Chase-Based Computation of Cores for Existential Rules

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Chase-Based Computation of Cores for Existential Rules

Lukas Gerlach

Knowledge-Based Systems Group Technische Universit ¨at Dresden

16.09.2021

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Existential Rules

∀ ~ x ∀ ~ y .Body( ~ x , ~ y ) → ∃ ~ z.Head( ~ x , ~ z )

• Body andHead: conjunctions of atoms

• ~x, ~y, ~z: pairwise disjoint lists of variables

Pizza(x) → ∃z.SameDeliverer(x , z) ∧ Pizza(z ) WeeklyOrder(y , x) → ∃z.WeeklyOrder(x , z)

Pizza(x) ∧ WeeklyOrder(x , y) → Pizza(y) ∧ SameDeliverer(x , y)

Lukas Gerlach Chase-Based Computation of Cores for Existential Rules 16.09.2021

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Existential Rules

∀ ~ x ∀ ~ y .Body( ~ x , ~ y ) → ∃ ~ z.Head( ~ x , ~ z )

Pizza(x) → ∃z.SameDeliverer(x , z) ∧ Pizza(z ) WeeklyOrder(y , x) → ∃z.WeeklyOrder(x , z)

Pizza(x) ∧ WeeklyOrder(x , y) → Pizza(y) ∧ SameDeliverer(x , y)

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Existential Rules

∀ ~ x ∀ ~ y .Body( ~ x , ~ y ) → ∃ ~ z.Head( ~ x , ~ z )

Pizza(x) → ∃z.SameDeliverer(x , z) ∧ Pizza(z)

WeeklyOrder(y , x) → ∃z.WeeklyOrder(x , z)

Pizza(x) ∧ WeeklyOrder(x , y) → Pizza(y) ∧ SameDeliverer(x , y)

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Existential Rules

∀ ~ x ∀ ~ y .Body( ~ x , ~ y ) → ∃ ~ z.Head( ~ x , ~ z )

Pizza(x) → ∃z.SameDeliverer(x , z) ∧ Pizza(z) WeeklyOrder(y , x) → ∃z.WeeklyOrder(x , z)

Pizza(x) ∧ WeeklyOrder(x , y) → Pizza(y) ∧ SameDeliverer(x , y)

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Existential Rules

∀ ~ x ∀ ~ y .Body( ~ x , ~ y ) → ∃ ~ z.Head( ~ x , ~ z )

Pizza(x) → ∃z.SameDeliverer(x , z) ∧ Pizza(z) WeeklyOrder(y , x) → ∃z.WeeklyOrder(x , z)

Pizza(x) ∧ WeeklyOrder(x , y) → Pizza(y) ∧ SameDeliverer(x , y)

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Chasing a Universal Core Model

Pizza(x)→ ∃z.SameDeliverer(x,z)∧Pizza(z) WeeklyOrder(y,x)→ ∃z.WeeklyOrder(x,z)

Pizza(x)∧WeeklyOrder(x,y)→Pizza(y)∧SameDeliverer(x,y)

order1:Pizza n₁:Pizza SameDeliverer

n2:Pizza SameDeliverer

order2

:Pizza

WeeklyOrder

n3

:Pizza

WeeklyOrder SameDeliverer

SameDeliverer

. . . .

Therestricted chaseand it yields auniversal model. Thecore chaseand it yields auniversal core model.

Withoutalternative matches, therestricted chasealso yields auniversal core model.

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Chasing a Universal Core Model

order1:Pizza

n₁:Pizza SameDeliverer

order2

:Pizza

WeeklyOrder

n3

:Pizza

SameDeliverer

. . . .

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Chasing a Universal Core Model

order2

:Pizza

WeeklyOrder

n3

:Pizza

SameDeliverer

. . . .

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Chasing a Universal Core Model

order2

:Pizza

WeeklyOrder

n3

:Pizza

SameDeliverer

. . . .

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Chasing a Universal Core Model

order2

:Pizza

WeeklyOrder

n3

:Pizza

WeeklyOrder

SameDeliverer

. . . .

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Chasing a Universal Core Model

order2:Pizza WeeklyOrder

n3

:Pizza

SameDeliverer

. . . .

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Chasing a Universal Core Model

n3

:Pizza

SameDeliverer

. . . .

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Chasing a Universal Core Model

n3:Pizza WeeklyOrder

SameDeliverer

. . . .

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Chasing a Universal Core Model

SameDeliverer

. . . .

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Chasing a Universal Core Model

SameDeliverer

. . . .

Therestricted chaseand it yields auniversal model.

Thecore chaseand it yields auniversal core model.

