A Sheaf Model of Contradictions and Disagreements. A (very) Preliminary Report

We introduce a new formal model – based on the mathematical construct of sheaves – for representing contradictory information in textual sources. This model has the advantage of letting us (a) identify the causes of the inconsistency; (b) measure how strong it is; (c) and do something about it, e.g. suggest ways to reconcile inconsistent advice. This model naturally represents the distinction between contradictions and disagreements. It is based on the idea of representing natural language sentences as formulas with parameters sitting on lattices, creating partial orders based on predicates shared by theories, and building sheaves on these partial orders with products of lattices as stalks. Degrees of disagreement are measured by the existence of global and local sections. Limitations of the sheaf approach and connections to recent work in natural language processing, as well as the topics of contextuality in physics, data fusion, topological data analysis and epistemology are also discussed.

[1]  Jennifer Chu-Carroll,et al.  Textual resource acquisition and engineering , 2012, IBM J. Res. Dev..

[2]  Luciana Garbayo,et al.  Epistemic Considerations on Expert Disagreement, Normative Justification, and Inconsistency Regarding Multi-criteria Decision Making , 2014, Constraint Programming and Decision Making.

[3]  Samson Abramsky,et al.  The sheaf-theoretic structure of non-locality and contextuality , 2011, 1102.0264.

[4]  Wlodek Zadrozny,et al.  Semantics of Paragraphs , 1991, Comput. Linguistics.

[5]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[6]  V. S. Subrahmanian,et al.  A General Framework for Reasoning On Inconsistency , 2013, SpringerBriefs in Computer Science.

[7]  Jerome R. Busemeyer,et al.  Quantum Models of Cognition and Decision , 2012 .

[8]  Sanja Fidler,et al.  MovieQA: Understanding Stories in Movies through Question-Answering , 2015, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[9]  Zoran Budimac,et al.  An overview of ontologies and data resources in medical domains , 2014, Expert Syst. Appl..

[10]  Michael Robinson,et al.  Sheaves are the canonical data structure for sensor integration , 2017, Inf. Fusion.

[11]  John Grant,et al.  Analysing inconsistent first-order knowledgebases , 2008, Artif. Intell..

[12]  Kohei Kishida.  Logic of Local Inference for Contextuality in Quantum Physics and Beyond , 2016, ICALP.

[13]  Michael Robinson,et al.  Sheaf and Duality Methods for Analyzing Multi-Model Systems , 2016, 1604.04647.

[14]  Richard Fikes,et al.  A Reusable Ontology for Fluents in OWL , 2006, FOIS.

[15]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[16]  Christopher D. Manning,et al.  Finding Contradictions in Text , 2008, ACL.

[17]  Wlodek Zadrozny,et al.  Towards Semantic Modeling of Contradictions and Disagreements: A Case Study of Medical Guidelines , 2017, IWCS.

[18]  Valeria C V de Paiva,et al.  Correcting Contradictions , 2017 .

[20]  Frédéric Chazal,et al.  Subsampling Methods for Persistent Homology , 2014, ICML.

[21]  Jennifer Chu-Carroll,et al.  Statistical source expansion for question answering , 2011, CIKM '11.

[22]  John Grant,et al.  Analysing inconsistent information using distance-based measures , 2017, Int. J. Approx. Reason..

[23]  Cliff Joslyn,et al.  Towards A Topological Framework for Integrating Semantic Information Sources , 2014, STIDS.

[24]  Diederik Aerts,et al.  Quantum Structure in Cognition , 2008, 0805.3850.

[25]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[26]  Samuel R. Bowman,et al.  A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference , 2017, NAACL.

[27]  Xiaojin Zhu,et al.  Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing , 2013, IJCAI.

[28]  Tim Fernando,et al.  Incremental semantic scales by strings , 2014, EACL 2014.

[29]  Simon Cox,et al.  Time Ontology in OWL , 2017 .

[30]  Kohei Kishida,et al.  Contextuality, Cohomology and Paradox , 2015, CSL.

[31]  Mathieu d'Aquin,et al.  Where to publish and find ontologies? A survey of ontology libraries , 2012, J. Web Semant..

[32]  Michael Robinson,et al.  Topological Signal Processing , 2014 .

[33]  Peter Bubenik,et al.  Statistical topological data analysis using persistence landscapes , 2012, J. Mach. Learn. Res..

[34]  David Christensen,et al.  Putting logic in its place - formal constraints on rational belief , 2005 .

[35]  Joel A. Henderson,et al.  On Modeling Multi-experts Multi-criteria Decision-Making Argumentation and Disagreement: Philosophical and Computational Approaches Reconsidered , 2018 .