domain restrictions.…”

Section: Example 15mentioning

confidence: 99%

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

2017

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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 ...

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“…The main induction: the forest factorization theorem of Simon The forest factorization theorem of Simon [8] is a powerful combinatorial tool for understanding the structure of finite semigroups. In this short abstract, we will not describe the original statement of this theorem, in terms of trees of factorizations, but rather a direct consequence of it which is central in our proof (the presentation of the theorem was used in a similar way in [2]). …”

Section: Presentable Sets Of Weighted Matricesmentioning

confidence: 99%