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Introduction to the sheaf-theoretic approach to contextuality Samson Abramsky Department of Computer Science, University of Oxford Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic

  1. Example (0 , 0) (0 , 1) (1 , 0) (1 , 1) ( a , b ) 1 / 2 0 0 1 / 2 ( a , b ′ ) 3 / 8 1 / 8 1 / 8 3 / 8 ( a ′ , b ) 3 / 8 1 / 8 1 / 8 3 / 8 ( a ′ , b ′ ) 1 / 8 3 / 8 3 / 8 1 / 8 In this table, the set of variables is X = { a , a ′ , b , b ′ } . The measurement contexts are: {{ a 1 , b 1 } , { a 2 , b 1 } , { a 1 , b 2 } , { a 2 , b 2 }} The outcomes are O = { 0 , 1 } Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 8 / 35

  2. The 18-vector Kochen-Specker construction (Cabello et al) Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 9 / 35

  3. The 18-vector Kochen-Specker construction (Cabello et al) This uses Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 9 / 35

  4. The 18-vector Kochen-Specker construction (Cabello et al) This uses A set X of 18 variables, { A , . . . , O } Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 9 / 35

  5. The 18-vector Kochen-Specker construction (Cabello et al) This uses A set X of 18 variables, { A , . . . , O } A measurement cover U = { U 1 , . . . , U 9 } , where the columns U i are the sets in the cover: Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 9 / 35

  6. The 18-vector Kochen-Specker construction (Cabello et al) This uses A set X of 18 variables, { A , . . . , O } A measurement cover U = { U 1 , . . . , U 9 } , where the columns U i are the sets in the cover: U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 A A H H B I P P Q B E I K E K Q R R C F C G M N D F M D G J L N O J L O Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 9 / 35

  7. The 18-vector Kochen-Specker construction (Cabello et al) This uses A set X of 18 variables, { A , . . . , O } A measurement cover U = { U 1 , . . . , U 9 } , where the columns U i are the sets in the cover: U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 A A H H B I P P Q B E I K E K Q R R C F C G M N D F M D G J L N O J L O The original K-S construction used 117 variables! Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 9 / 35

  8. Basic events are local sections Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 10 / 35

  9. Basic events are local sections A basic event is to measure all the variables in a context C ∈ M , and observe the outcomes. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 10 / 35

  10. Basic events are local sections A basic event is to measure all the variables in a context C ∈ M , and observe the outcomes. This is represented by a function s : C → O , i.e. s ∈ O C , or more generally s ∈ � x ∈ C O x . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 10 / 35

  11. Basic events are local sections A basic event is to measure all the variables in a context C ∈ M , and observe the outcomes. This is represented by a function s : C → O , i.e. s ∈ O C , or more generally s ∈ � x ∈ C O x . Example: if C = { a , b } , O = { 0 , 1 } , such an outcome might be s = { a �→ 0 , b �→ 1 } Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 10 / 35

  12. Basic events are local sections A basic event is to measure all the variables in a context C ∈ M , and observe the outcomes. This is represented by a function s : C → O , i.e. s ∈ O C , or more generally s ∈ � x ∈ C O x . Example: if C = { a , b } , O = { 0 , 1 } , such an outcome might be s = { a �→ 0 , b �→ 1 } This is a local section , since it is defined only on C , not on the whole of X ! Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 10 / 35

  13. Basic events are local sections A basic event is to measure all the variables in a context C ∈ M , and observe the outcomes. This is represented by a function s : C → O , i.e. s ∈ O C , or more generally s ∈ � x ∈ C O x . Example: if C = { a , b } , O = { 0 , 1 } , such an outcome might be s = { a �→ 0 , b �→ 1 } This is a local section , since it is defined only on C , not on the whole of X ! Basic operation of restriction: if C ⊆ C ′ , s ∈ O C ′ , then s | C ∈ O C . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 10 / 35

