Least solutions of equations over N

Seidl, Helmut

doi:10.1007/3-540-58201-0_85

Cited by 10 publications

(7 citation statements)

References 9 publications

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“…Together with a suitable strategy iteration, we thus obtain an exact method for solving integer equations. This method vastly generalizes the results from [9,15] which are only applicable to systems of equations without negative numbers. Along the lines of [6], our technique for systems of integer equations provides us with a precise algorithm for interval equations.…”

Section: Introductionmentioning

confidence: 81%

“…The set of variables of the system under consideration will be denoted by X in the following. In [15] polynomial algorithms for computing least solutions are presented for similar systems -but only when computing least solutions over nonnegative integers. In [6], also negative integers are allowed.…”

Section: Notation and Basic Conceptsmentioning

confidence: 99%

“…Since the linear ordering of integers has infinite ascending chains, ordinary fixpoint iteration will not result in terminating algorithms. For the lucky case where all numbers are non-negative, polynomial fixpoint algorithms are provided in [15]. In presence of general minima as well as negative numbers, no practical precise methods have been suggested so far.…”

Section: Introductionmentioning

confidence: 99%

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Precise Fixpoint Computation Through Strategy Iteration

Gawlitza¹,

Seidl²

2007

Programming Languages and Systems

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Abstract. We present a practical algorithm for computing least solutions of systems of equations over the integers with addition, multiplication with positive constants, maximum and minimum. The algorithm is based on strategy iteration. Its run-time (w.r.t. the uniform cost measure) is independent of the sizes of occurring numbers. We apply our technique to solve systems of interval equations. In particular, we show how arbitrary intersections as well as full interval multiplication in interval equations can be dealt with precisely.

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Section: Introductionmentioning

confidence: 81%

Section: Notation and Basic Conceptsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Precise Fixpoint Computation Through Strategy Iteration

Gawlitza¹,

Seidl²

2007

Programming Languages and Systems

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“…Their testbed is based on Emulab [45] as well, with a real-time connection simulator Simulink [26]. Seidl designend a Python [15]-environment that simulates user-defined industrial behaviour called VirtuaPlant [41]. This simulation can then be introduced to attacks and malware.…”

Section: Related Workmentioning

confidence: 99%

Evaluation of Machine Learning-based Anomaly Detection Algorithms on an Industrial Modbus/TCP Data Set

Antón

Kanoor

Fraunholz

et al. 2018

Proceedings of the 13th International Conference on Availability, Reliability and Security

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In the context of the Industrial Internet of Things, communication technology, originally used in home and office environments, is introduced into industrial applications. Commercial off-the-shelf products, as well as unified and well-established communication protocols make this technology easy to integrate and use. Furthermore, productivity is increased in comparison to classic industrial control by making systems easier to manage, set up and configure. Unfortunately, most attack surfaces of home and office environments are introduced into industrial applications as well, which usually have very few security mechanisms in place. Over the last years, several technologies tackling that issue have been researched. In this work, machine learning-based anomaly detection algorithms are employed to find malicious traffic in a synthetically generated data set of Modbus/TCP communication of a fictitious industrial scenario. The applied algorithms are Support Vector Machine (SVM), Random Forest, k-nearest neighbour and k-means clustering. Due to the synthetic data set, supervised learning is possible. Support Vector Machine and k-nearest neighbour perform well with different data sets, while k-nearest neighbour and k-means clustering do not perform satisfactorily.

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“…Note that we excluded general multiplication since multiplication with negative numbers is no longer monotonic. Similar systems of equations have been investigated in [10] where polynomial algorithms for computing least upper bounds are presented -but only when computing least solutions over nonnegative integers. A function µ : X → Z is called a variable assignment.…”

Section: Notation and Basic Conceptsmentioning

confidence: 99%

Polynomial Precise Interval Analysis Revisited

Gawlitza¹,

Leroux

Reineke

et al. 2009

Lecture Notes in Computer Science

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Abstract. We consider a class of arithmetic equations over the complete lattice of integers (extended with −∞ and ∞) and provide a polynomial time algorithm for computing least solutions. For systems of equations with addition and least upper bounds, this algorithm is a smooth generalization of the Bellman-Ford algorithm for computing the single source shortest path in presence of positive and negative edge weights. The method then is extended to deal with more general forms of operations as well as minima with constants. For the latter, a controlled widening is applied at loops where unbounded increase occurs. We apply this algorithm to construct a cubic time algorithm for the class of interval equations using least upper bounds, addition, intersection with constant intervals as well as multiplication.

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Least solutions of equations over N

Cited by 10 publications

References 9 publications

Precise Fixpoint Computation Through Strategy Iteration

Precise Fixpoint Computation Through Strategy Iteration

Evaluation of Machine Learning-based Anomaly Detection Algorithms on an Industrial Modbus/TCP Data Set

Polynomial Precise Interval Analysis Revisited

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