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Analyzing, Comparing and Debugging Schema Mappings Emanuel Sallinger Vienna University of Technology Institute of Information Systems Database and Artificial Intelligence Group DEIS10 11 November, 2010 Emanuel Sallinger DEIS10 11

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Analyzing, Comparing and Debugging Schema Mappings

Emanuel Sallinger

Vienna University of Technology Institute of Information Systems Database and Artificial Intelligence Group

DEIS’10

11 November, 2010

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 1

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Outline

Given a schema mapping M ... S T Σ

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 2

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Outline

Given a schema mapping M ... S T Σ What does it do? (analyzing) Are there any errors in it? (debugging)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 2

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Outline

Given a schema mapping M ... S T Σ What does it do? (analyzing) Are there any errors in it? (debugging) Is there a better one . . . (comparing/optimizing)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 2

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Outline

Given a schema mapping M ... S T Σ What does it do? (analyzing) Are there any errors in it? (debugging) Is there a better one . . . (comparing/optimizing)

. . . that is equivalent?

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 2

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Outline

Given a schema mapping M ... S T Σ What does it do? (analyzing) Are there any errors in it? (debugging) Is there a better one . . . (comparing/optimizing)

. . . that is equivalent? . . . that is equivalent for specific purposes, e.g. data exchange?

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 2

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Outline

Given a schema mapping M ... S T Σ What does it do? (analyzing) Are there any errors in it? (debugging) Is there a better one . . . (comparing/optimizing)

. . . that is equivalent? . . . that is equivalent for specific purposes, e.g. data exchange? What about other comparison criteria?

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 2

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Debugging with Routes

Manhattan Credit Cards SuppCards Fargo Finance Accounts Clients Σ

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 3

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Debugging with Routes

Manhattan Credit Cards SuppCards Fargo Finance Accounts Clients Σ σ1: Cards(cn, l, s, n, m, sal, loc) → ∃A (Accounts(cn, l, s) ∧ Clients(s, m, m, sal, A)) σ2: SuppCards(an, s, n, a) → ∃M, I Clients(s, n, M, I, a) σ3: Accounts(a, l, s) → ∃N, M, I, A Clients(s, N, M, I, A) σ4: Clients(s, n, m, i, a) → ∃N, L Accounts(N, L, s) σ5: Accounts(a, I, s) ∧ Accounts(a′, I ′, s) → I = I ′

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 3

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Debugging with Routes

Cards Accounts cardNo accNo limit limit ssn accHolder name mName Clients salary ssn location name mName SuppCards income accNo address ssn name address

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 4

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Debugging with Routes

Cards Accounts cardNo accNo limit limit ssn accHolder name mName Clients salary ssn location name mName SuppCards income accNo address ssn name address

σ1: Cards(cn, l, s, n, m, sal, loc) → ∃A (Accounts(cn, l, s) ∧ Clients(s, m, m, sal, A) σ2: SuppCards(an, s, n, a) → ∃M, I Clients(s, n, M, I, a)

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Debugging with Routes

I s1, s2 J s1: Cards(6689, 15K, 434, J.Long, Smith, 50K, Seattle) s2: SuppCards(6689, 234, A.Long, California)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 5

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Debugging with Routes

I s1, s2 J t1, t2, t3, t4 s1: Cards(6689, 15K, 434, J.Long, Smith, 50K, Seattle) s2: SuppCards(6689, 234, A.Long, California) t1: Accounts(6689, 15K, 434) t2: Accounts(N1, 50K, 234) t3: Clients(434, Smith, Smith, 50K, A1) t4: Clients(234, A.Long, M1, I1, California)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 5

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Debugging with Routes

I s1, s2 J t1, t2, t3, t4 s1: Cards(6689, 15K, 434, J.Long, Smith, 50K, Seattle) s2: SuppCards(6689, 234, A.Long, California) t1: Accounts(6689, 15K, 434) t2: Accounts(N1, 50K, 234) t3: Clients(434, Smith, Smith, 50K, A1) t4: Clients(234, A.Long, M1, I1, California)

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Debugging with Routes

{s1}, ∅ {s1}, {t1, t3} σ1, h s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards(cn, l, s, n, m, sal, loc) → ∃A (Accounts(cn, l, s) ∧ Clients(s, m, m, sal, A) h: {cn → 6689, l → 15K, s → 434, n → J.Long, m → Smith, sal → 50K, loc → Seattle, A → A1} t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 6