  14. Basic events are local sections A basic event is to measure all the variables in a context C ∈ M , and observe the outcomes. This is represented by a function s : C → O , i.e. s ∈ O C , or more generally s ∈ � x ∈ C O x . Example: if C = { a , b } , O = { 0 , 1 } , such an outcome might be s = { a �→ 0 , b �→ 1 } This is a local section , since it is defined only on C , not on the whole of X ! Basic operation of restriction: if C ⊆ C ′ , s ∈ O C ′ , then s | C ∈ O C . E.g. s | { a } = { a �→ 0 } . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 10 / 35

  15. Formalizing Contextuality: Empirical models Empirical model e : ( X , M , O ): e = { e C ∈ Prob( O C ) | C ∈ M} Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 11 / 35

  16. Formalizing Contextuality: Empirical models Empirical model e : ( X , M , O ): e = { e C ∈ Prob( O C ) | C ∈ M} In other words, the empirical model specifies a probability distribution over the events in each context. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 11 / 35

  17. Formalizing Contextuality: Empirical models Empirical model e : ( X , M , O ): e = { e C ∈ Prob( O C ) | C ∈ M} In other words, the empirical model specifies a probability distribution over the events in each context. These distributions are the rows of our probability tables. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 11 / 35

  18. Formalizing Contextuality: Empirical models Empirical model e : ( X , M , O ): e = { e C ∈ Prob( O C ) | C ∈ M} In other words, the empirical model specifies a probability distribution over the events in each context. These distributions are the rows of our probability tables. Thus we have a family of probability distributions over different , but coherently related , sample spaces. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 11 / 35

  19. Formalizing Contextuality: Empirical models Empirical model e : ( X , M , O ): e = { e C ∈ Prob( O C ) | C ∈ M} In other words, the empirical model specifies a probability distribution over the events in each context. These distributions are the rows of our probability tables. Thus we have a family of probability distributions over different , but coherently related , sample spaces. (The coherent relationship is functoriality!) Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 11 / 35

  20. Restriction and Compatibility Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 12 / 35

  21. Restriction and Compatibility We would like to express the condition that an empirical model is compatible , i.e. “locally consistent”. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 12 / 35

  22. Restriction and Compatibility We would like to express the condition that an empirical model is compatible , i.e. “locally consistent”. We want to do this by saying that the distributions “agree on overlaps”. For all C , C ′ ∈ M : e C | C ∩ C ′ = e C ′ | C ∩ C ′ . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 12 / 35

  23. Restriction and Compatibility We would like to express the condition that an empirical model is compatible , i.e. “locally consistent”. We want to do this by saying that the distributions “agree on overlaps”. For all C , C ′ ∈ M : e C | C ∩ C ′ = e C ′ | C ∩ C ′ . Cf. the usual notion of compatibility of a family of functions defined on subsets. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 12 / 35

  24. Restriction and Compatibility We would like to express the condition that an empirical model is compatible , i.e. “locally consistent”. We want to do this by saying that the distributions “agree on overlaps”. For all C , C ′ ∈ M : e C | C ∩ C ′ = e C ′ | C ∩ C ′ . Cf. the usual notion of compatibility of a family of functions defined on subsets. Marginalization of distributions: if C ⊆ C ′ , d ∈ Prob( O C ′ ), � d | C ( s ) := d ( t ) t ∈ O C ′ , t | C = s Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 12 / 35

  25. Restriction and Compatibility We would like to express the condition that an empirical model is compatible , i.e. “locally consistent”. We want to do this by saying that the distributions “agree on overlaps”. For all C , C ′ ∈ M : e C | C ∩ C ′ = e C ′ | C ∩ C ′ . Cf. the usual notion of compatibility of a family of functions defined on subsets. Marginalization of distributions: if C ⊆ C ′ , d ∈ Prob( O C ′ ), � d | C ( s ) := d ( t ) t ∈ O C ′ , t | C = s Compatibility is a general form of the important physical principle of No-Signalling ; this general form is also known as No Disturbance . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 12 / 35

