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Information Dynamics Samson Abramsky Department of Computer Science, Oxford University Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 1 / 29 Information Dynamics: the very idea Robins lecture at

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Information Dynamics

Samson Abramsky

Department of Computer Science, Oxford University

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 1 / 29

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Information Dynamics: the very idea

Robin’s lecture at MFPS 2000.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 2 / 29

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Information Dynamics: the very idea

Robin’s lecture at MFPS 2000. From Information, Processes and Games, in Handbook of Philosophy of Information, ed. Johan van Benthem and Pieter Adriaans, Elsevier 2008: What, then, is this nascent field? We would like to use the term Information Dynamics, which was proposed some time ago by Robin Milner, to suggest how the area of Theoretical Computer Science usually known as “Semantics” might emancipate itself from its traditional focus

  • n interpreting the syntax of pre-existing programming languages, and

become a more autonomous study of the fundamental structures of Informatics.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 2 / 29

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Computing as a science

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 3 / 29

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Computing as a science

From Robin’s essay on Semantic Ideas in Computing, in Computing Tomorrow,

  • ed. Ian Wand and Robin Milner, Cambridge 1996:

Are there distinct principles and concepts which underlie computing, so that we are justified in calling it an independent science? . . . In this essay I argue that a rich conceptual development is in progress, to which we cannot predict limits, and whose outcome will be a distinct science. . . . In the previous section we found that the domain model can be understood in terms of amounts of information, and also that sequential computation corresponds to a special discipline imposed on the flow of

  • information. In the present section, we have found that a key to

understanding concurrent or interactive computation lies in the structure

  • f this information flow.

. . . Thus both applications and theories converge upon the phenomena

  • f information flow; in my view this indicates a new scientific identity.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 3 / 29

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Mathematical structures for information flow

From the preface to Robin’s book The space and motion of communicating agents: Large informatic systems are complex, and any rigorous model must control this complexity by means of adequate structure. After many years seeking such models, I am convinced that categories provide this structure most convincingly.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 4 / 29

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Mathematical structures for information flow

From the preface to Robin’s book The space and motion of communicating agents: Large informatic systems are complex, and any rigorous model must control this complexity by means of adequate structure. After many years seeking such models, I am convinced that categories provide this structure most convincingly. We shall use the setting of monoidal categories to trace a path through quantum information, topology, logic, computation and linguistics, showing how common structures arise in all of these, and give rise to some core mathematics of information flow.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 4 / 29

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Mathematical structures for information flow

From the preface to Robin’s book The space and motion of communicating agents: Large informatic systems are complex, and any rigorous model must control this complexity by means of adequate structure. After many years seeking such models, I am convinced that categories provide this structure most convincingly. We shall use the setting of monoidal categories to trace a path through quantum information, topology, logic, computation and linguistics, showing how common structures arise in all of these, and give rise to some core mathematics of information flow. Diagrammatic representations (‘string diagrams’) will play a key role. The same pictures and the same diagrammatic transformations show up in all these, apparently very different contexts.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 4 / 29

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A crash course in qubits

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 5 / 29

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A crash course in qubits

Classical bit register: state is 0 or 1.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 5 / 29

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A crash course in qubits

Classical bit register: state is 0 or 1. Qubit: complex linear combination α0|0 + α1|1, |α0|2 + |α1|2 = 1.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 5 / 29

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A crash course in qubits

Classical bit register: state is 0 or 1. Qubit: complex linear combination α0|0 + α1|1, |α0|2 + |α1|2 = 1. Measurement (in |0, |1 basis): get |i with probability |αi|2.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 5 / 29

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A crash course in qubits

Classical bit register: state is 0 or 1. Qubit: complex linear combination α0|0 + α1|1, |α0|2 + |α1|2 = 1. Measurement (in |0, |1 basis): get |i with probability |αi|2. Geometric picture: the Bloch sphere

  • 1

1

  • i

i

N S P x z (a) (b) y P’ θ ϕ

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 5 / 29

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A crash course in qubits

Classical bit register: state is 0 or 1. Qubit: complex linear combination α0|0 + α1|1, |α0|2 + |α1|2 = 1. Measurement (in |0, |1 basis): get |i with probability |αi|2. Geometric picture: the Bloch sphere

