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Spatially Induced Concurrency within Presheaves of Labelled Transition Systems

Simon Fortier-Garceau May 28-June 2, 2019 Supervisors : P. Hofstra, P. Scott

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 1 / 28

Introduction

Spatially Induced Independence (intuitively)

Basic principle: Two actions (or events) are spatially independent of each other when each is “contained” in a region of space where the other does not interfere. In that case, the order of execution of such actions should have no impact

• n the final outcome.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 2 / 28

Labelled Transition Systems

Labelled Transition Systems (LTS)

A labelled transition systems is a tuple T = (S, L, δ) where S is a set of states; L is a set of labels (or actions); δ ⊆ S × L × S is a set of transitions.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 3 / 28

Labelled Transition Systems

Labelled Transition Systems (LTS)

A labelled transition systems is a tuple T = (S, L, δ) where S is a set of states; L is a set of labels (or actions); δ ⊆ S × L × S is a set of transitions.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 3 / 28

Labelled Transition Systems

Labelled Transition Systems (LTS)

A labelled transition systems is a tuple T = (S, L, δ) where S is a set of states; L is a set of labels (or actions); δ ⊆ S × L × S is a set of transitions. For a given triple (X, a, Y ) ∈ δ, we say that the system can make a transition from the state X to the state Y through the action a, and we write this as X

a

− → Y .

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 3 / 28

Labelled Transition Systems

Labelled Transition Systems (LTS)

A labelled transition systems is a tuple T = (S, L, δ) where S is a set of states; L is a set of labels (or actions); δ ⊆ S × L × S is a set of transitions. For a given triple (X, a, Y ) ∈ δ, we say that the system can make a transition from the state X to the state Y through the action a, and we write this as X

a

− → Y . In fact, for a fixed action a, we have a transition relation

a

− →, i.e. a binary relation on the set of states given by {(X, Y ) | X

a

− → Y }.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 3 / 28

Labelled Transition Systems

Labelled Transition Systems (LTS)

X Y Z W a a c b c a c

Figure: A labelled digraph example of a LTS

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 4 / 28

Labelled Transition Systems

Labelled Transition Systems (LTS)

X Y Z W a a c b c a c

Figure: A labelled digraph example of a LTS

Example of linear computation path: X

a

− → Y

c

− → Y

a

− → Z.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 4 / 28

Labelled Transition Systems

Morphisms of LTS

Given two LTS: T0 = (S0, L0, − →0) and T1 = (S1, L1, − →1), a morphism of labelled transitions systems f : T0 → T1 is a pair f = (σ, λ) where

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 5 / 28

Labelled Transition Systems

Morphisms of LTS

Given two LTS: T0 = (S0, L0, − →0) and T1 = (S1, L1, − →1), a morphism of labelled transitions systems f : T0 → T1 is a pair f = (σ, λ) where σ : S0 → S1 is a function that maps the set of states S0 to S1;

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 5 / 28

Labelled Transition Systems

Morphisms of LTS

Given two LTS: T0 = (S0, L0, − →0) and T1 = (S1, L1, − →1), a morphism of labelled transitions systems f : T0 → T1 is a pair f = (σ, λ) where σ : S0 → S1 is a function that maps the set of states S0 to S1; λ : L0 ⇀ L1 is a partial function on the labelling sets, which satisfies:

◮ if X

a

− →0 Y and λ(a) is defined, then σ(X)

λ(a)

− − − − →1 σ(Y );

◮ if X

a

− →0 Y and λ(a) is undefined, then σ(X) = σ(Y );

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 5 / 28

Labelled Transition Systems

Category of Labelled Transition Systems

The category of LTS, written T, consists of Objects: labelled transition systems (LTS) Morphisms: LTS morphisms as defined previously

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 6 / 28

Labelled Transition Systems

Category of Labelled Transition Systems

The category of LTS, written T, consists of Objects: labelled transition systems (LTS) Morphisms: LTS morphisms as defined previously Composition: morphisms are composed pairwise, with total functions on the set of states on the left and partial maps on sets of labels on the right : (σ1, λ1) ◦ (σ0, λ0) := (σ1 ◦ σ0, λ1 ◦ λ0)

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 6 / 28

Labelled Transition Systems

Category of Labelled Transition Systems

The category of LTS, written T, consists of Objects: labelled transition systems (LTS) Morphisms: LTS morphisms as defined previously Composition: morphisms are composed pairwise, with total functions on the set of states on the left and partial maps on sets of labels on the right : (σ1, λ1) ◦ (σ0, λ0) := (σ1 ◦ σ0, λ1 ◦ λ0) Identity morphism: For T = (S, L, δ), 1T := (1S, 1L) where 1S is the identity map on the set of states and 1L is the identity map on the labelling set.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 6 / 28