  26. Contextuality defined Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 13 / 35

  27. Contextuality defined An empirical model { e C } C ∈M on a measurement scenario ( X , M , O ) is non-contextual if there is a distribution d ∈ Prob( O X ) such that, for all C ∈ M : d | C = e C . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 13 / 35

  28. Contextuality defined An empirical model { e C } C ∈M on a measurement scenario ( X , M , O ) is non-contextual if there is a distribution d ∈ Prob( O X ) such that, for all C ∈ M : d | C = e C . That is, we can glue all the local information together into a global consistent description from which the local information can be recovered. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 13 / 35

  29. Contextuality defined An empirical model { e C } C ∈M on a measurement scenario ( X , M , O ) is non-contextual if there is a distribution d ∈ Prob( O X ) such that, for all C ∈ M : d | C = e C . That is, we can glue all the local information together into a global consistent description from which the local information can be recovered. We call such a d a global section . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 13 / 35

  30. Contextuality defined An empirical model { e C } C ∈M on a measurement scenario ( X , M , O ) is non-contextual if there is a distribution d ∈ Prob( O X ) such that, for all C ∈ M : d | C = e C . That is, we can glue all the local information together into a global consistent description from which the local information can be recovered. We call such a d a global section . If no such global section exists, the empirical model is contextual . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 13 / 35

  31. Contextuality defined An empirical model { e C } C ∈M on a measurement scenario ( X , M , O ) is non-contextual if there is a distribution d ∈ Prob( O X ) such that, for all C ∈ M : d | C = e C . That is, we can glue all the local information together into a global consistent description from which the local information can be recovered. We call such a d a global section . If no such global section exists, the empirical model is contextual . Thus contextuality arises where we have a family of data which is locally consistent but globally inconsistent . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 13 / 35

  32. Contextuality defined An empirical model { e C } C ∈M on a measurement scenario ( X , M , O ) is non-contextual if there is a distribution d ∈ Prob( O X ) such that, for all C ∈ M : d | C = e C . That is, we can glue all the local information together into a global consistent description from which the local information can be recovered. We call such a d a global section . If no such global section exists, the empirical model is contextual . Thus contextuality arises where we have a family of data which is locally consistent but globally inconsistent . The import of Bell’s theorem and similar results is that there are empirical models arising from quantum mechanics which are contextual. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 13 / 35

  33. Bundle Pictures Logical Contextuality Ignore precise probabilities Events are possible or not E.g. the Hardy model: 00 01 10 11 ab � � � � ab ′ × � � � a ′ b × � � � a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  34. Bundle Pictures Logical Contextuality Ignore precise probabilities Events are possible or not E.g. the Hardy model: 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  35. Bundle Pictures Logical Contextuality Ignore precise probabilities Events are possible or not E.g. the Hardy model: 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  36. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  37. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  38. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  39. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  40. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  41. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  42. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  43. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  44. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  45. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  46. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  47. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  48. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  49. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  50. Bundle Pictures Logical Contextuality • 0 • 0 Ignore precise probabilities • • Events are possible or not 0 • 1 • E.g. the Hardy model: • 1 • 1 00 01 10 11 ab � � � � b ′ • ab ′ × � � � • a ′ a ′ b × � � � • a • b a ′ b ′ � � � × Local consistency: We may extend from one context to the next Global inconsistency: Not all events extend to global valuations Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 14 / 35

  51. Strong Contextuality A B (0 , 0) (1 , 0) (0 , 1) (1 , 1) 1 0 0 1 a 1 b 1 1 0 0 1 a 1 b 2 1 0 0 1 a 2 b 1 a 2 b 2 0 1 1 0 The PR Box Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 15 / 35

  52. Bundle Pictures • 0 Strong Contextuality • 0 • E.g. the PR box: • 0 • 1 • 00 01 10 11 • 1 • 1 × × ab � � ab ′ � × × � • b ′ a ′ b × × � � • a ′ a ′ b ′ • × × a � � • b Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 16 / 35

