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=== Sheaf-theoretic framework ===
The [[Sheaf (mathematics)|sheaf]]-theoretic, or Abramsky–Brandenburger, approach to contextuality initiated by [[Samson Abramsky]] and [[Adam Brandenburger]] is theory-independent and can be applied beyond quantum theory to any situation in which empirical data arises in contexts. As well as being used to study forms of contextuality arising in quantum theory and other physical theories, it has also been used to study formally equivalent phenomena in [[logic]],<ref name=":7">{{Cite journal |last1=Abramsky |first1=Samson |last2=Soares Barbosa |first2=Rui |last3=Kishida |first3=Kohei |last4=Lal |first4=Raymond |last5=Mansfield |first5=Shane |date=2015 |title=Contextuality, Cohomology and Paradox |journal=Schloss Dagstuhl - |publisher=Leibniz-Zentrum für Informatik GMBH, Wadern/Saarbruecken, Germany |volume=41 |pages=211–228 |doi=10.4230/lipics.csl.2015.211 |isbn=9783939897903 |series=Leibniz International Proceedings in Informatics (LIPIcs) |doi-access=free |bibcode=2015arXiv150203097A |arxiv=1502.03097 |s2cid=2150387}}</ref> [[relational database]]s,<ref>{{Citation |last=Abramsky |first=Samson |title=Relational Databases and Bell's Theorem |volume=8000 |date=2013 |work=In Search of Elegance in the Theory and Practice of Computation: Essays Dedicated to Peter Buneman |pages=13–35 |editor-last=Tannen |editor-first=Val |series=Lecture Notes in Computer Science |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-41660-6_2 |isbn=9783642416606 |s2cid=18824713 |editor2-last=Wong |editor2-first=Limsoon |editor3-last=Libkin |editor3-first=Leonid |editor4-last=Fan |editor4-first=Wenfei |arxiv=1208.6416}}.</ref> [[natural language processing]],<ref>{{Citation |last1=Abramsky |first1=Samson |title=Semantic Unification |date=2014 |work=Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday |pages=1–13 |editor-last=Casadio |editor-first=Claudia |series=Lecture Notes in Computer Science |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-54789-8_1 |isbn=9783642547898 |last2=Sadrzadeh |first2=Mehrnoosh |editor2-last=Coecke |editor2-first=Bob |editor3-last=Moortgat |editor3-first=Michael |editor4-last=Scott |editor4-first=Philip |arxiv=1403.3351 |s2cid=462058}}.</ref> and [[constraint satisfaction]].<ref>{{Cite book |doi=10.1109/LICS.2017.8005129 |isbn=9781509030187 |arxiv=1704.05124 |chapter=The pebbling comonad in Finite Model Theory |title=2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) |pages=1–12 |year=2017 |last1=Abramsky |first1=Samson |last2=Dawar |first2=Anuj |last3=Wang |first3=Pengming |s2cid=11767737}}</ref>
In essence, contextuality arises when empirical data is ''locally consistent but globally inconsistent''.
This framework gives rise in a natural way to a qualitative hierarchy of contextuality.:
* '''(Probabilistic) contextuality''' may be witnessed in measurement statistics, e.g. by the violation of an inequality. A representative example is the [[KCBS pentagram|KCBS]] proof of contextuality.
* '''(Probabilistic)Logical contextuality''' may be witnessed in measurementthe statistics,"possibilistic" e.g.information byabout thewhich violationoutcome ofevents anare inequalitypossible and which are not possible. A representative example is the [[KCBSHardy's pentagramparadox|KCBSHardy's nonlocality]] proof of contextualitynonlocality.
* '''LogicalStrong contextuality''' mayis a maximal form of contextuality. Whereas (probabilistic) contextuality arises when measurement statistics cannot be witnessedreproduced inby thea 'possibilistic'mixture informationof aboutglobal whichvalue outcomeassignments, eventsstrong arecontextuality possiblearises andwhen whichno areglobal notvalue assignment is even compatible with the possible outcome events. A representative example is [[Hardy'sthe paradox|Hardy'soriginal nonlocality]]Kochen–Specker proof of nonlocalitycontextuality.
*'''Strong contextuality''' is a maximal form of contextuality. Whereas (probabilistic) contextuality arises when measurement statistics cannot be reproduced by a mixture of global value assignments, strong contextuality arises when no global value assignment is even compatible with the possible outcome events. A representative example is the original Kochen–Specker proof of contextuality.
Each level in this hierarchy strictly includes the next. An important intermediate level that lies strictly between the logical and strong contextuality classes is '''all-versus-nothing contextuality''',<ref name=":7" /> a representative example of which is the [[Greenberger–Horne–Zeilinger state|Greenberger–Horne–Zeilinger]] proof of nonlocality.
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