Labelled Transition Systems

Category of Labelled Transition Systems

The category of LTS, written T, consists of Objects: labelled transition systems (LTS) Morphisms: LTS morphisms as defined previously Composition: morphisms are composed pairwise, with total functions on the set of states on the left and partial maps on sets of labels on the right : (σ1, λ1) ◦ (σ0, λ0) := (σ1 ◦ σ0, λ1 ◦ λ0) Identity morphism: For T = (S, L, δ), 1T := (1S, 1L) where 1S is the identity map on the set of states and 1L is the identity map on the labelling set.

Proposition

T is bicomplete.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 6 / 28

Concurrency for Labelled Transition Systems

Models for Abstract Concurrency in LTS

Two previously studied models:

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 7 / 28

Concurrency for Labelled Transition Systems

Models for Abstract Concurrency in LTS

Two previously studied models:

1

Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): Use an independence relation I ⊆ L × L on actions, with axioms for alternative paths + co-amalgamation of transitions whenever a I b:

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 7 / 28

Concurrency for Labelled Transition Systems

Models for Abstract Concurrency in LTS

Two previously studied models:

1

Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): Use an independence relation I ⊆ L × L on actions, with axioms for alternative paths + co-amalgamation of transitions whenever a I b:

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 7 / 28

Concurrency for Labelled Transition Systems

Models for Abstract Concurrency in LTS

Two previously studied models:

1

Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): Use an independence relation I ⊆ L × L on actions, with axioms for alternative paths + co-amalgamation of transitions whenever a I b:

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 7 / 28

Concurrency for Labelled Transition Systems

Models for Abstract Concurrency in LTS

Two previously studied models:

1

Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): Use an independence relation I ⊆ L × L on actions, with axioms for alternative paths + co-amalgamation of transitions whenever a I b:

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 7 / 28

Concurrency for Labelled Transition Systems

Models for Abstract Concurrency in LTS

Two previously studied models:

1

Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): Use an independence relation I ⊆ L × L on actions, with axioms for alternative paths + co-amalgamation of transitions whenever a I b:

2

Labelled Transition Systems with Independence (LTSI) (Winskel and Nielsen): Same axioms as an ALTS basically, but use an independence relation directly on transitions, and derive an equivalence relation on transitions from it.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 7 / 28

Concurrency for Labelled Transition Systems

Asynchronous Labelled Transition System with Equivalence

An asynchronous labelled transition system with equivalence (ALTSE) is a tuple T = (S, L, δ, I, ∼) where : I ⊆ L × L is an irreflexive and symmetric binary relation on labels (i.e. the independence relation on actions); ∼ ⊆ δ × δ is an equivalence relation on transitions.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 8 / 28

Concurrency for Labelled Transition Systems

Asynchronous Labelled Transition System with Equivalence

An asynchronous labelled transition system with equivalence (ALTSE) is a tuple T = (S, L, δ, I, ∼) where : I ⊆ L × L is an irreflexive and symmetric binary relation on labels (i.e. the independence relation on actions); ∼ ⊆ δ × δ is an equivalence relation on transitions. And for any actions a, b ∈ L and any states X, Y , X ′, Y ′ ∈ S:

(X, a, Y ) ∼ (X ′, b, Y ′) ⇒ a = b (X, a, Y ) ∼ (X ′, a, Y ′) ⇒ (X = X ′ ⇔ Y = Y ′)

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 8 / 28

Concurrency for Labelled Transition Systems

Asynchronous Labelled Transition System with Equivalence

(Alternative Path) For any states X, Y , Z ∈ S and actions a, b ∈ L, If a I b and X

a

− → Y

b

− → Z, then there exists Y ′ ∈ S such that:

X

b

− → Y ′

a

− → Z and (X, a, Y ) ∼ (Y ′, a, Z) and (Y , b, Z) ∼ (X, b, Y ′)

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 9 / 28

Concurrency for Labelled Transition Systems

Asynchronous Labelled Transition System with Equivalence

(One-Step Co-amalgamation) For any states X, Y , Y ′ ∈ S and labels a, b ∈ L, If a I b and Y

a

← − X

b

− → Y ′, then there exists Z ∈ S such that: Y

b

− → Z

a

← − Y ′ and (X, a, Y ) ∼ (Y ′, a, Z) and (Y , b, Z) ∼ (X, b, Y ′)