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Debugging with Routes

K1 K2 σ, h

Definition [CT06]

A satisfaction step is given as K1

σ,h

− − → K2 K1 is an instance such that K1 ⊆ K and K satisfies σ σ is a tgd ϕ( x) → ∃ y ψ( x, y) h is a homomorphism from ϕ( x) ∧ ψ( x, y) to K such that h is also a homomorphism from ϕ( x) to K1 K2 is the result of satisfying σ on K1 with homomorphism h, where K2 = K1 ∪ h(ψ( x, y))

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 7

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Debugging with Routes

{s1}, ∅ {s1}, {t1, t3} σ1, h s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards(cn, l, s, n, m, sal, loc) → ∃A (Accounts(cn, l, s) ∧ Clients(s, m, m, sal, A) h: {cn → 6689, l → 15K, s → 434, n → J.Long, m → Smith, sal → 50K, loc → Seattle, A → A1} t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 8

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Debugging with Routes

{s1}, ∅ {s1}, {t1, t3} σ1, h s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards(cn, l, s, n, m, sal, loc) → ∃A (Accounts(cn, l, s) ∧ Clients(s, m, m, sal, A) h: {cn → 6689, l → 15K, s → 434, n → J.Long, m → Smith, sal → 50K, loc → Seattle, A → A1} t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 9

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Debugging with Routes

Cards Accounts cardNo accNo limit limit ssn accHolder name mName Clients salary ssn location name mName SuppCards income accNo address ssn name address

σ1: Cards(cn, l, s, n, m, sal, loc) → ∃A (Accounts(cn, l, s) ∧ Clients(s, m, m, sal, A)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 10

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Debugging with Routes

Cards Accounts cardNo accNo limit limit ssn accHolder name mName Clients salary ssn location name mName SuppCards income accNo address ssn name address

σ′

1: Cards(cn, l, s, n, m, sal, loc) →

(Accounts(cn, l, s) ∧ Clients(s, n, m, sal, loc))

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 10

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Debugging with Routes

I s1, s2 J t1, t2, t3, t4 s1: Cards(6689, 15K, 434, J.Long, Smith, 50K, Seattle) s2: SuppCards(6689, 234, A.Long, California) t1: Accounts(6689, 15K, 434) t2: Accounts(N1, 50K, 234) t3: Clients(434, Smith, Smith, 50K, A1) t4: Clients(234, A.Long, M1, I1, California)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 11

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Debugging with Routes

I s1, s2 J t1, t2, t3, t4 s1: Cards(6689, 15K, 434, J.Long, Smith, 50K, Seattle) s2: SuppCards(6689, 234, A.Long, California) t1: Accounts(6689, 15K, 434) t2: Accounts(N1, 50K, 234) t3: Clients(434, Smith, Smith, 50K, A1) t4: Clients(234, A.Long, M1, I1, California)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 11

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Debugging with Routes

I, ∅ I, {t4} I, {t4, t2} σ2, h σ3, h′ s2: SuppCards(6689, 234, A.Long, California) σ2: SuppCards(an, s, n, a) → ∃M, I Clients(s, n, M, I, a) t4: Clients(234, A.Long, M1, I1, California) σ4: Clients(s, n, m, i, a) → ∃N, L Accounts(N, L, s) t2: Accounts(N1, 50K, 234)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 12

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Debugging with Routes

I, ∅ I, J1 I, Jn σ1, h1 . . . σn, hn

Definition [CT06]

A route for Js with M, I and J is a sequence of satisfaction steps (I, ∅)

σ1,h1

− − − → (I, J1) . . .

σn,hn

− − − → (I, Jn) where J is a solution of I under M Ji ⊆ J and σi are from M Js ⊆ Jn

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Debugging with Routes

I, ∅ I, {t4} I, {t4, t2} σ2, h σ3, h′ s2: SuppCards(6689, 234, A.Long, California) σ2: SuppCards(an, s, n, a) → ∃M, I Clients(s, n, M, I, a) t4: Clients(234, A.Long, M1, I1, California) σ4: Clients(s, n, m, i, a) → ∃N, L Accounts(N, L, s) t2: Accounts(N1, 50K, 234)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 14

