Spatially Induced Concurrency within Presheaves of Labelled - PowerPoint PPT Presentation

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 on 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 a write this as X − → 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 a write this as X − → Y . a In fact, for a fixed action a , we have a transition relation − → , i.e. a binary a relation on the set of states given by { ( X , Y ) | X − → 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) c a Y a a X b c c Z W 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) c a Y a a X b c c Z W Figure: A labelled digraph example of a LTS a c a Example of linear computation path: X − → Y − → Y − → 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: T 0 = ( S 0 , L 0 , − → 0 ) and T 1 = ( S 1 , L 1 , − → 1 ), a morphism of labelled transitions systems f : T 0 → T 1 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: T 0 = ( S 0 , L 0 , − → 0 ) and T 1 = ( S 1 , L 1 , − → 1 ), a morphism of labelled transitions systems f : T 0 → T 1 is a pair f = ( σ, λ ) where σ : S 0 → S 1 is a function that maps the set of states S 0 to S 1 ; 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: T 0 = ( S 0 , L 0 , − → 0 ) and T 1 = ( S 1 , L 1 , − → 1 ), a morphism of labelled transitions systems f : T 0 → T 1 is a pair f = ( σ, λ ) where σ : S 0 → S 1 is a function that maps the set of states S 0 to S 1 ; λ : L 0 ⇀ L 1 is a partial function on the labelling sets, which satisfies: λ ( a ) a ◮ if X − → 0 Y and λ ( a ) is defined, then σ ( X ) − − − − → 1 σ ( Y ); a ◮ if X − → 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 : ( σ 1 , λ 1 ) ◦ ( σ 0 , λ 0 ) := ( σ 1 ◦ σ 0 , λ 1 ◦ λ 0 ) 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 : ( σ 1 , λ 1 ) ◦ ( σ 0 , λ 0 ) := ( σ 1 ◦ σ 0 , λ 1 ◦ λ 0 ) Identity morphism: For T = ( S , L , δ ), 1 T := (1 S , 1 L ) where 1 S is the identity map on the set of states and 1 L 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 : ( σ 1 , λ 1 ) ◦ ( σ 0 , λ 0 ) := ( σ 1 ◦ σ 0 , λ 1 ◦ λ 0 ) Identity morphism: For T = ( S , L , δ ), 1 T := (1 S , 1 L ) where 1 S is the identity map on the set of states and 1 L 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: Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): 1 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: Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): 1 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: Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): 1 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: Asynchronous Labelled Transition Systems (ALTS) (Bednarczyk and Shields): 1 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