From ‘paradox’ to ‘feature’: Teleportation | φ � x ∈ B 2 U x M Bell | 00 � + | 11 � | φ � Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 7 / 29
Entangled states as linear maps H 1 ⊗ H 2 is spanned by | 11 � · · · | 1 m � . . ... . . . . | n 1 � · · · | nm � hence α 11 · · · α 1 m . . ... � � . . α ij | ij � ← → ← → | i � �→ α ij | j � . . i , j · · · j α n 1 α nm Pairs | ψ 1 , ψ 2 � are a special case — | ij � in a well-chosen basis. This is Map-State Duality : = A ∗ ⊗ B . Hom ( A , B ) ∼ Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 8 / 29
Entangled states as linear maps H 1 ⊗ H 2 is spanned by | 11 � · · · | 1 m � . . ... . . . . | n 1 � · · · | nm � hence α 11 · · · α 1 m . . ... � � . . α ij | ij � ← → ← → | i � �→ α ij | j � . . i , j · · · j α n 1 α nm Pairs | ψ 1 , ψ 2 � are a special case — | ij � in a well-chosen basis. This is Map-State Duality : = A ∗ ⊗ B . Hom ( A , B ) ∼ Notation. Given a linear map f : H → H , we write P f for the projector on H ⊗ H determined by the vector corresponding to f under Map-State duality: Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 8 / 29
Entangled states as linear maps H 1 ⊗ H 2 is spanned by | 11 � · · · | 1 m � . . ... . . . . | n 1 � · · · | nm � hence α 11 · · · α 1 m . . ... � � . . α ij | ij � ← → ← → | i � �→ α ij | j � . . i , j · · · j α n 1 α nm Pairs | ψ 1 , ψ 2 � are a special case — | ij � in a well-chosen basis. This is Map-State Duality : = A ∗ ⊗ B . Hom ( A , B ) ∼ Notation. Given a linear map f : H → H , we write P f for the projector on H ⊗ H determined by the vector corresponding to f under Map-State duality: Does this remind you of λ -calculus a little bit? . . . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 8 / 29
What is the output? φ out ? f 4 f 3 f 2 f 1 φ in ( P f 4 ⊗ 1) ◦ (1 ⊗ P f 3 ) ◦ ( P f 2 ⊗ 1) ◦ (1 ⊗ P f 1 ) : H 1 ⊗ H 2 ⊗ H 3 − → H 1 ⊗ H 2 ⊗ H 3 Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 9 / 29
What is the output? φ out ? f 4 f 3 f 2 f 1 φ in ( P f 4 ⊗ 1) ◦ (1 ⊗ P f 3 ) ◦ ( P f 2 ⊗ 1) ◦ (1 ⊗ P f 1 ) : H 1 ⊗ H 2 ⊗ H 3 − → H 1 ⊗ H 2 ⊗ H 3 φ out = f 3 ◦ f 4 ◦ f † 2 ◦ f † 3 ◦ f 1 ◦ f 2 ( φ in ) Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 9 / 29
Follow the line! f 4 f 3 f 2 f 1 f 3 ◦ f 4 ◦ f † 2 ◦ f † 3 ◦ f 1 ◦ f 2 Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 10 / 29
Bipartite Projectors Information flow in entangled states can be captured mathematically by the isomorphism = A ∗ ⊗ B . Hom ( A , B ) ∼ This leads to a decomposition of bipartite projectors into “names” (preparations) and “conames” (measurements). In graphical notation: f f † f f † Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 11 / 29
Graphical Calculus for Information Flow Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 12 / 29
Graphical Calculus for Information Flow Compact Closure : The basic algebraic laws for units and counits. = = (1 A ∗ ⊗ ǫ A ) ◦ ( η A ⊗ 1 A ∗ ) = 1 A ∗ ( ǫ A ⊗ 1 A ) ◦ (1 A ⊗ η A ) = 1 A Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 12 / 29
Compositionality The key algebraic fact from which teleportation (and many other protocols) can be derived. g f = g f Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 13 / 29
Compositionality ctd f = g Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 14 / 29
Compositionality ctd g g = f f Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 15 / 29
Teleportation diagrammatically β − 1 β − 1 i i = = β i β i Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 16 / 29
Categorical Quantum Mechanics Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 17 / 29
Categorical Quantum Mechanics Work of many people, both in the Quantum Group at Oxford CS Dept and elsewhere. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 17 / 29
Categorical Quantum Mechanics Work of many people, both in the Quantum Group at Oxford CS Dept and elsewhere. Underlying mathematics: monoidal dagger categories, dagger compact structure, Frobenius algebras, bialgebras . . . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 17 / 29
Categorical Quantum Mechanics Work of many people, both in the Quantum Group at Oxford CS Dept and elsewhere. Underlying mathematics: monoidal dagger categories, dagger compact structure, Frobenius algebras, bialgebras . . . Diagrammatic representation. Connections to logic and category theory. Underpinning mathematics, effective visualization, making mathematical structures accessible. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 17 / 29
Categorical Quantum Mechanics Work of many people, both in the Quantum Group at Oxford CS Dept and elsewhere. Underlying mathematics: monoidal dagger categories, dagger compact structure, Frobenius algebras, bialgebras . . . Diagrammatic representation. Connections to logic and category theory. Underpinning mathematics, effective visualization, making mathematical structures accessible. Software tool support: Quantomatic. Tactics, graph rewriting, visual interface. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 17 / 29