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Debugging with Routes

I, ∅ I, {t4} I, {t4, t2} σ2, h σ3, h′ s2: SuppCards(6689, 234, A.Long, California) σ2: SuppCards(an, s, n, a) → ∃M, I Clients(s, n, M, I, a) t4: Clients(234, A.Long, M1, I1, California) σ4: Clients(s, n, m, i, a) → ∃N, L Accounts(N, L, s) t2: Accounts(N1, 50K, 234)

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Debugging with Routes

I, ∅ I, {t4} I, {t4, t2} σ2, h σ3, h′ s2: SuppCards(6689, 234, A.Long, California) σ2: SuppCards(an, s, n, a) → ∃M, I Clients(s, n, M, I, a) t4: Clients(234, A.Long, M1, I1, California) σ4: Clients(s, n, m, i, a) → ∃N, L Accounts(N, L, s) t2: Accounts(N1, 50K, 234)

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Debugging with Routes

I, ∅ I, {t4} I, {t4, t2} σ2, h σ3, h′ s1: Cards(6689, 15K, 434, J.Long, Smith, 50K, Seattle) s2: SuppCards(6689, 234, A.Long, California) σ′

2: Cards(cn, l, s1, n1, m, sal, loc) ∧ SuppCards(cn, s2, n2, a) →

∃M, I (Clients(s2, n2, M, I, a) ∧ Accounts(cn, l, s2) t′

2: Accounts(6689, 15K, 234)

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Computing Routes

In general, a single route is not sufficient for analyzing and debugging.

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Computing Routes

In general, a single route is not sufficient for analyzing and debugging.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 16

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Computing Routes

In general, a single route is not sufficient for analyzing and debugging.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 16

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Computing Routes

In general, a single route is not sufficient for analyzing and debugging.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 16

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Computing Routes

In general, a single route is not sufficient for analyzing and debugging.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 16

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Computing Routes

In general, a single route is not sufficient for analyzing and debugging.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 16

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Computing Routes

In general, a single route is not sufficient for analyzing and debugging.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 16

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Computing Routes

{s1}, ∅ {s1}, {t1, t3} σ1, h s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 17

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Computing Routes

s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1) h:

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 18

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Computing Routes

s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1) h: {cn → 6689, l → 15K, s → 434,

1 Map an atom from ψ(

x, y) to t. Add it to h.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 18

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Computing Routes

s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1) h: {cn → 6689, l → 15K, s → 434,

1 Map an atom from ψ(

x, y) to t. Add it to h.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 18

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Computing Routes

s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1) h: {cn → 6689, l → 15K, s → 434, n → J.Long, m → Smith, sal → 50K, loc → Seattle,

2 Map ϕ(

x)h to I/J. Add it to h.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 19

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Computing Routes

s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1) h: {cn → 6689, l → 15K, s → 434, n → J.Long, m → Smith, sal → 50K, loc → Seattle,

2 Map ϕ(

x)h to I/J. Add it to h.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 19

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Computing Routes

s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1) h: {cn → 6689, l → 15K, s → 434, n → J.Long, m → Smith, sal → 50K, loc → Seattle, A → A1}

3 Map ψ(

x, y)h to J. Add it to h.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 20

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Computing Routes

s1: Cards(6689, 15K, 434, J. Long, Smith, 50K, Seattle) σ1: Cards( cn , l , s , n , m , sal , loc ) → ∃A (Accounts( cn , l , s ) ∧ Clients( s , m , m , sal , A ) t1: Accounts(6689, 15K, 434) t3: Clients(434, Smith, Smith, 50K, A1) h: {cn → 6689, l → 15K, s → 434, n → J.Long, m → Smith, sal → 50K, loc → Seattle, A → A1}

4 return h

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 21

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Computing Routes

Algorithm [CT06] (sketch)

FindHom(I, J, t, σ)

1 Map an atom from ψ(

x, y) to t. Add it to h.

2 Map ϕ(

x)h to I/J. Add it to h.

3 Map ψ(

x, y)h to J. Add it to h.

