A compositional approach to networks Brendan Fong, University of Oxford Southampton ECS Seminar 4 February 2015
The big picture Network-style diagrammatic languages have developed to represent and reason about many different sciences: Why? Can we formalise their key features and relationships? Can we unify them?
The big picture Network-style diagrammatic languages have developed to represent and reason about many different sciences: Why? Can we formalise their key features and relationships? Can we unify them?
The big picture Jan C. Willems: . . . classical system-theoretic thinking is unsuitable for dealing. . . with the basic tenets at which system theory aims, namely, open and interconnected systems . The behavioural approach to open and interconnected systems, 2007. Joseph Goguen: Following clues from systems engineering, general systems theory and cybernetics. . . I decided that the most general concepts of engineering might be system, behavior, and interconnection . . . Tossing algebraic flowers down the great divide, 1999.
The big picture Jan C. Willems: . . . classical system-theoretic thinking is unsuitable for dealing. . . with the basic tenets at which system theory aims, namely, open and interconnected systems . The behavioural approach to open and interconnected systems, 2007. Joseph Goguen: Following clues from systems engineering, general systems theory and cybernetics. . . I decided that the most general concepts of engineering might be system, behavior, and interconnection . . . Tossing algebraic flowers down the great divide, 1999.
Categories ◦ Categories are a great algebraic framework for discussing interconnection, or composition. They first arose in the 1940s in algebraic topology. ◦ From the 1980s, it became clear they had a role to play in formalising uses of string/network diagrams, such as Feynman diagrams:
Categories ◦ Categories are a great algebraic framework for discussing interconnection, or composition. They first arose in the 1940s in algebraic topology. ◦ From the 1980s, it became clear they had a role to play in formalising uses of string/network diagrams, such as Feynman diagrams:
Categories ◦ Categories are a great algebraic framework for discussing interconnection, or composition. They first arose in the 1940s in algebraic topology. ◦ From the 1980s, it became clear they had a role to play in formalising uses of string/network diagrams, such as Feynman diagrams:
Categories ◦ A category C is the structure of one-dimensional flow charts. ◦ They comprise objects , or types : X , Y , etc . together with morphisms between these types: f X Y ◦ We can compose morphisms of matching types to get new morphisms: g f Y Z h X W The composition rule must have such pictures unambiguously describe a morphism.
Categories ◦ A category C is the structure of one-dimensional flow charts. ◦ They comprise objects , or types : X , Y , etc . together with morphisms between these types: f X Y ◦ We can compose morphisms of matching types to get new morphisms: g f Y Z h X W The composition rule must have such pictures unambiguously describe a morphism.
Categories ◦ A category C is the structure of one-dimensional flow charts. ◦ They comprise objects , or types : X , Y , etc . together with morphisms between these types: f X Y ◦ We can compose morphisms of matching types to get new morphisms: g f Y Z h X W The composition rule must have such pictures unambiguously describe a morphism.
Categories ◦ There are various types of categories that allow extra operations. ◦ A monoidal category is the structure of two-dimensional flow charts. f W X X h X Z k g V Y V The key point is that we have a notion of ‘parallel’ or tensor composition. ◦ A symmetric monoidal category further allows you to cross wires: X Y Y X
Categories ◦ There are various types of categories that allow extra operations. ◦ A monoidal category is the structure of two-dimensional flow charts. f W X X h X Z k g V Y V The key point is that we have a notion of ‘parallel’ or tensor composition. ◦ A symmetric monoidal category further allows you to cross wires: X Y Y X
Categories ◦ There are various types of categories that allow extra operations. ◦ A monoidal category is the structure of two-dimensional flow charts. f W X X h X Z k g V Y V The key point is that we have a notion of ‘parallel’ or tensor composition. ◦ A symmetric monoidal category further allows you to cross wires: X Y Y X
Categories ◦ A functor F : C → D is a map between categories. ◦ It turns morphisms f X Y in C into morphisms Ff FX FY in D . This assignment must preserve composition. That is, the diagram Ff FY Fg FX FZ must be unambiguous.
Categories ◦ A functor F : C → D is a map between categories. ◦ It turns morphisms f X Y in C into morphisms Ff FX FY in D . This assignment must preserve composition. That is, the diagram Ff FY Fg FX FZ must be unambiguous.
� � � Categories ◦ Guiding principle : each diagrammatic language draws morphisms in some symmetric monoidal category. ◦ In this talk we will consider symmetric monoidal categories of electrical circuits , signal flow graphs , and their behaviours , and functors between them: Circ SigFlow LinRel (This is known as a commutative diagram.)