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 10 / 28

Concurrency for Labelled Transition Systems

Asynchronous Labelled Transition System with Equivalence

(One-Step Amalgamation) For any states Y , Y ′, Z ∈ S and labels a, b ∈ L, If a I b and Y

b

− → Z

a

← − Y ′, then there exists X ∈ S such that: Y

a

← − X

b

− → Y ′ and (X, a, Y ) ∼ (Y ′, a, Z) and (Y , b, Z) ∼ (X, b, Y ′)

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 11 / 28

Concurrency for Labelled Transition Systems

ALTSE morphisms and category

Given two ALTSE: T0 = (S0, L0, − →0, I0, ∼0) and T1 = (S1, L1, − →1, I1, ∼1), a morphism of ALSTE f : T0 → T1 is a LTS morphism f = (σ, λ) as before, with the additional conditions that

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 12 / 28

Concurrency for Labelled Transition Systems

ALTSE morphisms and category

Given two ALTSE: T0 = (S0, L0, − →0, I0, ∼0) and T1 = (S1, L1, − →1, I1, ∼1), a morphism of ALSTE f : T0 → T1 is a LTS morphism f = (σ, λ) as before, with the additional conditions that

1

Preservation of independence: ∀a, b ∈ L0, a I0 b with λ(a) and λ(b) defined ⇒ λ(a) I1 λ(b)

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 12 / 28

Concurrency for Labelled Transition Systems

ALTSE morphisms and category

Given two ALTSE: T0 = (S0, L0, − →0, I0, ∼0) and T1 = (S1, L1, − →1, I1, ∼1), a morphism of ALSTE f : T0 → T1 is a LTS morphism f = (σ, λ) as before, with the additional conditions that

1

Preservation of independence: ∀a, b ∈ L0, a I0 b with λ(a) and λ(b) defined ⇒ λ(a) I1 λ(b)

2

Preservation of equivalence: ∀ (X, a, Y ), (W , a, Z) ∈ δ0, (X, a, Y ) ∼0 (W , a, Z) and λ(a) defined ⇒ (σ(X), λ(a), σ(Y )) ∼1 (σ(W ), λ(a), σ(Z))

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 12 / 28

Concurrency for Labelled Transition Systems

ALTSE morphisms and category

Given two ALTSE: T0 = (S0, L0, − →0, I0, ∼0) and T1 = (S1, L1, − →1, I1, ∼1), a morphism of ALSTE f : T0 → T1 is a LTS morphism f = (σ, λ) as before, with the additional conditions that

1

Preservation of independence: ∀a, b ∈ L0, a I0 b with λ(a) and λ(b) defined ⇒ λ(a) I1 λ(b)

2

Preservation of equivalence: ∀ (X, a, Y ), (W , a, Z) ∈ δ0, (X, a, Y ) ∼0 (W , a, Z) and λ(a) defined ⇒ (σ(X), λ(a), σ(Y )) ∼1 (σ(W ), λ(a), σ(Z)) Using ALTSE as objects with ALTSE morphisms yields the category of asynchronous labelled transition systems with equivalence, written A∼. Composition and identity morphisms are the same as for T.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 12 / 28

Spatially Distributed Systems

Complete Heyting algebra for space

Complete Heyting algebras (H, , , 0, 1, ≤) were used to represent spaces where processes are distributed.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 13 / 28

Spatially Distributed Systems

Complete Heyting algebra for space

Complete Heyting algebras (H, , , 0, 1, ≤) were used to represent spaces where processes are distributed. Terminology : Elements U ∈ H are regions of space; V ≤ U in H means that V is a subregion of U;

• j∈J

Vj and

j∈J

Vj are the join and meet of the indexed family of regions {Vj}j∈J respectively; 1 is the global region, and 0 is the empty region.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 13 / 28

Spatially Distributed Systems

Complete Heyting algebra for space

Complete Heyting algebras (H, , , 0, 1, ≤) were used to represent spaces where processes are distributed. Terminology : Elements U ∈ H are regions of space; V ≤ U in H means that V is a subregion of U;

• j∈J

Vj and

j∈J

Vj are the join and meet of the indexed family of regions {Vj}j∈J respectively; 1 is the global region, and 0 is the empty region. Given regions V1, V2 ≤ U, we will say that V1 is a join-complement of V2 in U if V1 ∨ V2 = U.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 13 / 28