  • 1

1

  • i

i

N S P x z (a) (b) y P’ θ ϕ

Things get interesting with n-qubit registers

  • i

αi|i, i ∈ {0, 1}n.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 5 / 29

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Quantum Entanglement

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 6 / 29

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Quantum Entanglement

Bell state: |00 + |11 EPR state: |01 + |10

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 6 / 29

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Quantum Entanglement

Bell state: |00 + |11 EPR state: |01 + |10 Compound systems are represented by tensor product: H1 ⊗ H2. Typical element:

  • i

λi · φi ⊗ ψi Superposition encodes correlation.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 6 / 29

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Quantum Entanglement

Bell state: |00 + |11 EPR state: |01 + |10 Compound systems are represented by tensor product: H1 ⊗ H2. Typical element:

  • i

λi · φi ⊗ ψi Superposition encodes correlation. Einstein’s ‘spooky action at a distance’. Even if the particles are spatially separated, measuring one has an effect on the state of the other.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 6 / 29

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Quantum Entanglement

Bell state: |00 + |11 EPR state: |01 + |10 Compound systems are represented by tensor product: H1 ⊗ H2. Typical element:

  • i

λi · φi ⊗ ψi Superposition encodes correlation. Einstein’s ‘spooky action at a distance’. Even if the particles are spatially separated, measuring one has an effect on the state of the other. Bell’s theorem: QM is essentially non-local.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 6 / 29

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From ‘paradox’ to ‘feature’: Teleportation

MBell Ux |00 + |11 x ∈ B2 |φ |φ

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 7 / 29

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Entangled states as linear maps

H1 ⊗ H2 is spanned by |11 · · · |1m . . . ... . . . |n1 · · · |nm hence

  • i,j

αij|ij ← →    α11 · · · α1m . . . ... . . . αn1 · · · αnm    ← → |i →

  • j

αij|j Pairs |ψ1, ψ2 are a special case — |ij in a well-chosen basis. This is Map-State Duality: Hom(A, B) ∼ = A∗ ⊗ B.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 8 / 29

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Entangled states as linear maps

H1 ⊗ H2 is spanned by |11 · · · |1m . . . ... . . . |n1 · · · |nm hence

  • i,j

αij|ij ← →    α11 · · · α1m . . . ... . . . αn1 · · · αnm    ← → |i →

  • j

αij|j Pairs |ψ1, ψ2 are a special case — |ij in a well-chosen basis. This is Map-State Duality: Hom(A, B) ∼ = A∗ ⊗ B.

  • Notation. Given a linear map f : H → H, we write Pf 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

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Entangled states as linear maps

H1 ⊗ H2 is spanned by |11 · · · |1m . . . ... . . . |n1 · · · |nm hence

  • i,j

αij|ij ← →    α11 · · · α1m . . . ... . . . αn1 · · · αnm    ← → |i →

  • j

αij|j Pairs |ψ1, ψ2 are a special case — |ij in a well-chosen basis. This is Map-State Duality: Hom(A, B) ∼ = A∗ ⊗ B.

  • Notation. Given a linear map f : H → H, we write Pf 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

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What is the output?

f1 f2 f3 f4 φin φout? (Pf4 ⊗ 1) ◦ (1 ⊗ Pf3) ◦ (Pf2 ⊗ 1) ◦ (1 ⊗ Pf1) : H1 ⊗ H2 ⊗ H3 − → H1 ⊗ H2 ⊗ H3

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 9 / 29

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What is the output?

f1 f2 f3 f4 φin φout? (Pf4 ⊗ 1) ◦ (1 ⊗ Pf3) ◦ (Pf2 ⊗ 1) ◦ (1 ⊗ Pf1) : H1 ⊗ H2 ⊗ H3 − → H1 ⊗ H2 ⊗ H3 φout = f3 ◦ f4 ◦ f †

2 ◦ f † 3 ◦ f1 ◦ f2(φin)

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 9 / 29

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Follow the line!

f1 f2 f3 f4 f3 ◦ f4 ◦ f †

2 ◦ f † 3 ◦ f1 ◦ f2

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 10 / 29

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Bipartite Projectors

Information flow in entangled states can be captured mathematically by the isomorphism Hom(A, B) ∼ = 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

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Graphical Calculus for Information Flow

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 12 / 29

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Graphical Calculus for Information Flow

Compact Closure: The basic algebraic laws for units and counits.