4 return h

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 22

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Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence Optimizing with Relaxed Notions of Equivalence Comparing Schema Mappings

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 23

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Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence Optimizing with Relaxed Notions of Equivalence Comparing Schema Mappings

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 23

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Comparing and Optimizing

Optimization

Finding a “better” schema mapping that is still “equivalent”. S T S T Σ Σ′

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 24

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Comparing and Optimizing

Optimization

Finding a “better” schema mapping that is still “equivalent”. S T

≡log

S T Σ Σ′

Definition [FKNP08]

M and M′ are logically equivalent, if for every instance (I, J) (I, J) M ⇔ (I, J) M′

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 24

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 25

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Example

L(x1, x2, x3) → ∃ y1 , y2 C(y1, y2) ∧ C(x1, y2) L(x1, x2, x3) ∧ L(x4, x5, x6) → ∃y2 C(x1, y2)

Optimality Criteria

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 25

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Example

L(x1, x2, x3) → ∃ y1 , y2 C(y1, y2) ∧ C(x1, y2) L(x1, x2, x3) ∧ L(x4, x5, x6) → ∃y2 C(x1, y2)

Optimality Criteria

Minimize the number of atoms in each conclusion

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 25

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Example

L(x1, x2, x3) → ∃ y1 , y2 C(y1, y2) ∧ C(x1, y2) L(x1, x2, x3) ∧ L(x4, x5, x6) → ∃y2 C(x1, y2)

Optimality Criteria

Minimize the number of atoms in each conclusion Minimize the number of existentially quantified variables

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 25

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Example

L(x1, x2, x3) → ∃ y1 , y2 C(y1, y2) ∧ C(x1, y2) L(x1, x2, x3) ∧ L(x4, x5, x6) → ∃y2 C(x1, y2)

Optimality Criteria

Minimize the number of atoms in each conclusion Minimize the number of existentially quantified variables Minimize the number of atoms in each antecedent

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 25

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Example

L(x1, x2, x3) → ∃ y1 , y2 C(y1, y2) ∧ C(x1, y2) L(x1, x2, x3) ∧ L(x4, x5, x6) → ∃y2 C(x1, y2)

Optimality Criteria

Minimize the number of atoms in each conclusion Minimize the number of existentially quantified variables Minimize the number of atoms in each antecedent Minimize the number of dependencies

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 25

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Example

Consider the set Σ of s-t tgds: L(x1, x2, x3) → ∃y C(x1, y) L(x1, x2, x3) ∧ L(x4, x2, x5) → E(x1, x4)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 26

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Optimality Criteria

S Lecture(title, year, prof ) T Course(title, prof -area) Equal-Year(course1, course2) Σ

Example

Consider the set Σ of s-t tgds: L(x1, x2, x3) → ∃y C(x1, y) L(x1, x2, x3) ∧ L(x4, x2, x5) → E(x1, x4) Equivalent set of s-t tgds Σ′: L(x1, x2, x3) ∧ L(x4, x2, x5) → ∃y C(x1, y) ∧ E(x1, x4)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 26

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Example (continued)

Consider the set Σ of s-t tgds: L(x1, x2, x3) → ∃y C(x1, y) L(x1, x2, x3) ∧ L(x4, x2, x5) → E(x1, x4) Equivalent set of s-t tgds Σ′: L(x1, x2, x3) ∧ L(x4, x2, x5) → ∃y C(x1, y) ∧ E(x1, x4)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 27

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Example (continued)

Consider the set Σ of s-t tgds: L(x1, x2, x3) → ∃y C(x1, y) L(x1, x2, x3) ∧ L(x4, x2, x5) → E(x1, x4) Equivalent set of s-t tgds Σ′: L(x1, x2, x3) ∧ L(x4, x2, x5) → ∃y C(x1, y) ∧ E(x1, x4)

Observation

Canonical universal solution: for Σ: one tuple in the C-relation per tuple in the L-relation for Σ′: in total, quadratically many tuples in the C-relation

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 27

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Example (continued)

Consider the set Σ of s-t tgds: L(x1, x2, x3) → ∃y C(x1, y) L(x1, x2, x3) ∧ L(x4, x2, x5) → E(x1, x4) Equivalent set of s-t tgds Σ′: L(x1, x2, x3) ∧ L(x4, x2, x5) → ∃y C(x1, y) ∧ E(x1, x4)

Observation

Canonical universal solution: for Σ: one tuple in the C-relation per tuple in the L-relation for Σ′: in total, quadratically many tuples in the C-relation

Optimality Criteria

Splitting should be applied whenever possible.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 27

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SLIDE 60

Optimality Criteria

Optimality Criteria

Splitting: Splitting should be applied whenever possible.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 28

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SLIDE 61

Optimality Criteria

Optimality Criteria

Splitting: Splitting should be applied whenever possible. Optimization goals: cardinality-minimality: the number of dependencies shall be minimal antecedent-minimality: the total size of the antecedents shall be minimal conclusion-minimality: the total size of the conclusions shall be minimal variable-minimality: the total number of existentially quantified variables shall be minimal