Spatially Distributed Systems

Distributed Systems (or Distributed Processes)

I chose T-valued presheaves on a complete Heyting algebra H to represent a process distributed over H. We write [Hop, T] for the category of T-valued presheaves on H, with natural transformations as morphisms.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 14 / 28

Spatially Distributed Systems

Distributed Systems (or Distributed Processes)

I chose T-valued presheaves on a complete Heyting algebra H to represent a process distributed over H. We write [Hop, T] for the category of T-valued presheaves on H, with natural transformations as morphisms. Such a system is given by a functor T = (S, L, δ) : Hop → T where : S : Hop → Set represents the states presheaf of T L : Hop → Par represents the labelling presheaf of T . δ : Hop → Par represents the transitions presheaf of T .

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 14 / 28

Spatially Distributed Systems

Distributed Systems (or Distributed Processes)

I chose T-valued presheaves on a complete Heyting algebra H to represent a process distributed over H. We write [Hop, T] for the category of T-valued presheaves on H, with natural transformations as morphisms. Such a system is given by a functor T = (S, L, δ) : Hop → T where : S : Hop → Set represents the states presheaf of T L : Hop → Par represents the labelling presheaf of T . δ : Hop → Par represents the transitions presheaf of T . Given a region U ∈ H, the transition system local to U is T (U) = (S(U), L(U), δ(U)).

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 14 / 28

Spatially Distributed Systems

Petri Net Example

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 15 / 28

Spatially Distributed Systems

Petri Net Example

Discrete space H = P(P), where P = {x, y, z, w};

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 15 / 28

Spatially Distributed Systems

Petri Net Example

Discrete space H = P(P), where P = {x, y, z, w}; For V ⊆ U ⊆ P, define S(U) := NU and S(V ⊆ U) is the obvious restriction from NU to NV ;

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 15 / 28

Spatially Distributed Systems

Petri Net Example

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 16 / 28

Spatially Distributed Systems

Petri Net Example

We define containing regions for actions L = {t1, t2, t3} by ψ(t1) = {x, y, z}, ψ(t2) = {z, w} and ψ(t3) = {w}

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 16 / 28

Spatially Distributed Systems

Petri Net Example

We define containing regions for actions L = {t1, t2, t3} by ψ(t1) = {x, y, z}, ψ(t2) = {z, w} and ψ(t3) = {w} and let L(U) := {a ∈ L | ψ(a) ∩ U = ∅} with the labelling restriction L(V ⊆ U) acting as the identity on all elements of V , and undefined on every other element of U;

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 16 / 28

Spatially Distributed Systems

Petri Net Example

Example of transition in U = {y, z, w}, X

t1

− →U Y ⇐ ⇒ [ X(y) ≥ 1 and Y (y) = X(y) − 1 and Y (z) = X(z) + 2 and Y (w) = X(w)]

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 17 / 28

Spatially Distributed Systems

Petri Net Example

Example of transition in U = {y, z, w}, X

t1

− →U Y ⇐ ⇒ [ X(y) ≥ 1 and Y (y) = X(y) − 1 and Y (z) = X(z) + 2 and Y (w) = X(w)]

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 17 / 28

Spatially Distributed Systems

Spatially induced independence relation

Given a T-valued presheaf T = (S, L, δ) on a compl. Heyting algebra H, and a region U ∈ H, we define a binary relation I(U) on the set of actions L(U) such that for any actions a, b ∈ L(U):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 18 / 28

Spatially Distributed Systems

Spatially induced independence relation

Given a T-valued presheaf T = (S, L, δ) on a compl. Heyting algebra H, and a region U ∈ H, we define a binary relation I(U) on the set of actions L(U) such that for any actions a, b ∈ L(U): a I(U) b if and only if there exists a proper cover {V1, V2} of U such that a vanishes (is undefined) in V2 and b vanishes in V1.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 18 / 28

Spatially Distributed Systems

Spatially induced independence relation

Given a T-valued presheaf T = (S, L, δ) on a compl. Heyting algebra H, and a region U ∈ H, we define a binary relation I(U) on the set of actions L(U) such that for any actions a, b ∈ L(U): a I(U) b if and only if there exists a proper cover {V1, V2} of U such that a vanishes (is undefined) in V2 and b vanishes in V1.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 18 / 28

