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Reliance-Based Optimization of Existential Rule Reasoning

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Reliance-Based Optimization of Existential Rule Reasoning

Alex Ivliev

TU Dresden

(2)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

leadRole A 2 R 1 M 2

bigBudget M 1 M 2

stars A 1 M 1 A 2 M 1

A 1 M 2

famous A 2

costar

(3)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

leadRole A 2 R 1 M 2

bigBudget M 1

stars A 1 M 1 A 2 M 1

A 1 M 2

famous A 2

costar

(4)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

h = {m 7→ M 1 } leadRole

A 2 R 1 M 2 bigBudget

M 1 M 2

stars A 1 M 1 A 2 M 1

A 1 M 2

famous A 2

costar

(5)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

h = {m 7→ M 1 } leadRole

A 2 R 1 M 2 bigBudget

M 1

stars A 1 M 1 A 2 M 1

A 1 M 2

famous A 2

costar

(6)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

h = {m 7→ M 2 } leadRole

A 2 R 1 M 2 bigBudget

M 1 M 2

stars A 1 M 1 A 2 M 1

A 1 M 2

famous A 2

costar

(7)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

leadRole A 2 R 1 M 2

bigBudget M 1

stars A 1 M 1 A 2 M 1

A 1 M 2 n M

famous A 2

n

costar

(8)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

leadRole A 2 R 1 M 2

bigBudget M 1 M 2

stars A 1 M 1 A 2 M 1

A 1 M 2 n M 2

famous A 2

n

costar

(9)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

stars A 1 M 1 A 2 M 1 A 1 M 2 n M

./

stars A 1 M 1 A 2 M 1 A 1 M 2 n M

=

costar A 1 A 1 M 1 A 1 A 2 M 1 A 2 A 1 M 1

. . .

(10)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

leadRole A 2 R 1 M 2

bigBudget M 1 M 2

stars A 1 M 1 A 2 M 1

A 1 M 2 n M 2

famous A 2

n

costar A 1 A 1 M 1 A 1 A 2 M 1

A 2 A 1 M 1

. . .

(11)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

leadRole A 2 R 1 M 2

bigBudget M 1

stars A 1 M 1 A 2 M 1

A 1 M 2 n M

famous A 2

n

costar A 1 A 1 M 1 A 1 A 2 M 1

A 2 A 1 M 1

. . .

(12)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

h alt = {m 7→ M 1 , a 7→ A 2 } leadRole

A 2 R 1 M 2 bigBudget

M 1 M 2

stars A 1 M 1 A 2 M 1

A 1 M 2 n M 2 A 2 M 2

famous A 2

n

costar A 1 A 1 M 1 A 1 A 2 M 1

A 2 A 1 M 1

. . .

(13)

Introduction

leadRole(a, r , m) → stars(a, m) stars(a, m) ∧ stars(b, m) → costar(a, b, m)

bigBudget(m) → ∃a. stars(a, m) ∧ famous(a)

costar(a, b, m) ∧ famous(a) → famousCostar(a, b)

costar A

1

A

1

M

1

A

1

A

2

M

1

. . . A

2

A

2

M

2

./

famous A

2

n

(14)

Optimization Goals

Optimization 1: Minimize number of alternative Matches Avoids introducing redundant facts

Might lead to core models

Optimization 2: Minimize number of rule applications Avoids splitting facts across multiple tables

Reduces number of temporary unions of blocks

Current approach by VLog: Cycle through rules but prefer datalog rules

(15)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1 author 1 book 2 . . .

writtenBy authorOf writtenBy writtenBy authorOf

authorOf

(16)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1

author 1 book 2 . . .

writtenBy authorOf writtenBy writtenBy authorOf

authorOf

(17)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1 author 1

book 2 . . .

writtenBy

authorOf writtenBy writtenBy authorOf

authorOf

(18)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1 author 1 book 2

. . . writtenBy

authorOf

writtenBy

writtenBy

authorOf

authorOf

(19)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1 author 1 book 2 . . .