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 28

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SLIDE 62

Optimizing Schema Mappings

Rewrite System for s-t tgds [GPS09]

1 Simplification of the conclusion (core computation) 2 Simplification of the antecedent (core computation) 3 Splitting 4 Deletion of an s-t tgd (implication test) 5 Simplification of the conclusion using other tgds (implication test)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 29

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SLIDE 63

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) ∧ R(y1, x2, y2) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) ∧ R(x2, y4, x3) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 30

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SLIDE 64

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) ∧ R(y1, x2, y2) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) ∧ R(x2, y4, x3) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

1 Simplification of the conclusion (core computation)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 30

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SLIDE 65

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) ∧ R(x2, y4, x3) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

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SLIDE 66

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) ∧ R(x2, y4, x3) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 31

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SLIDE 67

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) ∧ R(x2, y4, x3) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 30

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SLIDE 68

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) ∧ R(x2, y4, x3) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

3 Splitting

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 30

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SLIDE 69

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

3 Splitting

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 30

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SLIDE 70

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

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SLIDE 71

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) ∧ L(x1, x2, x3) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

2 Simplification of the antecedent (core computation)

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SLIDE 72

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

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SLIDE 73

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) → P(x1, y2, y1) ∧ Q(y1, y3, x2) ∧ Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

5 Simplification of the conclusion using other tgds (implication test)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 32

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SLIDE 74

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) → Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 33

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SLIDE 75

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) → Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

4 Deletion of an s-t tgd (implication test)

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SLIDE 76

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x1, x1) → P(x1, y1, y2) ∧ Q(y2, y3, x1) ∧ R(y1, x1, y2) L(x1, x2, x2) → Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 33

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SLIDE 77

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x2, x2) → Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 34

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SLIDE 78

Optimizing Schema Mappings

L(x1, x2, x3) → P(x1, y1, 3) ∧ R(y1, x2, 3) L(x1, x2, x2) → Q(3, y3, x2) L(x1, x2, x2) ∧ L(x1, x2, x3) → R(x2, y4, x3)

Result of rewrite system

Is among all logically equivalent split-reduced mappings cardinality/antecedent/conclusion/variable-minimal Is a unique normal form

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 34

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SLIDE 79

Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence

– Optimality Criteria – Optimization and Normalization

Optimizing with Relaxed Notions of Equivalence Comparing Schema Mappings

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 35

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SLIDE 80

Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence

– Optimality Criteria – Optimization and Normalization

Optimizing with Relaxed Notions of Equivalence Comparing Schema Mappings

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 35

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SLIDE 81

Equivalence

S T

≡log

S T Σ Σ′

Definition [FKNP08]

M and M′ are logically equivalent, if for every source instance I Sol(I, M) = Sol(I, M′)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 36

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SLIDE 82

Equivalence

S T S T Σ Σ′ S(x) → T(x) T ′(x, y) → T ′(y, x) S(x) → T(x)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 37

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SLIDE 83

Equivalence

S T ≡log S T Σ Σ′ S(x) → T(x) T ′(x, y) → T ′(y, x) S(x) → T(x)

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SLIDE 84

Equivalence

S T ≡log S T Σ Σ′ S(x) → T(x) T ′(x, y) → T ′(y, x) S(x) → T(x)

Observation

If we are interested in typical data exchange, i.e. the universal solutions, M′ is “just as good as” M, and has smaller cardinality.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 37

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SLIDE 85

Relaxed Notions of Equivalence

DE equivalence

Data-exchange (DE) equivalence does not distinguish mappings which behave in the same way for data exchange. S T

≡DE

S T Σ Σ′

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 38

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SLIDE 86

Relaxed Notions of Equivalence

DE equivalence

Data-exchange (DE) equivalence does not distinguish mappings which behave in the same way for data exchange. S T

≡DE

S T Σ Σ′

Definition [FKNP08]

M and M′ are data-exchange equivalent, if for every source instance I UnivSol(I, M) = UnivSol(I, M′)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 38

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SLIDE 87

Relaxed Notions of Equivalence

CQ equivalence

Conjunctive-query (CQ) equivalence does not distinguish mappings which behave similarly for answering conjunctive queries. S T

≡CQ

S T Σ Σ′

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 39

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SLIDE 88

Relaxed Notions of Equivalence

CQ equivalence

Conjunctive-query (CQ) equivalence does not distinguish mappings which behave similarly for answering conjunctive queries. S T