Spatially Distributed Systems

Spatially induced independence relation

Given a T-valued presheaf T = (S, L, δ) on a compl. Heyting algebra H, and a region U ∈ H, we define a binary relation I(U) on the set of actions L(U) such that for any actions a, b ∈ L(U): a I(U) b if and only if there exists a proper cover {V1, V2} of U such that a vanishes (is undefined) in V2 and b vanishes in V1.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 18 / 28

Spatially Distributed Systems

Spatially induced independence relation

Given a T-valued presheaf T = (S, L, δ) on a compl. Heyting algebra H, and a region U ∈ H, we define a binary relation I(U) on the set of actions L(U) such that for any actions a, b ∈ L(U): a I(U) b if and only if there exists a proper cover {V1, V2} of U such that a vanishes (is undefined) in V2 and b vanishes in V1.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 18 / 28

Spatially Distributed Systems

Action Containment

Consider any T-valued presheaf, and any regions V ≤ U and action a ∈ L(U), and write resU

V := S(V ≤ U) and ρU V := L(V ≤ U).

We say that a is contained in V (or that V contains a) if there exists a vanishing location (say K) of a that join-complements V in U such that:

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 19 / 28

Spatially Distributed Systems

Action Containment

Consider any T-valued presheaf, and any regions V ≤ U and action a ∈ L(U), and write resU

V := S(V ≤ U) and ρU V := L(V ≤ U).

We say that a is contained in V (or that V contains a) if there exists a vanishing location (say K) of a that join-complements V in U such that: ρU

V (a)

− − − →V ◦ resU

V = resU V ◦

a

− →U

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 19 / 28

Adapted Presheaves

T-adapted presheaves

A T-adapted presheaf on a complete Heyting algebra H is a T-valued presheaf T = (S, L, δ) on H, where

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 20 / 28

Adapted Presheaves

T-adapted presheaves

A T-adapted presheaf on a complete Heyting algebra H is a T-valued presheaf T = (S, L, δ) on H, where

1

Sheaf of states: S is a set-valued sheaf.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 20 / 28

Adapted Presheaves

T-adapted presheaves

A T-adapted presheaf on a complete Heyting algebra H is a T-valued presheaf T = (S, L, δ) on H, where

1

Sheaf of states: S is a set-valued sheaf.

2

Separated presheaf of labels: L is a Par-valued separated presheaf.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 20 / 28

Adapted Presheaves

T-adapted presheaves

A T-adapted presheaf on a complete Heyting algebra H is a T-valued presheaf T = (S, L, δ) on H, where

1

Sheaf of states: S is a set-valued sheaf.

2

Separated presheaf of labels: L is a Par-valued separated presheaf.

3

Well-Contained Actions (WCA): For any locations V ≤ U in H and any action b ∈ L(U) : If V is a join-complement of a vanishing region of b in U, then V contains b.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 20 / 28

Adapted Presheaves

T-adapted presheaves

A T-adapted presheaf on a complete Heyting algebra H is a T-valued presheaf T = (S, L, δ) on H, where

1

Sheaf of states: S is a set-valued sheaf.

2

Separated presheaf of labels: L is a Par-valued separated presheaf.

3

Well-Contained Actions (WCA): For any locations V ≤ U in H and any action b ∈ L(U) : If V is a join-complement of a vanishing region of b in U, then V contains b. We write [Hop, T]adapt for the category of T-adapted presheaves on H, which is the full subcategory of [Hop, T] where the objects are T-adapted presheaves.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 20 / 28

Adapted Presheaves

Spatially Induced Equivalences (to become ALTSE equivalences)

For a given T-adapted presheaf and a region U, we define the SI-equivalence on transitions in U, denoted ∼U, as the binary relation on δ(U) such that for all transitions (X, b, Y ), (X ′, c, Y ′) ∈ δ(U), we have (X, b, Y ) ∼U (X ′, c, Y ′) if and only if

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 21 / 28

Adapted Presheaves

Spatially Induced Equivalences (to become ALTSE equivalences)

For a given T-adapted presheaf and a region U, we define the SI-equivalence on transitions in U, denoted ∼U, as the binary relation on δ(U) such that for all transitions (X, b, Y ), (X ′, c, Y ′) ∈ δ(U), we have (X, b, Y ) ∼U (X ′, c, Y ′) if and only if (1) b = c , and (2) there exists a region V that contains b with respect to U such that : X|V = X ′|V and Y |V = Y ′|V

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 21 / 28

Adapted Presheaves

SI-Independence Functors

Recall : A∼ is the category of asynch. LTS with equivalence (ALTSE). Given a complete Heyting algebra H, we define a functor KH : [Hop, T]adapt → [Hop, A∼] as follows :