Categorical Quantum Mechanics Work of many people, both in the Quantum Group at Oxford CS Dept and elsewhere. Underlying mathematics: monoidal dagger categories, dagger compact structure, Frobenius algebras, bialgebras . . . Diagrammatic representation. Connections to logic and category theory. Underpinning mathematics, effective visualization, making mathematical structures accessible. Software tool support: Quantomatic. Tactics, graph rewriting, visual interface. Applications. Formalization of quantum protocols, QKD, measurement-based quantum computation, etc. Analysis of determinism in MBQC, compositional structure of multipartite entanglement. Foundational topics: e.g. analysis of non-locality. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 17 / 29
String Diagrams Are Everywhere Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
String Diagrams Are Everywhere This graphical formalism, with the underlying mathematics of monoidal categories, compact closure etc., turns up in (at least) the following places: Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
String Diagrams Are Everywhere This graphical formalism, with the underlying mathematics of monoidal categories, compact closure etc., turns up in (at least) the following places: Quantum mechanics, quantum information. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
String Diagrams Are Everywhere This graphical formalism, with the underlying mathematics of monoidal categories, compact closure etc., turns up in (at least) the following places: Quantum mechanics, quantum information. Logic: (linear version of) cut-elimination Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
String Diagrams Are Everywhere This graphical formalism, with the underlying mathematics of monoidal categories, compact closure etc., turns up in (at least) the following places: Quantum mechanics, quantum information. Logic: (linear version of) cut-elimination Computation: (linear version of) λ -calculus, feedback, processes, game semantics. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
String Diagrams Are Everywhere This graphical formalism, with the underlying mathematics of monoidal categories, compact closure etc., turns up in (at least) the following places: Quantum mechanics, quantum information. Logic: (linear version of) cut-elimination Computation: (linear version of) λ -calculus, feedback, processes, game semantics. Linguistics: Lambek pregroup grammars, lifting vector space models of word meaning Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
String Diagrams Are Everywhere This graphical formalism, with the underlying mathematics of monoidal categories, compact closure etc., turns up in (at least) the following places: Quantum mechanics, quantum information. Logic: (linear version of) cut-elimination Computation: (linear version of) λ -calculus, feedback, processes, game semantics. Linguistics: Lambek pregroup grammars, lifting vector space models of word meaning Topology, knot theory: Temperley-Lieb algebra, braided, pivotal and ribbon categories. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
String Diagrams Are Everywhere This graphical formalism, with the underlying mathematics of monoidal categories, compact closure etc., turns up in (at least) the following places: Quantum mechanics, quantum information. Logic: (linear version of) cut-elimination Computation: (linear version of) λ -calculus, feedback, processes, game semantics. Linguistics: Lambek pregroup grammars, lifting vector space models of word meaning Topology, knot theory: Temperley-Lieb algebra, braided, pivotal and ribbon categories. We will trace a path through some of these . . . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29
The Temperley-Lieb Algebra Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 19 / 29
The Temperley-Lieb Algebra Generators: 1 2 3 n 1 n · · · · · · · · · · · · · · · 1 ′ 2 ′ 3 ′ n ′ 1 ′ n ′ U 1 U n − 1 Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 19 / 29
The Temperley-Lieb Algebra Generators: 1 2 3 n 1 n · · · · · · · · · · · · · · · 1 ′ 2 ′ 3 ′ n ′ 1 ′ n ′ U 1 U n − 1 Relations: = = = U 2 1 = δ U 1 U 1 U 3 = U 3 U 1 U 1 U 2 U 1 = U 1 Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 19 / 29
Structure of Temperley-Lieb category Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29
Structure of Temperley-Lieb category General form of composition: · · · Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29
Structure of Temperley-Lieb category General form of composition: · · · Compact closure/rigidity: = = Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29
Structure of Temperley-Lieb category General form of composition: · · · Compact closure/rigidity: = = The same structure which accounts for teleportation: = = ψ ψ ψ Alice Bob Alice Bob Alice Bob Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29
Temperley-Lieb: expressiveness of the generators Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 21 / 29