= =

(ǫA ⊗ 1A) ◦ (1A ⊗ ηA) = 1A (1A∗ ⊗ ǫA) ◦ (ηA ⊗ 1A∗) = 1A∗

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 12 / 29

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Compositionality

The key algebraic fact from which teleportation (and many other protocols) can be derived. f g

=

f g

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 13 / 29

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Compositionality ctd

f g

=

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 14 / 29

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Compositionality ctd

f g

=

g f

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 15 / 29

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Teleportation diagrammatically

βi β−1

i

=

βi β−1

i

=

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 16 / 29

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Categorical Quantum Mechanics

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 17 / 29

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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

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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

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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

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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

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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

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String Diagrams Are Everywhere

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 18 / 29

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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

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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

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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

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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

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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

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SLIDE 46

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

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SLIDE 47

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

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The Temperley-Lieb Algebra

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 19 / 29

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The Temperley-Lieb Algebra

Generators: · · · · · · 1 2 3 n 1′ 2′ 3′ n′ U1 · · · · · · · · · 1 n 1′ n′ Un−1

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 19 / 29

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The Temperley-Lieb Algebra

Generators: · · · · · · 1 2 3 n 1′ 2′ 3′ n′ U1 · · · · · · · · · 1 n 1′ n′ Un−1 Relations: = U1U2U1 = U1 = U2

1 = δU1

= U1U3 = U3U1

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 19 / 29

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Structure of Temperley-Lieb category

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29

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Structure of Temperley-Lieb category

General form of composition: · · ·

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29

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Structure of Temperley-Lieb category

General form of composition: · · · Compact closure/rigidity: = =

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29

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Structure of Temperley-Lieb category

General form of composition: · · · Compact closure/rigidity: = = The same structure which accounts for teleportation:

Alice Bob

=

ψ ψ

Alice Bob Alice Bob

=

ψ

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 20 / 29

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Temperley-Lieb: expressiveness of the generators

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 21 / 29

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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

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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 U2U1:

=

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 21 / 29

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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 U2U1:

=

Diagrammatic trace:

=

The Ear is a Circle

=

Trace of Identity is the Dimension

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 21 / 29

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The Connection to Knots

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 22 / 29

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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

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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

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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

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Computation: back to the λ-calculus

We shall consider the bracketing combinator B ≡ λx.λy.λz. x(yz) : (B → C) → (A → B) → (A → C). This is characterized by the equation Babc = a(bc).

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 23 / 29

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Computation: back to the λ-calculus

We shall consider the bracketing combinator B ≡ λx.λy.λz. x(yz) : (B → C) → (A → B) → (A → C). This is characterized by the equation Babc = 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+

  • Samson Abramsky (Department of Computer Science, Oxford University)

Information Dynamics 23 / 29

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SLIDE 65

Computation: back to the λ-calculus

We shall consider the bracketing combinator B ≡ λx.λy.λz. x(yz) : (B → C) → (A → B) → (A → C). This is characterized by the equation Babc = 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+

  • 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

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SLIDE 66

Diagrammatic Simplification as β-Reduction

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 24 / 29

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SLIDE 67

Diagrammatic Simplification as β-Reduction

When we abstract the variables, we obtain the following caps-only diagram: x+ x− y + y − z+

  • Samson Abramsky (Department of Computer Science, Oxford University)

Information Dynamics 24 / 29

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SLIDE 68

Diagrammatic Simplification as β-Reduction

When we abstract the variables, we obtain the following caps-only diagram: x+ x− y + y − z+

