Diagrammatic Algebra: from Linear to Concurrent Systems Filippo - - PowerPoint PPT Presentation

Diagrammatic Algebra: from Linear to Concurrent Systems

Filippo Bonchi, Joshua Holland, Robin Piedeleu, Pawe l Soboci´ nski and Fabio Zanasi PoPL ’19, Cascais, Portugal

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Question

Electronic Notes in Theoretical Computer Science, 2006

Process algebras vs. Petri nets

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In this talk

Towards bridging the gap between the two approaches: ◮ Start from a compositional diagrammatic language for linear dynamical systems. ◮ Give it a resource-conscious semantics by changing the domain from a field to the semiring N. ◮ Provide a sound and complete equational theory for this new semantics. ◮ Showcase the expressiveness of the calculus by embedding Petri nets with their usual operational semantics.

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A simple graphical language

Section 1 A simple graphical language

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A simple graphical language

Drawing open systems

c , d , . . .

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A simple graphical language

Drawing open systems

c , d , . . .

→

c X ⊆ X × X 2, d X ⊆ X × X . . . for some fixed set X.

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A simple graphical language

Parallel composition

c d

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A simple graphical language

Parallel composition

c d

→

     x1 x2

• ,

  y1 y2 y3    

• x1,

y1 y2

• ∈ c X , (x2, y3) ∈ d X

  

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A simple graphical language

Synchronising composition

c d

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A simple graphical language

Synchronising composition

c d

→

• x,

z1 z2

• ∃y (x, y) ∈ d X ,
• y,

z1 z2

• ∈ c X
• Bonchi et al.

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A simple graphical language

More complex networks

g d c f e

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A simple graphical language

Only the connectivity matters

g d c f e

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A simple graphical language

Only the connectivity matters

g d c f e

• X

= x y

• ,

y x

• x, y ∈ X
• Bonchi et al.

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A simple graphical language

Multiple connections

g d c f e

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A simple graphical language

Frobenius monoids

Special boxes/systems: , , , satisfying: = = = = = = = ◮ form a special commutative Frobenius monoid.

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A simple graphical language

Interpreted as:

• X =
• x,

x x

• | x ∈ X
• X = {(x, •) | x ∈ X}
• X =

x x

• , x
• | x ∈ X
• X = {(•, x) | x ∈ X}

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A simple graphical language

More algebraic structure

If X = R is a semiring we buy ourselves more structure: , and

r

for r ∈ R

→

• R =

x y

• , x + y
• | (x, y) ∈ R2
• R = {(0, •)}
• r

R = {(x, rx) | x ∈ R}

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A simple graphical language

More algebraic structure

If X = R is a semiring we buy ourselves more structure: , and

r

for r ∈ R and tranposes for free: := and :=

r

:=

r

Bonchi et al. From Linear to Concurrent Systems 18 Jan 2019 13 / 25

A simple graphical language

More algebraic structure

If X = R is a semiring we buy ourselves more structure: , and

r

for r ∈ R satisfying: = = = = ◮ , , , form a bimonoid.

Bonchi et al. From Linear to Concurrent Systems 18 Jan 2019 13 / 25

A simple graphical language

More algebraic structure

If X = R is a semiring we buy ourselves more structure: , and

r

for r ∈ R satisfying:

r r

=

r

=

r r

=

r r r

=

r s

=

rs s r

=

r + s

= ◮ Encode the additive and multiplicative operations of R.

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A simple graphical language

Adding state

◮ Introduce

x

that we interpret as a state-holding register.

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A simple graphical language

Adding state

◮ Introduce

x

that we interpret as a state-holding register. ◮ A stateful diagram c : k → l is interpreted as a relation c ⊆ Rs+k × Rs+l where s is the number of

x

in c. ◮ Semantics extended inductively with

• x

R = x y

• ,

y x

• x, y ∈ R
• Bonchi et al.

