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Some Thoughts on LFCS at 30 Samson Abramsky Department of Computer Science, University of Oxford Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 1 / 20 Sense And/Or Sensibility? Samson

  1. My first Close Encounter Working at GEC Computers, 1977, on operating system development. Came across two orange booklets: George Milne and Robin Milner, Concurrent Processes and their Syntax Robin Milner, Flowgraphs and Flow Algebras . This was the opening quotation from Flowgraphs and Flow Algebras: ”You could get an infinite number of wires in this . . . junction box, but we don’t usually go that far in practice” – Man from London Electricity Board, 1959. It presents an algebra of nets, and proves its completeness under a simple equational theory, using the methods of initial algebra semantics (ADJ). There are many pictures of nets: precursor of many subsequent diagrammatic calculi. Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 4 / 20

  2. Concurrent Processes and their Syntax Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 5 / 20

  3. Concurrent Processes and their Syntax Concurrency among computations which communicate with each other presents some of the most exciting challenges in the study of computation. Not only is it of great practical relevance, but it also appears to demand a theory of its own, which at present exists only tentatively. One aspect of this theory must be quantitative . . . . Another (perhaps the other) aspect is qualitative; what conceptual framework is needed to understand and discuss the behavior of concurrent computations? Although these two aspects should presumably interact, it appears to be sound methodology to separate them initially. This paper deals with the qualitative aspect. We attempt to provide a part of the necessary conceptual framework by attacking the question: What sort of mathematical object is a process? Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 5 / 20

  4. The Cunning of Reason I Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  5. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  6. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  7. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  8. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Probability Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  9. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Probability Weighted systems of various kinds Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  10. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Probability Weighted systems of various kinds Real-time Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  11. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Probability Weighted systems of various kinds Real-time Quantum Information Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  12. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Probability Weighted systems of various kinds Real-time Quantum Information Computational Systems Biology Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  13. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Probability Weighted systems of various kinds Real-time Quantum Information Computational Systems Biology Computational Game Theory and Economics Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  14. The Cunning of Reason I What is a fruitful guiding principle at a certain stage in the development of a scientific field does not always stay that way! This separation of concerns, precursor of Track A vs. Track B Eurotheory vs. US theory was an important and necessary step in its time. As was foreseen in the quotation, these two strands, quantitive and qualitative, must eventually interact, and are increasingly doing so: Probability Weighted systems of various kinds Real-time Quantum Information Computational Systems Biology Computational Game Theory and Economics and more . . . Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 6 / 20

  15. The Cunning of Reason II Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 7 / 20

  16. The Cunning of Reason II Another important feature, part of the spirit of LFCS: Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 7 / 20

  17. The Cunning of Reason II Another important feature, part of the spirit of LFCS: Explicitly marking Computer Science as a serious scientific discipline in its own right; not just a service subject, an auxiliary, a branch of engineering . . . Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 7 / 20

  18. The Cunning of Reason II Another important feature, part of the spirit of LFCS: Explicitly marking Computer Science as a serious scientific discipline in its own right; not just a service subject, an auxiliary, a branch of engineering . . . Crucial for me, when I picked up those orange booklets: ”Science is nothing more that doing your damnedest with your mind, no holds barred” Percy W Bridgman Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 7 / 20

  19. The Cunning of Reason II Another important feature, part of the spirit of LFCS: Explicitly marking Computer Science as a serious scientific discipline in its own right; not just a service subject, an auxiliary, a branch of engineering . . . Crucial for me, when I picked up those orange booklets: ”Science is nothing more that doing your damnedest with your mind, no holds barred” Percy W Bridgman Again, the picture looks different now. CS has arrived on the scene; now we must avoid remaining in splendid isolation! Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 7 / 20

  20. The Cunning of Reason II Another important feature, part of the spirit of LFCS: Explicitly marking Computer Science as a serious scientific discipline in its own right; not just a service subject, an auxiliary, a branch of engineering . . . Crucial for me, when I picked up those orange booklets: ”Science is nothing more that doing your damnedest with your mind, no holds barred” Percy W Bridgman Again, the picture looks different now. CS has arrived on the scene; now we must avoid remaining in splendid isolation! More positively: exciting things are happening on the boundaries with other disciplines. The methods and modes of thought of CS can – and should – be applied much more widely. Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 7 / 20