≡CQ

S T Σ Σ′

Definition [FKNP08]

M and M′ are conjunctive-query equivalent, if for every source instance I and every CQ q, either Sol(I, M) = Sol(I, M′) = ∅ or cert(q, I, M) = cert(q, I, M′)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 39

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SLIDE 89

Relaxed Notions of Equivalence

S T

≡CQ

S T Σ Σ′

Proposition [FKNP08]

Assume that the following holds for every source instance I: Sol(I, M) = ∅ ⇒ UnivSol(I, M) = ∅ Then M and M′ are conjunctive-query equivalent, if for every source instance I, either Sol(I, M) = Sol(I, M′) = ∅ or core(I, M) = core(I, M′)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 40

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SLIDE 90

Hierarchy of Equivalences

Proposition [FKNP08]

Let M = (S, T, Σ) and M′ = (S, T, Σ′) be two schema mappings. M ≡log M′ ⇒ M ≡DE M′ ⇒ M ≡CQ M′

log DE CQ

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 41

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SLIDE 91

Hierarchy of Equivalences

But is this hierarchy of optimization potential proper, or does it collapse?

log DE CQ log DE CQ

This of course depends on the class of schema mappings.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 42

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SLIDE 92

Hierarchy of Equivalences

S T S T Σ Σ′ ∅ T(x, y) → T(y, x)

(s-t tgds and target tgds)

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SLIDE 93

Hierarchy of Equivalences

S T ≡DE ≡log S T Σ Σ′ ∅ T(x, y) → T(y, x)

(s-t tgds and target tgds)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 43

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SLIDE 94

Hierarchy of Equivalences

S T ≡DE ≡log S T Σ Σ′ ∅ T(x, y) → T(y, x)

(s-t tgds and target tgds)

Observation

UnivSol(I, M) = UnivSol(I, M′) = {∅}, however for any I the solution J = {T(a, b)} is a solution under M but not under M′

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 43

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SLIDE 95

Hierarchy of Equivalences

S T S T Σ Σ′ S(x) → T(x, x) T(x, y) ∧ T(y, x) → T(x, x) S(x) → T(x, x) T(x, y) ∧ T(y, z) ∧ T(z, x) → T(x, x)

(s-t tgds and target tgds)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 44

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SLIDE 96

Hierarchy of Equivalences

S T ≡CQ ≡DE S T Σ Σ′ S(x) → T(x, x) T(x, y) ∧ T(y, x) → T(x, x) S(x) → T(x, x) T(x, y) ∧ T(y, z) ∧ T(z, x) → T(x, x)

(s-t tgds and target tgds)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 44

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SLIDE 97

Hierarchy of Equivalences

S T ≡CQ ≡DE S T Σ Σ′ S(x) → T(x, x) T(x, y) ∧ T(y, x) → T(x, x) S(x) → T(x, x) T(x, y) ∧ T(y, z) ∧ T(z, x) → T(x, x)

(s-t tgds and target tgds)

Observation

This is a universal solution for I = {S(1)} under M, but not M′: J = {T(1, 1), T(x, y), T(y, z), T(z, x)} While J is universal for M and M′, J is no solution for I under M′.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 44

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SLIDE 98

Hierarchy of Equivalences

log DE CQ

s-t tgds

– no additional optimization power – all three equivalences decidable

log DE CQ

s-t tgds and target tgds

– additional optimization power – DE- and CQ-equivalence undecidable

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 45

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SLIDE 99

CQ-Equivalence to s-t tgds

Theorem [FKNP08]

If M is specified by full s-t tgds and full target tgds, then the following statements are equivalent: M has bounded parallel chase There is an M′ ≡CQ M specified by full s-t tgds There is an M′ ≡CQ M specified by s-t tgds There is an M′ ≡CQ M specified by an SO tgd

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 46

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SLIDE 100

CQ-Equivalence to s-t tgds

S T ≡CQ S T Σ Σ′ ∃f ∀x, y S(x, y) → T(x, y) ∧ ∀x S(x, x) → T(x, f (x)) ∧ ∀x S(x, x)∧x = f (x) → W (x)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 47

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SLIDE 101

CQ-Equivalence to s-t tgds

S {S} T {T, W } Σ S(x, y) → T(x, y) S(x, x) → T(x, f (x)) S(x, x) ∧ x = f (x) → W (x)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 48