writtenBy authorOf

writtenBy writtenBy authorOf

authorOf

(20)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1

author 1 book 2 . . .

writtenBy authorOf writtenBy writtenBy authorOf

authorOf

(21)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1 author 1

book 2 . . .

writtenBy

authorOf writtenBy writtenBy authorOf

authorOf

(22)

Datalog First

book(x) → ∃v . writtenBy(x, v ) ∧ author(v ) author(x) → ∃w . authorOf(x, w ) ∧ book(w ) authorOf(x, y ) → writtenBy(y , x)

writtenBy(x, y ) → authorOf(y , x )

book 1 author 1

book 2 . . . writtenBy authorOf writtenBy

writtenBy

authorOf

authorOf

(23)

Datalog First

book(x ) → ∃v , i . writtenBy(x , v , i ) ∧ author(v ) author(x ) → ∃w , i . authorOf(x , w , i ) ∧ book(w ) authorOf(x , y , j ) → ∃i . writtenBy(y , x , i )

writtenBy(x , y , j ) → ∃i . authorOf(y , x , i)

book 1 author 1 book 2 . . .

writtenBy authorOf writtenBy writtenBy authorOf

authorOf

(24)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

J

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(25)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

J

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(26)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

J

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 )

h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(27)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(28)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

J

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(29)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 )

h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(30)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

J

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(31)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(32)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

J

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

I b = I a ∪ {stars(A 2 , M 2 )}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m)

bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(33)

Reliances

Positive Reliances: ρ 1 ≺ + ρ 2 Restraint Reliances: ρ 1 ≺ ρ 2

I a

I b

@ J @

h 1 (ρ 1 )

h 2 (ρ 2 ) h 2 (ρ 2 )

I a = {leadRole(A 2 , R 1 , M 2 )}

I b = I a ∪ {stars(A 2 , M 2 )}

leadRole(a, r , m) → stars(a, m)

stars(a, m) ∧ stars(b, m)

→ costar(a, b, m)

J

I a

I b

h 2 (ρ 2 )

h 1 (ρ 1 )

h 2 alt

I a = J ∪ {stars(n, M 2 ), famous(n)}

J = {bigBudget(M 2 ), famous(A 2 )}

leadRole(a, r, m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

(34)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(35)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(36)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(37)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(38)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(39)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(40)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(41)

Calculating Reliances – Unification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

(42)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z }

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(43)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z }

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(44)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z }

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(45)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy ,

v 7→ n v , y 2 7→ n v , z 1 7→ c z }

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(46)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v ,

z 1 7→ c z }

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(47)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z } I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(48)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z } I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(49)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z } I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

(50)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z } I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

I b 6| = ∃w. a(w , w) ∧ b(c xy )

(51)

Calculating Reliances – Verification

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

1 Compute unifer η

2 I a = ϕ 1 η ∪ ϕ 22 η

3 I b = I a ∪ ψ 1 η

4 Check I b 6| = ψ 2 η

η = {x 2 /x 1 /y 1 7→ c xy , v 7→ n v , y 2 7→ n v , z 1 7→ c z } I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

I b | = ∃w. a(w , w)

(52)

Calculating Reliances – Completeness

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

J a = {r (7), a(1, 1), b(2), p(1, 1)}

J b = J a ∪ {p(1, n v ), q(n v , 1)}

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

h(ρ 1 ) η(ρ 1 )

τ = {c xy 7→ 1, c z 7→ 2}

(53)

Calculating Reliances – Completeness

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

J a = {r (7), a(1, 1), b(2), p(1, 1)}

J b = J a ∪ {p(1, n v ), q(n v , 1)}

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

h(ρ 1 )

η(ρ 1 )

τ = {c xy 7→ 1, c z 7→ 2}

(54)

Calculating Reliances – Completeness

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

J a = {r (7), a(1, 1), b(2), p(1, 1)}

J b = J a ∪ {p(1, n v ), q(n v , 1)}

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

h(ρ 1 ) η(ρ 1 )

τ = {c xy 7→ 1, c z 7→ 2}

(55)