  21. The Cunning of Reason II Another important feature, part of the spirit of LFCS: Explicitly marking Computer Science as a serious scientific discipline in its own right; not just a service subject, an auxiliary, a branch of engineering . . . Crucial for me, when I picked up those orange booklets: ”Science is nothing more that doing your damnedest with your mind, no holds barred” Percy W Bridgman Again, the picture looks different now. CS has arrived on the scene; now we must avoid remaining in splendid isolation! More positively: exciting things are happening on the boundaries with other disciplines. The methods and modes of thought of CS can – and should – be applied much more widely. Cautionary tales: LiCS and other battlegrounds . . . Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 7 / 20

  22. The Cunning of Reason III Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  23. The Cunning of Reason III What do we need to know? Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  24. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  25. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  26. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) How it looked back then: logic, order theory, universal algebra, some (general/point-set) topology, increasingly category theory . . . Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  27. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) How it looked back then: logic, order theory, universal algebra, some (general/point-set) topology, increasingly category theory . . . A clear remove from ”mainstream mathematics” Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  28. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) How it looked back then: logic, order theory, universal algebra, some (general/point-set) topology, increasingly category theory . . . A clear remove from ”mainstream mathematics” Another aspect of the separation of qualitative from quantitative. Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  29. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) How it looked back then: logic, order theory, universal algebra, some (general/point-set) topology, increasingly category theory . . . A clear remove from ”mainstream mathematics” Another aspect of the separation of qualitative from quantitative. It looks very different now! Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  30. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) How it looked back then: logic, order theory, universal algebra, some (general/point-set) topology, increasingly category theory . . . A clear remove from ”mainstream mathematics” Another aspect of the separation of qualitative from quantitative. It looks very different now! My own (modest, call-by-need) experience from quantum information and foundations. Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  31. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) How it looked back then: logic, order theory, universal algebra, some (general/point-set) topology, increasingly category theory . . . A clear remove from ”mainstream mathematics” Another aspect of the separation of qualitative from quantitative. It looks very different now! My own (modest, call-by-need) experience from quantum information and foundations. Mathematics which has been unavoidable : probability, linear algebra, functional analysis, linear programming, convex geometry, projective geometry, logic, algorithms and complexity, sheaves and cohomology, . . . Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  32. The Cunning of Reason III What do we need to know? Those papers by Robin defined an initial syllabus for me. Later, I learned domain theory by reading Gordon’s papers (!!) How it looked back then: logic, order theory, universal algebra, some (general/point-set) topology, increasingly category theory . . . A clear remove from ”mainstream mathematics” Another aspect of the separation of qualitative from quantitative. It looks very different now! My own (modest, call-by-need) experience from quantum information and foundations. Mathematics which has been unavoidable : probability, linear algebra, functional analysis, linear programming, convex geometry, projective geometry, logic, algorithms and complexity, sheaves and cohomology, . . . Potentially, there is much more. Look at the students! Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 8 / 20

  33. Show, don’t tell Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 9 / 20

  34. Show, don’t tell I will illustrate these themes: Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 9 / 20

  35. Show, don’t tell I will illustrate these themes: Reconnecting the qualitative and structural with the quantitative Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 9 / 20

  36. Show, don’t tell I will illustrate these themes: Reconnecting the qualitative and structural with the quantitative CS interconnecting with other fields of science, being influenced by – and daring to (try to) influence! Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 9 / 20

  37. Show, don’t tell I will illustrate these themes: Reconnecting the qualitative and structural with the quantitative CS interconnecting with other fields of science, being influenced by – and daring to (try to) influence! Broadening the mathematical tools, connecting with the ”mainstream” Illustrate literally, with some pictures to give an impression of some current work on contextuality Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 9 / 20