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SLIDE 102

CQ-Equivalence to s-t tgds

S {S} T {T, W } Σ S(x, y) → T(x, y) S(x, x) → T(x, f (x)) S(x, x) ∧ x = f (x) → W (x)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 48

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SLIDE 103

CQ-Equivalence to s-t tgds

S {S} T {T, W } Σ S(x, y) → T(x, y) S(x, x) → T(x, f (x))

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 48

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SLIDE 104

CQ-Equivalence to s-t tgds

S {S} T {T, W } Σ S(x, y) → T(x, y) S(x, x) → T(x, f (x))

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 48

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SLIDE 105

CQ-Equivalence to s-t tgds

S {S} T {T, W } Σ S(x, y) → T(x, y)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 48

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SLIDE 106

CQ-Equivalence to s-t tgds

S T ≡CQ S T Σ Σ′ ∃f ∀x, y S(x, y) → T(x, y) ∧ ∀x S(x, x) → T(x, f (x)) ∧ ∀x S(x, x)∧x = f (x) → W (x) S(x, y) → T(x, y)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 49

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SLIDE 107

CQ-Equivalence to s-t tgds

Some further results [FKNP08]

Characterization for CQ-equivalence to s-t tgds: full s-t tgds and full target tgds: bounded parallel chase SO tgds: bounded f-block size s-t tgds and target tgds: bounded core chase as well as bounded f-block size

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 50

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SLIDE 108

Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence

– Optimality Criteria – Optimization and Normalization

Optimizing with Relaxed Notions of Equivalence

– Data-Exchange Equivalence – Conjunctive-Query Equivalence

Comparing Schema Mappings

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 51

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SLIDE 109

Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence

– Optimality Criteria – Optimization and Normalization

Optimizing with Relaxed Notions of Equivalence

– Data-Exchange Equivalence – Conjunctive-Query Equivalence

Comparing Schema Mappings

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 51

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SLIDE 110

Information Transfer

S T S T′ Σ Σ′ S(x, y) → T(x) S(x, y) → U(x, y)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 52

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SLIDE 111

Information Transfer

S T ≡CQ ≡DE ≡log S T′ Σ Σ′ S(x, y) → T(x) S(x, y) → U(x, y)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 52

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SLIDE 112

Information Transfer

S T S S T′ Σ Σ′ S(x, y) → T(x) S(x, y) → U(x, y)

Observation

Intuitively, M′ transfers more information than M, since with U(x, y) → T(x) the information transferred by M can be obtained from the target of M′.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 52

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SLIDE 113

Information Transfer

S T S S T′ Σ Σ′

Definition [APRR10]

M S M′ if there exists a mapping N from T′ to T s.t. M = M′ ◦ N We say that M′ transfers as much source information as M. S T′ T M′ N M

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 53

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SLIDE 114

Information Transfer

S T T S′ T Σ Σ′

Definition [APRR10]

M T M′ if there exists a mapping N from S to S′ s.t. M = N ◦ M′ We say that M′ covers as much target information as M. S S′ T N M′ M

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 54

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SLIDE 115

Redundancy

S T S T′ Σ Σ′ S(x) → T(x) S(x) → U(x, x)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 55

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SLIDE 116

Redundancy

S T ≡S S T′ Σ Σ′ S(x) → T(x) S(x) → U(x, x)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 55

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SLIDE 117

Redundancy

S T ≡S S T′ Σ Σ′ S(x) → T(x) S(x) → U(x, x)

Definition [APRR10]

M is target redundant if there exists an instance J∗ of T s.t. M∗ = {(I, J) ∈ M | J = J∗} satisfies M∗ ≡S M. Similarly defined for source redundancy.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 55

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SLIDE 118

Redundancy

S T ≡S S T′ Σ Σ′ S(x) → T(x) target non-redundant S(x) → U(x, x) target redundant

Definition [APRR10]

M is target redundant if there exists an instance J∗ of T s.t. M∗ = {(I, J) ∈ M | J = J∗} satisfies M∗ ≡S M. Similarly defined for source redundancy.

