{"type":"ScholarlyArticle","title":"Measuring the Likelihood of Numerical Constraints","authors":[{"name":"Marco Console","slug":"marco-console","given":"Marco","family":"Console","affiliation":"School of Informatics , University of Edinburgh"},{"name":"Matthias Hofer","slug":"matthias-hofer","given":"Matthias","family":"Hofer","affiliation":"School of Informatics , University of Edinburgh"},{"name":"Leonid Libkin","slug":"leonid-libkin","given":"Leonid","family":"Libkin","affiliation":"School of Informatics , University of Edinburgh"}],"abstract":"Our goal is to measure the likelihood of the satisfaction of numerical\r\nconstraints in the absence of prior information.  We study expressive\r\nconstraints, involving arithmetic and complex numerical functions, and\r\neven quantification over numbers. Such problems arise in processing\r\nincomplete data, or analyzing conditions in programs without a priori\r\nbounds on variables.  We show that for constraints on n variables,\r\nthe proper way to define such a measure is as the limit of the part of\r\nthe n-dimensional ball that consists of points satisfying the\r\nconstraints, when the radius increases. We prove that the existence of\r\nsuch a limit is closely related to the notion of o-minimality from\r\nmodel theory. Thus, for constraints definable with the usual\r\narithmetic and exponentiation, the likelihood is well defined, but\r\nadding trigonometric functions is problematic. We look at computing\r\nand approximating such likelihoods for order and linear\r\nconstraints, and prove an impossibility result for approximating with\r\nmultiplicative error. However, as the\r\nlikelihood is a number between 0 and 1, an approximation scheme with\r\nadditive error is acceptable, and we give it for arbitrary\r\nlinear constraints.","doi":"10.24963/ijcai.2019/229","url":"https://ijcai.org/proceedings/2019/229","pdf":"https://ijcai.org/proceedings/2019/0229.pdf","markdown":"https://ijcai.org/proceedings/2019/229.md","bibtex":"https://ijcai.org/proceedings/2019/bibtex/229","volume":{"slug":"2019","title":"Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2019","url":"https://ijcai.org/proceedings/2019/","year":2019,"location":"Macao","dates":"10-16 August 2019","editors":"Sarit Kraus","isbn":"978-0-9992411-4-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2019"},"section":"Knowledge Representation and Reasoning","track":"Main track","number":229,"pages":{"first":1654,"last":1660},"keywords":["Constraints and SAT: Constraints: Evaluation and Analysis","Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning","Knowledge Representation and Reasoning: Computational Complexity of Reasoning","Constraints and SAT: Constraints: Applications"],"videos":[],"references":[{"position":1,"raw":"[Arora and Barak, 2009] Sanjeev Arora and Boaz Barak. 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