Characterisation of Parallel Independence in AGREE-Rewriting Michael - - PowerPoint PPT Presentation
Characterisation of Parallel Independence in AGREE-Rewriting Michael Lwe (FHDW Hannover) ICGT 2018, Toulouse June 26, 2018 Contents Partial arrow classifier AGREE-rewriting Gluing construction Residual Parallel independence
ICGT 2018, Toulouse June 26, 2018
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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ICGT 2018
A category has partial arrow classifiers, if the following
For every object O, there is a monomorphism πO: O β Oβ which satisfies the following universal property: For every pair of morphisms (i : D β X, f : D β O) with monic i, there is a unique morphism (i, f )β : X β Oβ such that the pair (i, f ) is pullback of the pair (πO, (i, f )β ).
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A category has partial arrow classifiers, if for every object O there is object Oβ and partial morphism πO: Oβ β O such that for every object X and partial morphism (p : X β O) there is unique total morphism pβ : X β Oβ with πO β¦ π (pβ) = p, where functor π is given by: π O : O βΌ O and π M : m βΌ [id, m]. The embedding from category C (with total arrows) into the category of partial arrows over C is a free construction!
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A category has partial arrow classifiers, if for every object O there is object Oβ and partial morphism πO: Oβ β O such that for every object X and partial morphism (p : X β O) there is unique total morphism pβ : X β Oβ with πO β¦ π (pβ) = p, where functor π is given by: π O : O βΌ O and π M : m βΌ [id, m]. The embedding from category C (with total arrows) into the category of partial arrows over C is a free construction!
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1 2 3 4
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β
1 2 3 4 1 2 3 4
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β
1 2 3 4 1 2 3 4
5 6 31 42 32 41 43 7 31 42 32 41 43
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β
1 2 3 4 1 2 3 4
5 6 31 42 32 41 43 7 31 42 32 41 43
β β β
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1 2
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1 2
1 2
β
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1 2
1 2
β
1 21 21 1 21 21
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β β
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1 2
1 2
β
1 21 21 1 21 21
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A E C
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A E C A E C β
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A E C A E C β
A1 E C A2
A1 E C A2
S1 S2
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A E C A E C β
A1 E C A2
A1 E C A2
β β
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A E C A E C β
A1 E C A2
A1 E C A2
β β
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A E C A E C β
A E C
A E C β
β β β
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A E C A E C β
A E C A E C β β β β
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A E C A E C β
C A E
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A E C A E C β
β β β β C A E
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T t L K R l r Rule:
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T t L K R l r Rule: Lβ πL (t, l)β (PB)
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T t L K R l r Rule: Lβ πL (t, l)β (PB) G Base Match: m mβ Inverse Match:
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T t L K R l r Rule: Lβ πL (t, l)β (PB) G Base Match: m mβ Inverse Match: nβ D g n
(PB)
(PB)
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T t L K R l r Rule: Lβ πL (t, l)β (PB) G Base Match: m mβ Inverse Match: nβ D g n pβ h H (PO)
(PB)
(PB)
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T t L K R l r Rule: Lβ πL (t, l)β Rβ tβ rβ Context Rule: (PB) (PO) G Base Match: m mβ Inverse Match: nβ D g n pβ h H (PO)
(PB) (FPC)
Trace: p h H (PB)
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Lβ
x
L K T
x
π l t (t, l)β r n
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Lβ
x
L K T
x
π l t (t, l)β
G
m r mβ
x x
n
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Lβ
x
L K T
x
π l t (t, l)β
G
m r mβ
x x
n g
D
x x
nβ
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Lβ
x
L K T
x
π l t (t, l)β r n
x
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Lβ
x
L K T
x
π l t (t, l)β
G
m r mβ
x x
n g
D
x x
nβ
x x x
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Lβ
x
L K T
x
π l t (t, l)β r n
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Lβ
x
L K T
x
π l t (t, l)β
G
m r mβ
x x
n g
D
nβ
x x
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Lβ
x
L K T
π l t (t, l)β r n
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Lβ