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 55

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SLIDE 119

Redundancy

S T ≡S S T′ Σ Σ′ S(x) → T(x) target non-redundant S(x) → U(x, x) target redundant

Observation

Σ′ is target redundant, since a solution can contain an atom U(a, b) with a = b

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 55

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SLIDE 120

Redundancy

S T ≡S S T′ Σ Σ′ S(x) → T(x) target non-redundant S(x) → U(x, x) target redundant

Observation

Σ′ is target redundant, since a solution can contain an atom U(a, b) with a = b But the schema mapping given as follows is target non-redundant: S(x) → U(x, x) U(x, y) → x = y

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 55

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SLIDE 121

Comparing Schema Mappings

Extract operator

Given mapping M, create a new source schema S′ that captures exactly the information participating in M. S S′ T M1 M2 M

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Comparing Schema Mappings

Extract operator

Given mapping M, create a new source schema S′ that captures exactly the information participating in M. P(x, y) → ∃u T(x, u) ∧ U(x, x) P(x, y) ∧ R(y, z) → ∃v V (x, y, v) S S′ T M1 M2 M

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 56

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SLIDE 123

Comparing Schema Mappings

Extract operator

Given mapping M, create a new source schema S′ that captures exactly the information participating in M. P(x, y) → ∃u T(x, u) ∧ U(x, x) P(x, y) ∧ R(y, z) → ∃v V (x, y, v) S S′ T M1 M2 M P(x, y) → P1(x) P(x, y) ∧ R(y, z) → P2(x, y)

Emanuel Sallinger DEIS’10 – 11 November, 2010 Page 56

slide-124
SLIDE 124

Comparing Schema Mappings

Extract operator

Given mapping M, create a new source schema S′ that captures exactly the information participating in M. P(x, y) → ∃u T(x, u) ∧ U(x, x) P(x, y) ∧ R(y, z) → ∃v V (x, y, v) S S′ T M1 M2 M P(x, y) → P1(x) P(x, y) ∧ R(y, z) → P2(x, y) P1(x) → ∃u T(x, u) ∧ U(x, x) P2(x, y) → ∃v V (x, y, v)

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SLIDE 125

Comparing Schema Mappings

Extract operator

Given mapping M, create a new source schema S′ that captures exactly the information participating in M. S S′ T M1 M2 M

Characterization [APRR10]

(M1, M2) is an extract of M iff M1 ◦ M2 = M M1 ≡S M and M1 is target non-redundant M2 ≡T M and M2 is source non-redundant

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Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence

– Optimality Criteria – Optimization and Normalization

Optimizing with Relaxed Notions of Equivalence

– Data-Exchange Equivalence – Conjunctive-Query Equivalence

Comparing Schema Mappings

– Information Transfer – Redundancy

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SLIDE 127

Outline

S T Σ Analyzing and Debugging

– Debugging with Routes – Computing Routes

Optimizing with Logical Equivalence

– Optimality Criteria – Optimization and Normalization

Optimizing with Relaxed Notions of Equivalence

– Data-Exchange Equivalence – Conjunctive-Query Equivalence

Comparing Schema Mappings

– Information Transfer – Redundancy

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Further Results

Analyzing and Debugging

– Computing a single route fast – Application to XML-based settings

Optimizing with Logical Equivalence

– Extending normalization to target egds

Optimizing with Relaxed Notions of Equivalence

– Boundary between DE- and CQ-equivalence – Full characterization of CQ-equivalence

Comparing Schema Mappings

– Characterization of the inverse operator, schema evolution

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Next Steps

Support debugging with routes for target egds Extend optimization to target egds Normalize and optimize target tgds Create a full characterization for DE-equivalence Find useful decidable fragments for the relaxed notions Develop heuristic approaches to optimization Expand results beyond the relational setting

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References

Marcelo Arenas, Jorge P´ erez, Juan L. Reutter, and Cristian Riveros. Foundations of schema mapping management. In Jan Paredaens and Dirk Van Gucht, editors, PODS, pages 227–238. ACM, 2010. Laura Chiticariu and Wang Chiew Tan. Debugging schema mappings with routes. In Umeshwar Dayal, Kyu-Young Whang, David B. Lomet, Gustavo Alonso, Guy M. Lohman, Martin L. Kersten, Sang Kyun Cha, and Young-Kuk Kim, editors, VLDB, pages 79–90. ACM, 2006. Ronald Fagin, Phokion G. Kolaitis, Alan Nash, and Lucian Popa. Towards a theory of schema-mapping optimization. In Maurizio Lenzerini and Domenico Lembo, editors, PODS, pages 33–42. ACM, 2008. Georg Gottlob, Reinhard Pichler, and Vadim Savenkov. Normalization and optimization of schema mappings. PVLDB, 2(1):1102–1113, 2009.

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