  53. Visualizing Contextuality 0 0 • • • 0 • 0 • • • • 0 0 • 1 • 1 • • • • 1 1 • 1 • 1 b 2 b 2 • • • a 2 • a 2 a 1 • a 1 • • b 1 • b 1 The Hardy table and the PR box as bundles Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 17 / 35

  54. Comparison with the graph-theoretic CSW approach Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 18 / 35

  55. Comparison with the graph-theoretic CSW approach Deriving an orthogonality graph G e = ( V , E ) from an empirical model e V = { ( C , s ) | C ∈ M , s ∈ O C } ( C , s ) ⌢ ( C ′ , s ′ ) ⇐ ⇒ ∃ x ∈ C ∩ C ′ . s ( x ) � = s ′ ( x ) Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 18 / 35

  56. Comparison with the graph-theoretic CSW approach Deriving an orthogonality graph G e = ( V , E ) from an empirical model e V = { ( C , s ) | C ∈ M , s ∈ O C } ( C , s ) ⌢ ( C ′ , s ′ ) ⇐ ⇒ ∃ x ∈ C ∩ C ′ . s ( x ) � = s ′ ( x ) (de Silva 2016): e is strongly contextual iff the independence number of G e is less than |M| . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 18 / 35

  57. Comparison with the graph-theoretic CSW approach Deriving an orthogonality graph G e = ( V , E ) from an empirical model e V = { ( C , s ) | C ∈ M , s ∈ O C } ( C , s ) ⌢ ( C ′ , s ′ ) ⇐ ⇒ ∃ x ∈ C ∩ C ′ . s ( x ) � = s ′ ( x ) (de Silva 2016): e is strongly contextual iff the independence number of G e is less than |M| . There is more structure in an empirical model e than in G e . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 18 / 35

  58. Contextuality, Logic and Paradoxes Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 19 / 35

  59. Contextuality, Logic and Paradoxes Liar cycles . A Liar cycle of length N is a sequence of statements S 1 : S 2 is true, S 2 : S 3 is true, . . . S N − 1 : S N is true, S N : S 1 is false. For N = 1, this is the classic Liar sentence S : S is false. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 19 / 35

  60. Contextuality, Logic and Paradoxes Liar cycles . A Liar cycle of length N is a sequence of statements S 1 : S 2 is true, S 2 : S 3 is true, . . . S N − 1 : S N is true, S N : S 1 is false. For N = 1, this is the classic Liar sentence S : S is false. Following Cook, Walicki et al. we can model the situation by boolean equations: x 1 = x 2 , . . . , x n − 1 = x n , x n = ¬ x 1 Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 19 / 35

  61. Contextuality, Logic and Paradoxes Liar cycles . A Liar cycle of length N is a sequence of statements S 1 : S 2 is true, S 2 : S 3 is true, . . . S N − 1 : S N is true, S N : S 1 is false. For N = 1, this is the classic Liar sentence S : S is false. Following Cook, Walicki et al. we can model the situation by boolean equations: x 1 = x 2 , . . . , x n − 1 = x n , x n = ¬ x 1 The “paradoxical” nature of the original statements is now captured by the inconsistency of these equations. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 19 / 35

  62. Contextuality in the Liar; Liar cycles in the PR Box Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 20 / 35

  63. Contextuality in the Liar; Liar cycles in the PR Box We can regard each of these equations as fibered over the set of variables which occur in it: { x 1 , x 2 } : x 1 = x 2 { x 2 , x 3 } : x 2 = x 3 . . . { x n − 1 , x n } : x n − 1 = x n { x n , x 1 } : x n = ¬ x 1 Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 20 / 35

  64. Contextuality in the Liar; Liar cycles in the PR Box We can regard each of these equations as fibered over the set of variables which occur in it: { x 1 , x 2 } : x 1 = x 2 { x 2 , x 3 } : x 2 = x 3 . . . { x n − 1 , x n } : x n − 1 = x n { x n , x 1 } : x n = ¬ x 1 Any subset of up to n − 1 of these equations is consistent; while the whole set is inconsistent. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 20 / 35