Temperley-Lieb: expressiveness of the generators All planar diagrams can be expressed as products of generators. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 21 / 29
Temperley-Lieb: expressiveness of the generators All planar diagrams can be expressed as products of generators. E.g. the ‘left wave’ can be expressed as the product U 2 U 1 : = Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 21 / 29
Temperley-Lieb: expressiveness of the generators All planar diagrams can be expressed as products of generators. E.g. the ‘left wave’ can be expressed as the product U 2 U 1 : = Diagrammatic trace: = = The Ear is a Trace of Identity Circle is the Dimension Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 21 / 29
The Connection to Knots Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 22 / 29
The Connection to Knots How does this connect to knots? A key conceptual insight is due to Kauffman, who saw how to recast the Jones polynomial in elementary combinatorial form in terms of his bracket polynomial . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 22 / 29
The Connection to Knots How does this connect to knots? A key conceptual insight is due to Kauffman, who saw how to recast the Jones polynomial in elementary combinatorial form in terms of his bracket polynomial . The basic idea of the bracket polynomial is expressed by the following equation: = + A B Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 22 / 29
The Connection to Knots How does this connect to knots? A key conceptual insight is due to Kauffman, who saw how to recast the Jones polynomial in elementary combinatorial form in terms of his bracket polynomial . The basic idea of the bracket polynomial is expressed by the following equation: = + A B Each over-crossing in a knot or link is evaluated to a weighted sum of the two possible planar smoothings in the Temperley-Lieb algebra. With suitable choices for the coefficients A and B (as Laurent polynomials), this is invariant under the second and third Reidemeister moves. With an ingenious choice of normalizing factor, it becomes invariant under the first Reidemeister move — and yields the Jones polynomial! Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 22 / 29
Computation: back to the λ -calculus We shall consider the bracketing combinator B ≡ λ x .λ y .λ z . x ( yz ) : ( B → C ) → ( A → B ) → ( A → C ). This is characterized by the equation B abc = a ( bc ). Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 23 / 29
Computation: back to the λ -calculus We shall consider the bracketing combinator B ≡ λ x .λ y .λ z . x ( yz ) : ( B → C ) → ( A → B ) → ( A → C ). This is characterized by the equation B abc = a ( bc ). We take A = B = C = 1 in TL . The interpretation of the open term x : B → C , y : A → B , z : A ⊢ x ( yz ) : C is as follows: x + x − y + y − z + o Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 23 / 29
Computation: back to the λ -calculus We shall consider the bracketing combinator B ≡ λ x .λ y .λ z . x ( yz ) : ( B → C ) → ( A → B ) → ( A → C ). This is characterized by the equation B abc = a ( bc ). We take A = B = C = 1 in TL . The interpretation of the open term x : B → C , y : A → B , z : A ⊢ x ( yz ) : C is as follows: x + x − y + y − z + o Here x + is the output of x , and x − the input, and similarly for y . The output of the whole expression is o . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 23 / 29
Diagrammatic Simplification as β -Reduction Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 24 / 29
Diagrammatic Simplification as β -Reduction When we abstract the variables, we obtain the following caps-only diagram: z + y − y + x − x + o Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 24 / 29
Diagrammatic Simplification as β -Reduction When we abstract the variables, we obtain the following caps-only diagram: z + y − y + x − x + o Now we consider an application B abc (where application is represented by cups): a c = a c b b z + y + x + o y − x − o Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 24 / 29
A Non-Planar Example Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29
A Non-Planar Example We shall consider the commuting combinator C ≡ λ x .λ y .λ z . xzy : ( A → B → C ) → B → A → C . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29
A Non-Planar Example We shall consider the commuting combinator C ≡ λ x .λ y .λ z . xzy : ( A → B → C ) → B → A → C . This is characterized by the equation C abc = acb . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29
A Non-Planar Example We shall consider the commuting combinator C ≡ λ x .λ y .λ z . xzy : ( A → B → C ) → B → A → C . This is characterized by the equation C abc = acb . The interpretation of the open term x : A → B → C , y : B , z : A ⊢ xzy : C is as follows: x + x 1 x 2 y z o Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29