Calculating Reliances – Completeness

a(x 1 , y 1 ) ∧ b(z 1 ) →∃v . p(x 1 , v ) ∧ q(v , y 1 )

p(x 2 , x 2 ) ∧ p(x 2 , y 2 ) ∧ q(y 2 , x 2 ) → ∃w . a(w , w ) ∧ b(x 2 )

J a = {r (7), a(1, 1), b(2), p(1, 1)}

J b = J a ∪ {p(1, n v ), q(n v , 1)}

I a = {a(c xy , c xy ), b(c z ), p(c xy , c xy )}

I b = I a ∪ {p(c xy , n v ), q(n v , c xy )}

h(ρ 1 ) η(ρ 1 )

(56)

Rule Order – Ideal Case

stars(a, m) ∧ stars(b, m) → costar(a, b, m) leadRole(a, r , m) → stars(a, m) bigBudget(m)

→ ∃a. stars(a, m) ∧ famous(a)

+ +

Core-stratified = No cycle containing a ≺ -reliance Possible to compute a core model

Possible to apply every rule only once

(57)

Rule Order – Non-Stratified Case

+

+

+

+

+ +

+

+

rule

applicable

selected

(58)

Rule Order – Non-Stratified Case

+

+

+

+

+ +

+

+

rule

applicable

selected

(59)

Rule Order – Non-Stratified Case

+

+

+

+

+ +

+

+

rule

applicable

selected

(60)

Rule Order – Non-Stratified Case

+

+

+ +

+

+

rule

applicable

selected

(61)

Rule Order – Non-Stratified Case

+ +

+

+

rule

applicable

selected

(62)

Rule Order – Non-Stratified Case

+ +

+

+

rule

applicable

selected

(63)

Rule Order – Non-Stratified Case

+ +

+

+

rule

applicable

selected

(64)

Rule Order – Non-Stratified Case

rule

applicable

selected

(65)

Rule Order – Non-Stratified Case

+ +

+

+

rule

applicable

selected

(66)

Unrestrained First

book(x ) → ∃v , i . writtenBy(x , v , i ) ∧ author(v ) author(x ) → ∃w , i . authorOf(x , w , i ) ∧ book(w ) authorOf(x , y , j ) → ∃i . writtenBy(y , x , i )

writtenBy(x , y , j ) → ∃i . authorOf(y , x , i)

(67)

Evaluation

Implemented reliance calculating and ordering strategy into VLog Measured performance on several knowledge bases

ChaseBench: Deep , LUBM , Ontology-256 , STB-128 , Doctors , UOBM

Real World: UniProt , Reactome

Created: Cycle

(68)

Average Rule Applications

Cycle Deep LUBM UniProt

0 5 10 15

Avg. #Rule A p plications

VLog

PosFirst

UnresFirst

(69)

Time Measurements

1 2 3 4 5

Sp eedup

PosFirst

(70)

Alternative Matches

Dataset Strat. Facts PosFst UnresFst

Sel. Delta Sel. Delta

Cycle 3 0.4M 0 0 0 0

Doctors 3 0.8M 0 0 0 0

Reactome 3 11.4M 0 0 0 0

STB 3 1.9M 0 0 0 0

UniProt 3 24.2M 0 0 0 0

LUBM 7 18.7M 3 0 0 0

Deep 7 0.9M 1148 +19K 2032 -0.5K

Ontology 7 5.7M 123 0 64 0

UOMB 7 18.6M 21 +35K 6 -12K

(71)

Cost of Computing Reliances

96.4 % VLog

3.6 %

Reliances

(72)

Summary

Summary:

We optimized the order of rule execution by analysing the relationships between rules of a given program

Positive reliance: Avoids unnecessary rule executions Restraint reliance: Avoids redundant facts

Strategy shows potential for improving run times No significant reduction in the number of derived facts Computation reliances is cheap even on larger rule sets Future work:

Repeat experiments with larger input facts or other rule sets Problem: Strategy might exclude terminating runs

Analyze factors that influence run time outside of reliances

Referenzen

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