  • Now we consider an application Babc (where application is represented by cups):

x+ x− y + y − z+

  • a

b c a b c

  • =

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 24 / 29

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SLIDE 69

A Non-Planar Example

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29

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SLIDE 70

A Non-Planar Example

We shall consider the commuting combinator C ≡ λx.λy.λz. xzy : (A → B → C) → B → A → C.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29

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SLIDE 71

A Non-Planar Example

We shall consider the commuting combinator C ≡ λx.λy.λz. xzy : (A → B → C) → B → A → C. This is characterized by the equation Cabc = acb.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29

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SLIDE 72

A Non-Planar Example

We shall consider the commuting combinator C ≡ λx.λy.λz. xzy : (A → B → C) → B → A → C. This is characterized by the equation Cabc = acb. The interpretation of the open term x : A → B → C, y : B, z : A ⊢ xzy : C is as follows: x+ x1 x2 y z

  • Samson Abramsky (Department of Computer Science, Oxford University)

Information Dynamics 25 / 29

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SLIDE 73

A Non-Planar Example

We shall consider the commuting combinator C ≡ λx.λy.λz. xzy : (A → B → C) → B → A → C. This is characterized by the equation Cabc = acb. The interpretation of the open term x : A → B → C, y : B, z : A ⊢ xzy : C is as follows: x+ x1 x2 y z

  • Here x+ is the output of x, x1 the first input, and x2 the second input. The
  • utput of the whole expression is o.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 25 / 29

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SLIDE 74

Diagrammatic Simplification as β-Reduction

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 26 / 29

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SLIDE 75

Diagrammatic Simplification as β-Reduction

When we abstract the variables, we obtain the following caps-only diagram: x+ x1 x2 y z

  • Samson Abramsky (Department of Computer Science, Oxford University)

Information Dynamics 26 / 29

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SLIDE 76

Diagrammatic Simplification as β-Reduction

When we abstract the variables, we obtain the following caps-only diagram: x+ x1 x2 y z

  • Now we consider an application Cabc:

x+ x1 x2 y z

  • a

b c a b c

  • =

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 26 / 29

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SLIDE 77

Diagrammatic Simplification as β-Reduction

When we abstract the variables, we obtain the following caps-only diagram: x+ x1 x2 y z

  • Now we consider an application Cabc:

x+ x1 x2 y z

  • a

b c a b c

  • =

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

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SLIDE 78

Linguistics

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29

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SLIDE 79

Linguistics

Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning.

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29

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SLIDE 80

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

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SLIDE 81

Linguistics

Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . s ilπl π i ol

  • →

s

Does he like her? question

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 27 / 29

slide-82
SLIDE 82

Linguistics

Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . s ilπl π i ol

  • →

s

Does he like her? question

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

slide-83
SLIDE 83

Linguistics

Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . s ilπl π i ol

  • →

s

Does he like her? question

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

slide-84
SLIDE 84

Linguistics

Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . s ilπl π i ol

  • →

s

Does he like her? question

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

slide-85
SLIDE 85

Linguistics

Clark, Coecke and Sadrzadeh: Compositional Distributional Models of Meaning. Lambek grammars: π pronoun, i infinitive, o direct object, . . . s ilπl π i ol

  • →

s

Does he like her? question

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

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SLIDE 86

Final Remarks

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 28 / 29

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SLIDE 87

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

slide-88
SLIDE 88

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

slide-89
SLIDE 89

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

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SLIDE 90

Some lessons we can learn from Robin

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29

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SLIDE 91

Some lessons we can learn from Robin

No Stone Tablets!

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29

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SLIDE 92

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

slide-93
SLIDE 93

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

slide-94
SLIDE 94

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

slide-95
SLIDE 95

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

slide-96
SLIDE 96

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

slide-97
SLIDE 97

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

slide-98
SLIDE 98

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

slide-99
SLIDE 99

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

slide-100
SLIDE 100

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

slide-101
SLIDE 101

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 . . . There are many more . . .

Samson Abramsky (Department of Computer Science, Oxford University) Information Dynamics 29 / 29