A Non-Planar Example We shall consider the commuting combinator C ≡ λ x .λ y .λ z . xzy : ( A → B → C ) → B → A → C . This is characterized by the equation C abc = acb . The interpretation of the open term x : A → B → C , y : B , z : A ⊢ xzy : C is as follows: x + x 1 x 2 y z o Here x + is the output of x , x 1 the first input, and x 2 the second input. The output of the whole expression is o . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29
Diagrammatic Simplification as β -Reduction Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 26 / 29
Diagrammatic Simplification as β -Reduction When we abstract the variables, we obtain the following caps-only diagram: x 2 x 1 x + o z y Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 26 / 29
Diagrammatic Simplification as β -Reduction When we abstract the variables, we obtain the following caps-only diagram: x 2 x 1 x + o z y Now we consider an application C abc : = a b c a c b x 2 x 1 x + o z y o Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 26 / 29
Diagrammatic Simplification as β -Reduction When we abstract the variables, we obtain the following caps-only diagram: x 2 x 1 x + o z y Now we consider an application C abc : = a b c a c b x 2 x 1 x + o z y o With BCI combinators one can interpret Linear λ -calculus . With just BI one has planar λ -calculus . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 26 / 29
Linguistics Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Linguistics Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Linguistics Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Linguistics Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . question Does he like her? s i l π l i o l → π o s Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Linguistics Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . question Does he like her? s i l π l i o l → π o s Distributional models: words interpreted as vectors of frequency counts of co-occurrences of a set of reference words (the basis) within a fixed (small) word radius in a large text corpus. Widely used in information retrieval. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Linguistics Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . question Does he like her? s i l π l i o l → π o s Distributional models: words interpreted as vectors of frequency counts of co-occurrences of a set of reference words (the basis) within a fixed (small) word radius in a large text corpus. Widely used in information retrieval. These seem very different: but they have the same categorical/diagrammatic structure — vector spaces treated as in the quantum information setting! Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Linguistics Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . question Does he like her? s i l π l i o l → π o s Distributional models: words interpreted as vectors of frequency counts of co-occurrences of a set of reference words (the basis) within a fixed (small) word radius in a large text corpus. Widely used in information retrieval. These seem very different: but they have the same categorical/diagrammatic structure — vector spaces treated as in the quantum information setting! So we can functorially map Lambek pregroup parses into vector spaces to lift the distributional word meanings compositionally to meanings for phrases and sentences. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Linguistics Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . question Does he like her? s i l π l i o l → π o s Distributional models: words interpreted as vectors of frequency counts of co-occurrences of a set of reference words (the basis) within a fixed (small) word radius in a large text corpus. Widely used in information retrieval. These seem very different: but they have the same categorical/diagrammatic structure — vector spaces treated as in the quantum information setting! So we can functorially map Lambek pregroup parses into vector spaces to lift the distributional word meanings compositionally to meanings for phrases and sentences. Implementations and benchmarks look promising: see recent work by Sadrzadeh and Graefenstette. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29
Final Remarks Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 28 / 29
Final Remarks Structures in monoidal categories, involving compact structure, trace etc., which support the diagrammatic calculus we have illustrated seem to provide a canonical setting for discussing processes . Have been widely used as such, implicitly or explicitly, in Computer Science. Recent work has emphasized their relevance in quantum information and quantum foundations. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 28 / 29
Final Remarks Structures in monoidal categories, involving compact structure, trace etc., which support the diagrammatic calculus we have illustrated seem to provide a canonical setting for discussing processes . Have been widely used as such, implicitly or explicitly, in Computer Science. Recent work has emphasized their relevance in quantum information and quantum foundations. As we have seen, the same structures reach into a wide range of other disciplines. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 28 / 29