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 22 / 28

Adapted Presheaves

SI-Independence Functors

Recall : A∼ is the category of asynch. LTS with equivalence (ALTSE). Given a complete Heyting algebra H, we define a functor KH : [Hop, T]adapt → [Hop, A∼] as follows :

1

A T-adapted presheaf T = (S, L, δ) is sent to the A∼-valued presheaf KH(T ) given by : KH(T )(U) = (S(U), L(U), δ(U), I(U), ∼U) for a region U in H, and where I(U) and ∼U are the spatially induced independence relation on actions and equivalence on transitions as defined for T (U) respectively. (The restriction morphisms are unchanged: KH(T )(U ≤ V ) = T (U ≤ V ).)

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 22 / 28

Adapted Presheaves

SI-Independence Functors

Recall : A∼ is the category of asynch. LTS with equivalence (ALTSE). Given a complete Heyting algebra H, we define a functor KH : [Hop, T]adapt → [Hop, A∼] as follows :

1

A T-adapted presheaf T = (S, L, δ) is sent to the A∼-valued presheaf KH(T ) given by : KH(T )(U) = (S(U), L(U), δ(U), I(U), ∼U) for a region U in H, and where I(U) and ∼U are the spatially induced independence relation on actions and equivalence on transitions as defined for T (U) respectively. (The restriction morphisms are unchanged: KH(T )(U ≤ V ) = T (U ≤ V ).)

2

The natural transformations T

θ

− → T ′ are given by the same families of LTS morphisms, i.e. KH(θ) = θ.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 22 / 28

Adapted Presheaves

Theorem of Spatially Induced Independence

KH(T )(U) = (S(U), L(U), δ(U), I(U), ∼U) for a region U in H KH(θ) = θ, i.e. natural transf. use the same families of LTS morphisms.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 23 / 28

Adapted Presheaves

Theorem of Spatially Induced Independence

KH(T )(U) = (S(U), L(U), δ(U), I(U), ∼U) for a region U in H KH(θ) = θ, i.e. natural transf. use the same families of LTS morphisms.

Theorem of Spatially Induced Independence :

For any (complete) Heyting algebra H, we have that KH : [Hop, T]adapt → [Hop, A∼] as previously defined is indeed a functor.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 23 / 28

Conclusion

Conclusion

A well-established presheaf structure from a space to an adequate category of process models yields a natural form of spatially induced independence and concurrency.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 24 / 28

Conclusion

Conclusion

A well-established presheaf structure from a space to an adequate category of process models yields a natural form of spatially induced independence and concurrency. In particular, the case of T-adapted presheaves subsumes the case of T-valued sheaves.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 24 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

It is possible to characterize regions of dependencies and regions of effects for an action (that may be distinct), creating a flow of dependencies from one region to the other (ex. Concurrent Register Machines):

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 25 / 28

Conclusion

Future Directions

Characterize continuous and smooth transition systems LTS and extend the theory of (spatial) concurrency for them.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 26 / 28

Conclusion

Future Directions

Characterize continuous and smooth transition systems LTS and extend the theory of (spatial) concurrency for them.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 26 / 28

Conclusion

Future Directions

Characterize continuous and smooth transition systems LTS and extend the theory of (spatial) concurrency for them.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 26 / 28

Conclusion

Future Directions

Characterize continuous and smooth transition systems LTS and extend the theory of (spatial) concurrency for them.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 26 / 28

Conclusion

Bibliography

• S. Fortier-Garceau, Spatially Induced Independence within Presheaves of Labelled

Transition Systems. M.Sc. thesis, Dept. of Math, U. Ottawa, 2015

• G. Malcolm, Sheaves and Structures of Transition Systems, Algebra, Meaning and

Computation: Essays dedicated to Joseph A. Goguen on the occasion of his 65th Birthday, Springer Lecture Notes in Computer Science, Vol. 4060, Springer, 2006, pages 405-419.

• G. Winskel and M. Nielsen, Models for Concurrency, 1993
• G. Winskel, M. Nielsen, and A. Joyal, Bisimulation from open maps, LICS ’93

special issue of Information and Computation, 127(2), In BRICS reports series RS-94-7, June 1996, pages 164-185.

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 27 / 28

Conclusion

Questions

Questions?

Simon Fortier-Garceau Spatially Induced Concurrency within Presheaves of Labelled Transition Systems May 28-June 2, 2019 28 / 28