� � � Categories ◦ Guiding principle : each diagrammatic language draws morphisms in some symmetric monoidal category. ◦ In this talk we will consider symmetric monoidal categories of electrical circuits , signal flow graphs , and their behaviours , and functors between them: Circ SigFlow LinRel (This is known as a commutative diagram.)
� � � Categories ◦ Guiding principle : each diagrammatic language draws morphisms in some symmetric monoidal category. ◦ In this talk we will consider symmetric monoidal categories of electrical circuits , signal flow graphs , and their behaviours , and functors between them: Circ SigFlow LinRel (This is known as a commutative diagram.)
Circuits ◦ A (closed) circuit, for us, is a graph with edges labelled by resistances. 2 Ω 3 Ω 1 Ω 1 Ω ◦ Given potentials on each node, the circuit induces pointwise currents. The behaviour of a circuit is the collection of possible potential–current readings. ◦ For linear resistors, this is governed by Ohm’s law : the induced current along an edge is equal to its potential difference divided by its resistance. Letting N be the set of nodes, the behaviour is thus a linear subspace of R N ⊕ R N .
Circuits ◦ A (closed) circuit, for us, is a graph with edges labelled by resistances. 2 Ω 3 Ω 1 Ω 1 Ω ◦ Given potentials on each node, the circuit induces pointwise currents. The behaviour of a circuit is the collection of possible potential–current readings. ◦ For linear resistors, this is governed by Ohm’s law : the induced current along an edge is equal to its potential difference divided by its resistance. Letting N be the set of nodes, the behaviour is thus a linear subspace of R N ⊕ R N .
Circuits ◦ A (closed) circuit, for us, is a graph with edges labelled by resistances. 2 Ω 3 Ω 1 Ω 1 Ω ◦ Given potentials on each node, the circuit induces pointwise currents. The behaviour of a circuit is the collection of possible potential–current readings. ◦ For linear resistors, this is governed by Ohm’s law : the induced current along an edge is equal to its potential difference divided by its resistance. Letting N be the set of nodes, the behaviour is thus a linear subspace of R N ⊕ R N .
Circuits But what about interconnections of circuits? To compose circuits, we first mark input and output terminals : 2 Ω 3 Ω 1 Ω 1 Ω
Circuits But what about interconnections of circuits? To compose circuits, we first mark input and output terminals : 2 Ω 3 Ω 1 Ω 1 Ω
Circuits But what about interconnections of circuits? To compose circuits, we first mark input and output terminals : 2 Ω 3 Ω 1 Ω 1 Ω X Y
Circuits Then compose by gluing along identified points: 2 Ω 5 Ω 3 Ω 1 Ω 1 Ω 8 Ω X Y Z ⇓ 5 Ω 2 Ω 3 Ω 8 Ω 1 Ω 1 Ω X Z
Circuits Then compose by gluing along identified points: 2 Ω 5 Ω 3 Ω 1 Ω 1 Ω 8 Ω X Y Z ⇓ 5 Ω 2 Ω 3 Ω 8 Ω 1 Ω 1 Ω X Z
Circuits Then compose by gluing along identified points: 2 Ω 5 Ω 3 Ω 1 Ω 1 Ω 8 Ω X Y Z ⇓ 5 Ω 2 Ω 3 Ω 8 Ω 1 Ω 1 Ω X Z
Circuits We also have monoidal composition, by placing circuits side-by-side: 2 Ω 2 Ω 3 Ω 1 Ω 1 Ω 3 Ω 1 Ω 1 Ω ⇒ X Y ⊗ 5 Ω 5 Ω 1 Ω 1 Ω X + X ′ Y + Y ′ X ′ Y ′
Circuits We also have monoidal composition, by placing circuits side-by-side: 2 Ω 2 Ω 3 Ω 1 Ω 1 Ω 3 Ω 1 Ω 1 Ω ⇒ X Y ⊗ 5 Ω 5 Ω 1 Ω 1 Ω X + X ′ Y + Y ′ X ′ Y ′
Circuits We also have monoidal composition, by placing circuits side-by-side: 2 Ω 2 Ω 3 Ω 1 Ω 1 Ω 3 Ω 1 Ω 1 Ω ⇒ X Y ⊗ 5 Ω 5 Ω 1 Ω 1 Ω X + X ′ Y + Y ′ X ′ Y ′
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