Final Remarks Structures in monoidal categories, involving compact structure, trace etc., which support the diagrammatic calculus we have illustrated seem to provide a canonical setting for discussing processes . Have been widely used as such, implicitly or explicitly, in Computer Science. Recent work has emphasized their relevance in quantum information and quantum foundations. As we have seen, the same structures reach into a wide range of other disciplines. There are other promising ingredients for a general theory of information flow. In particular, sheaves as a general ‘logic of contextuality’. See my paper with Adam Brandenburger in New Journal of Physics (2011). Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 28 / 29
Some lessons we can learn from Robin Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Four major books on concurrency: 1980, 1989, 1999, 2009. Led, inspired and encouraged a broad body of work, from theory to tool support and applications. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Four major books on concurrency: 1980, 1989, 1999, 2009. Led, inspired and encouraged a broad body of work, from theory to tool support and applications. Be open to the work of others, and learn from them . (Although in Robin’s hands, the ideas were usually transformed in some way!) Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Four major books on concurrency: 1980, 1989, 1999, 2009. Led, inspired and encouraged a broad body of work, from theory to tool support and applications. Be open to the work of others, and learn from them . (Although in Robin’s hands, the ideas were usually transformed in some way!) SOS (Plotkin), bisimulation (Park), structural congruence (Berry and Boudol), monoidal categories (Meseguer and Montanari), . . . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Four major books on concurrency: 1980, 1989, 1999, 2009. Led, inspired and encouraged a broad body of work, from theory to tool support and applications. Be open to the work of others, and learn from them . (Although in Robin’s hands, the ideas were usually transformed in some way!) SOS (Plotkin), bisimulation (Park), structural congruence (Berry and Boudol), monoidal categories (Meseguer and Montanari), . . . Use the right mathematics to realize your scientific vision; don’t tailor your aproach to suit some preconceived mathematics you happen to like. Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Four major books on concurrency: 1980, 1989, 1999, 2009. Led, inspired and encouraged a broad body of work, from theory to tool support and applications. Be open to the work of others, and learn from them . (Although in Robin’s hands, the ideas were usually transformed in some way!) SOS (Plotkin), bisimulation (Park), structural congruence (Berry and Boudol), monoidal categories (Meseguer and Montanari), . . . Use the right mathematics to realize your scientific vision; don’t tailor your aproach to suit some preconceived mathematics you happen to like. Domains, labelled transition systems and SOS, categories, . . . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Four major books on concurrency: 1980, 1989, 1999, 2009. Led, inspired and encouraged a broad body of work, from theory to tool support and applications. Be open to the work of others, and learn from them . (Although in Robin’s hands, the ideas were usually transformed in some way!) SOS (Plotkin), bisimulation (Park), structural congruence (Berry and Boudol), monoidal categories (Meseguer and Montanari), . . . Use the right mathematics to realize your scientific vision; don’t tailor your aproach to suit some preconceived mathematics you happen to like. Domains, labelled transition systems and SOS, categories, . . . Think it through . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
Some lessons we can learn from Robin No Stone Tablets! Having created a paradigm, he recast it and made it new, not once, but in three major phases. Always with a purpose, moving the subject to a higher level. Follow through! Four major books on concurrency: 1980, 1989, 1999, 2009. Led, inspired and encouraged a broad body of work, from theory to tool support and applications. Be open to the work of others, and learn from them . (Although in Robin’s hands, the ideas were usually transformed in some way!) SOS (Plotkin), bisimulation (Park), structural congruence (Berry and Boudol), monoidal categories (Meseguer and Montanari), . . . Use the right mathematics to realize your scientific vision; don’t tailor your aproach to suit some preconceived mathematics you happen to like. Domains, labelled transition systems and SOS, categories, . . . Think it through . The speed of the long-distance runner . . . Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29
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