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Detour: the Linear Interpretation

Section 2 Detour: the Linear Interpretation

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Detour: the Linear Interpretation

Linear relations

As relations over a field K:

• K =
• x,

x x

• | x ∈ K
• K = {(x, •) | x ∈ K}
• K =

x y

• , x + y
• | (x, y) ∈ K2
• K = {(0, •)}
• r

K = {(x, rx) | x ∈ K} ◮ For a diagram c : k → l, c K is a linear subspace of Kk × Kl, i.e., a relation closed under K-linear combinations.

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Detour: the Linear Interpretation

Complete equational theory

◮ Addition is also a special commutative Frobenius monoid: = = = = ◮ Scalars are invertible:

r r

= =

r r

for r = 0

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Detour: the Linear Interpretation

Linear dynamical systems

For the stateful linear case: ◮ K = R(x) susbumes the notion of state we introduced. ◮ Semantics in terms of generalised streams (Laurent series). ◮ Models linear dynamical systems (e.g., filters, amplifiers) ◮ Generalisation of Shannon’s signal flow graphs. ◮ Reformulates control-theory in diagrammatic terms (e.g., controllability, observability).

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The Resource Interpretation

Section 3 The Resource Interpretation

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The Resource Interpretation

Motivating example

x

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The Resource Interpretation

Motivating example

x

s l r t → s l

• ,

s′ r

• ∃t

s = t + r, s′ = t + l

• Bonchi et al.

From Linear to Concurrent Systems 18 Jan 2019 20 / 25

The Resource Interpretation

Motivating example

x

s l r t → s l

• ,

s′ r

• ∃t

s = t + r, s′ = t + l

• Over a field K, we can relate any two l and r:
• x
• K

= K

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The Resource Interpretation

Motivating example

x

s l r t → s l

• ,

s′ r

• ∃t

s = t + r, s′ = t + l

• Over N, we must have r ≤ s so:
• x
• N

= N

Bonchi et al. From Linear to Concurrent Systems 18 Jan 2019 20 / 25

The Resource Interpretation

Motivating example

x

s l r t → s l

• ,

s′ r

• ∃t

s = t + r, s′ = t + l

• Over N, we must have r ≤ s so:
• x
• N

= N

Intuition

Without additive inverses, we cannot borrow arbitrary quantities.

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The Resource Interpretation

Motivating example

x

s l r t → s l

• ,

s′ r

• ∃t

s = t + r, s′ = t + l

• Over N, we must have r ≤ s so:
• x
• N

= N ◮ This diagram behaves like the place of a Petri net!

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The Resource Interpretation

Additive relations

For a diagram c : k → l, c N is an additive relation: a finitely-generated submonoid of Nk × Nl, i.e., a relation closed under addition and containing (0, 0).

Proposition

Finitely-generated additive relations form a symmetric monoidal category, AddRel. ◮ The proof that they compose is non-trivial and relies on Dickson’s lemma.

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The Resource Interpretation

Complete equational theory

The Resource Calculus (RC) := every equation in Section I + ◮ , and , form an idempotent bimonoid: = = = = = ◮ And = =

n n

= for n = 0

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The Resource Interpretation

Embedding Petri nets

→ With new syntactic sugar: :=

x

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The Resource Interpretation

Embedding Petri nets

→ With new syntactic sugar: :=

x

Theorem

Firing semantics of Petri nets = semantics of corresponding diagram ◮ We can use RC to (de)compose Petri nets and reason equationally about their behaviour.

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The Resource Interpretation

Moral of the story

Seemingly diverse computational models can be studied within the same algebraic/categorical framework. This is not just another process algebra, with pictures.

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The Resource Interpretation

More coming: a graphical assembly language

◮ Affine extension (done): discrete polyhedral relations to capture non-additive phenomena like mutual exclusion. ◮ RC is strictly more expressive than Petri nets: compile other process algebras into it. ◮ Coarser semantics:

◮ trace equivalence ◮ bisimulation

◮ Compositional reachability checking.

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