  65. Contextuality in the Liar; Liar cycles in the PR Box We can regard each of these equations as fibered over the set of variables which occur in it: { x 1 , x 2 } : x 1 = x 2 { x 2 , x 3 } : x 2 = x 3 . . . { x n − 1 , x n } : x n − 1 = x n { x n , x 1 } : x n = ¬ x 1 Any subset of up to n − 1 of these equations is consistent; while the whole set is inconsistent. Up to rearrangement, the Liar cycle of length 4 corresponds exactly to the PR box . Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 20 / 35

  66. Contextuality in the Liar; Liar cycles in the PR Box We can regard each of these equations as fibered over the set of variables which occur in it: { x 1 , x 2 } : x 1 = x 2 { x 2 , x 3 } : x 2 = x 3 . . . { x n − 1 , x n } : x n − 1 = x n { x n , x 1 } : x n = ¬ x 1 Any subset of up to n − 1 of these equations is consistent; while the whole set is inconsistent. Up to rearrangement, the Liar cycle of length 4 corresponds exactly to the PR box . The usual reasoning to derive a contradiction from the Liar cycle corresponds precisely to the attempt to find a univocal path in the bundle diagram. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 20 / 35

  67. Paths to contradiction 0 • • 0 • • 0 • 1 • • 1 • 1 b 2 • • a 2 a 1 • • b 1 Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 21 / 35

  68. Paths to contradiction 0 • • 0 • • 0 • 1 • • 1 • 1 b 2 • • a 2 a 1 • • b 1 Suppose that we try to set a 2 to 1. Following the path on the right leads to the following local propagation of values: a 2 = 1 � b 1 = 1 � a 1 = 1 � b 2 = 1 � a 2 = 0 a 2 = 0 � b 1 = 0 � a 1 = 0 � b 2 = 0 � a 2 = 1 Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 21 / 35

  69. Paths to contradiction 0 • • 0 • • 0 • 1 • • 1 • 1 b 2 • • a 2 a 1 • • b 1 Suppose that we try to set a 2 to 1. Following the path on the right leads to the following local propagation of values: a 2 = 1 � b 1 = 1 � a 1 = 1 � b 2 = 1 � a 2 = 0 a 2 = 0 � b 1 = 0 � a 1 = 0 � b 2 = 0 � a 2 = 1 We have discussed a specific case here, but the analysis can be generalised to a large class of examples. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 21 / 35

  70. Constraint Satisfaction Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 22 / 35

  71. Constraint Satisfaction Constraint satisfaction is an important paradigm in AI, algorithms and complexity. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 22 / 35

  72. Constraint Satisfaction Constraint satisfaction is an important paradigm in AI, algorithms and complexity. A (possibilistic) empirical model is a constraint satisfaction problem! Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 22 / 35

  73. Constraint Satisfaction Constraint satisfaction is an important paradigm in AI, algorithms and complexity. A (possibilistic) empirical model is a constraint satisfaction problem! Represent e C ⊆ O C as a formula. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 22 / 35

  74. Constraint Satisfaction Constraint satisfaction is an important paradigm in AI, algorithms and complexity. A (possibilistic) empirical model is a constraint satisfaction problem! Represent e C ⊆ O C as a formula. Example : the PR Box 00 01 10 11 ab × × a ↔ b � � ab ′ a ↔ b ′ × × � � a ′ ↔ b a ′ b � × × � a ′ ⊕ b ′ a ′ b ′ × × � � Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 22 / 35

  75. Constraint Satisfaction Constraint satisfaction is an important paradigm in AI, algorithms and complexity. A (possibilistic) empirical model is a constraint satisfaction problem! Represent e C ⊆ O C as a formula. Example : the PR Box 00 01 10 11 ab × × a ↔ b � � ab ′ a ↔ b ′ × × � � a ′ ↔ b a ′ b � × × � a ′ ⊕ b ′ a ′ b ′ × × � � Local consistency is well-studied in (classical) CSP. Samson Abramsky (Department of Computer Science, University of Oxford) Introduction to the sheaf-theoretic approach to contextuality 22 / 35

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