  38. Show, don’t tell I will illustrate these themes: Reconnecting the qualitative and structural with the quantitative CS interconnecting with other fields of science, being influenced by – and daring to (try to) influence! Broadening the mathematical tools, connecting with the ”mainstream” Illustrate literally, with some pictures to give an impression of some current work on contextuality Rui Soares Barbosa, Kohei Kishida, Ray Lal and Shane Mansfield Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 9 / 20

  39. Show, don’t tell I will illustrate these themes: Reconnecting the qualitative and structural with the quantitative CS interconnecting with other fields of science, being influenced by – and daring to (try to) influence! Broadening the mathematical tools, connecting with the ”mainstream” Illustrate literally, with some pictures to give an impression of some current work on contextuality Rui Soares Barbosa, Kohei Kishida, Ray Lal and Shane Mansfield The slogan: 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) Some Thoughts on LFCS at 30 9 / 20

  40. Contextuality Analogy: Local Consistency b ′ a a ′ b Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 10 / 20

  41. Contextuality Analogy: Local Consistency b ′ a a ′ b Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 10 / 20

  42. Contextuality Analogy: Global Inconsistency Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 11 / 20

  43. 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) Some Thoughts on LFCS at 30 12 / 20

  44. 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) Some Thoughts on LFCS at 30 12 / 20

  45. 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) Some Thoughts on LFCS at 30 12 / 20

  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 ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 12 / 20

  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 ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 12 / 20

  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 ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 12 / 20

  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 ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 12 / 20

  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 ′ � � � × Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 12 / 20

  51. 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) Some Thoughts on LFCS at 30 12 / 20

  52. 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) Some Thoughts on LFCS at 30 12 / 20

  53. 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) Some Thoughts on LFCS at 30 12 / 20

  54. 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) Some Thoughts on LFCS at 30 12 / 20

  55. 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) Some Thoughts on LFCS at 30 12 / 20

  56. 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) Some Thoughts on LFCS at 30 12 / 20

  57. 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) Some Thoughts on LFCS at 30 12 / 20

  58. 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) Some Thoughts on LFCS at 30 12 / 20

  59. 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) Some Thoughts on LFCS at 30 12 / 20

  60. 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) Some Thoughts on LFCS at 30 12 / 20

  61. 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 • a ′ b ′ × × � � • b Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 13 / 20

  62. Contextuality, Logic and Paradoxes Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 14 / 20

  63. 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) Some Thoughts on LFCS at 30 14 / 20

  64. 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) Some Thoughts on LFCS at 30 14 / 20

  65. 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) Some Thoughts on LFCS at 30 14 / 20

  66. Contextuality in the Liar; Liar cycles in the PR Box Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 15 / 20

  67. 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) Some Thoughts on LFCS at 30 15 / 20

  68. 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) Some Thoughts on LFCS at 30 15 / 20

  69. 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) Some Thoughts on LFCS at 30 15 / 20

  70. 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) Some Thoughts on LFCS at 30 15 / 20

  71. 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) Some Thoughts on LFCS at 30 16 / 20

  72. 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) Some Thoughts on LFCS at 30 16 / 20

  73. 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) Some Thoughts on LFCS at 30 16 / 20

  74. Topological Characterisation • 0 • 0 ˘ Cech cohomology • • (on a presheaf derived from the 0 • 1 model) • • 1 Topological invariant γ ( s ) for every • 1 possible event s γ ( s ) � = ∅ implies contextuality • Cohomological witnesses for PR b ′ box, GHZ, Peres-Mermin, Specker’s • a ′ triangle, Kochen-Specker, All AvN a • models, . . . • b Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 17 / 20

  75. Edinburgh Memories Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 18 / 20

  76. Edinburgh Memories The Edinburgh interaction group Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 18 / 20

  77. Edinburgh Memories The Edinburgh interaction group Kohei Honda, Nobuko Yoshida, Paul-And´ e Melli` es, Juliusz Chroboczek, Jim Laird Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 18 / 20

  78. A Parting Shot Some good advice: ”Avoid the temptation to work so hard that there is no time left for serious thinking”. Francis Crick Samson Abramsky (Department of Computer Science, University of Oxford) Some Thoughts on LFCS at 30 19 / 20

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