x
L K T
π l t (t, l)β
G
m r mβ
x x
n g
D
nβ
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Lβ
x
L K T
π l t (t, l)β r
x
R
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Lβ
x
L K T
π l t (t, l)β
G
m r mβ
x x x
R
n g
D
nβ
x x
D
x x
pβ
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L K R l r G X m n
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
Isomorphism Monomorphism Pushout Complement
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
Isomorphism Monomorphism
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
Isomorphism
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
Special monomorphism Monomorphism Induced by base rule Induced by monic base match
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
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L K R l r G X m n Z x u
(PB)
Y y p
(FPC)
D g v
(FPC)
H h q
(PO)
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L K R G X Z Y D H
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L K R G X Z Y D H Kβ Rβ Kββ Zββ Y Hβ Dββ
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L K R G X Z Y D H Kβ Rβ Zβ Dβ Kββ Zββ Y Hβ Dββ
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L K R G X Z Y D H Kβ Rβ Zβ Dβ Kββ Zββ Y Hβ Dββ Gluing diagrams compose and decompose like pushouts
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(l, r) (mβ, nβ) (m, n) (lβ, rβ) (g, h) (gβ, hβ) (p, q) (pβ, qβ)
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(l, r) (mβ, nβ) (m, n) (lβ, rβ) (g, h) (gβ, hβ) (p, q) (pβ, qβ)
(gβ, hβ) β¦ (m, n) (g, h) β¦ (mβ, nβ)
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(l, r) (mβ, nβ) (m, n) (lβ, rβ) (g, h) (gβ, hβ) (p, q) (pβ, qβ)
(gβ, hβ) β¦ (m, n) (g, h) β¦ (mβ, nβ)
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(l, r) (mβ, nβ) (m, n) (lβ, rβ) (g, h) (gβ, hβ) (p, q) (pβ, qβ)
(gβ, hβ) β¦ (m, n) (g, h) β¦ (mβ, nβ)
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G H D g h
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G H D g h πL L Lβ m mβ mghβ mgh
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G H D g h πL L Lβ m mβ mβ g β¦ mβ = m, h β¦ mβ = mgh mghβ mgh
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G H D g h πL L Lβ m mβ mβ g β¦ mβ = m, h β¦ mβ = mgh mββ mghβ mβ β¦ g = mββ, mghβ β¦ h = mββ mgh
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G H D g h πL L Lβ m mβ mβ mββ mghβ mgh
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G L Lβ
x
πL m mβ
H D
mghβ mgh g h
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G L Lβ
x
πL m mβ
H D
mghβ mgh g h mβ g β¦ mβ = m, h β¦ mβ = mgh
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G L Lβ
x
πL m mβ
H D
mghβ mgh g h mβ mββ
x
g β¦ mβ = m, h β¦ mβ = mgh mβ β¦ g β mββ, mghβ β¦ h β mββ
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G H D g h πL L Lβ m mβ mβ mββ mghβ mgh (mβ, idL) pullback of (m, g) and (h, mgh)
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(mβ, idL) pullback of (m, g) and h β¦ mβ = mgh
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G H D g h πL L Lβ m mβ mβ mββ
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G H D g h
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G H D g h mβ mββ (h β¦ mβ)β L πL Lβ m mβ K T l t (t, l)β
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G H D g h mβ mββ (h β¦ mβ)β L πL Lβ m mβ K T l t (t, l)β hβ w Y v X x y
(PB) (PB)
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G H D g h mβ mββ (h β¦ mβ)β L πL Lβ m mβ K T l t (t, l)β hβ w Y v X x y
(PB) (PB)
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 (π2, π1)β
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 (π2, π1)β
(PB)
L1
idLI
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 (π2, π1)β
(PB)
L1
idLI
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 (π2, π1)β
(PB)
L1
idLI
add copy (globally) add copy (globally)
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 K1 l1 r1 R1 D1 g1 nβ1 mβ2 mββ2 r1
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 K1 l1 r1 R1 D1 g1 nβ1 mβ2 mββ2 (r1 β¦ mββ2 , π1)β r1
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 K1 l1 r1 R1 D1 g1 nβ1 mβ2 mββ2 (r1 β¦ mββ2 , π1)β R1
idRI
(PB)
r1
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 K1 l1 r1 R1 D1 g1 nβ1 mβ2 mββ2 (r1 β¦ mββ2 , π1)β R1
idRI
(PB)
r1
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L1 G m1 m2 L2 K2 l2 L2β π2 T2 t2 (t2, l2)β L* π1 π2 K1 l1 r1 R1 D1 g1 nβ1 mβ2 (π2, π1)β
(PB)
L1
idLI
mββ2 (r1 β¦ mββ2 , π1)β R1
idRI
(PB)
ICGT 2018
AGREE-rewriting is instance of the Gluing Construction! There is a precise notion of residual! Gluing and mutual residuals provides Church-Rosser! Residuals can be characterized syntactically! βββββββββββββββββββββββ Are global effects useful?
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