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Chasing a Universal Core Model

SameDeliverer

. . . .

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Chasing a Universal Core Model

SameDeliverer

. . . .

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Some Rule Set Classifications

R∈CT^res_∀∃

For the rule setRandallstarting fact sets,somerestrictedchase sequence terminates.

CT^res_∀∀ ⊂CT^res_∀∃⊂CT^core_∀ Each of these classes is undecidable.

R∈AM∀∃

For the rule setRandallstarting fact sets,somerestricted chase sequence does not have an alternative match.

AM∀∀⊂AM∀∃

AM∀∃is undecidable whereas AM∀∀is decidable.

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Some Rule Set Classifications

R∈CT^res_∀∃

CT^res_∀∀ ⊂CT^res_∀∃⊂CT^core_∀

Each of these classes is undecidable.

R∈AM∀∃

AM∀∀⊂AM∀∃

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Some Rule Set Classifications

R∈CT^res_∀∃

R∈AM∀∃

AM∀∀⊂AM∀∃

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Some Rule Set Classifications

R∈CT^res_∀∃

R∈AM∀∃

AM∀∀⊂AM∀∃

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Some Rule Set Classifications

R∈CT^res_∀∃

R∈AM∀∃

AM∀∀⊂AM∀∃

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Some Rule Set Classifications

R∈CT^res_∀∃

R∈AM∀∃

AM∀∀⊂AM∀∃

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Relations between Rules [Kr ¨otzsch, 2020]

Definition

A ruleρrestrainsa ruleρ⁰, writtenρ≺ρ⁰, if the application ofρafterρ⁰may introduce an alternative match forρ⁰.

Proposition

Consider a chase. If for each ruleρ, all rulesρ⁰withρ⁰≺ρare applied exhaustively beforeρ, then the chase does not have an alternative match.

Proposition

A rule setRdoes not have restraining relations iffR∈AM∀∀.

If we respect restraining relations, we find a restricted chase sequence that yields a core.

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Relations between Rules [Kr ¨otzsch, 2020]

Definition

Proposition

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Relations between Rules [Kr ¨otzsch, 2020]

Definition

Proposition

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Relations between Rules [Kr ¨otzsch, 2020]

Definition

Proposition

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Relations between Rules [Kr ¨otzsch, 2020]

Definition

A ruleρ⁰positively relieson a ruleρ, writtenρ≺⁴ρ⁰, if the application ofρmay allowρ⁰to be applied.

Definition

Thedownward closureρ↓of a ruleρis the set containing each ruleρ⁰for that we find ρ⁰((≺⁴)^∗◦ ≺)⁺ρ, i.e.ρ⁰directly or indirectly restrainsρ.

Definition

A rule set iscore-stratifiedif for every ruleρ, we haveρ /∈ρ↓.

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Relations between Rules [Kr ¨otzsch, 2020]

Definition

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Relations between Rules [Kr ¨otzsch, 2020]

Definition

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The Power of Core Stratification

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Core-Stratification is sufficient for AM _∀∃

Pizza(x)∧WeeklyOrder(x,y)→Pizza(y)∧SameDeliverer(x,y) We findρ₃≺ρ₁,ρ₂≺⁴ρ₃,ρ₃≺⁴ρ₁andρ_i ≺⁴ρ_i for everyi ∈ {1,2,3}.

Thus, we haveρ₁↓ ={ρ₂,ρ₃}andρ₂↓=ρ₃↓ =∅.

Theorem

If a rule set R is core stratified, then R∈AM∀∃.

(OriginallyR∈CT^res_∀∀is also required [Kr ¨otzsch, 2020].)

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Core-Stratification is sufficient for AM _∀∃

Pizza(x)∧WeeklyOrder(x,y)→Pizza(y)∧SameDeliverer(x,y) We findρ₃≺ρ₁,ρ₂≺⁴ρ₃,ρ₃≺⁴ρ₁andρ_i ≺⁴ρ_i for everyi ∈ {1,2,3}.

Thus, we haveρ₁↓ ={ρ₂,ρ₃}andρ₂↓=ρ₃↓ =∅.

Theorem

If a rule set R is core stratified, then R∈AM∀∃.

(OriginallyR∈CT^res_∀∀is also required [Kr ¨otzsch, 2020].)

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Avoiding Alternative Matches when Chasing

Pizza(x)→ ∃z.SameDeliverer(x,z)∧Pizza(z) (R₂) WeeklyOrder(y,x)→ ∃z.WeeklyOrder(x,z) (R1) Pizza(x)∧WeeklyOrder(x,y)→Pizza(y)∧SameDeliverer(x,y) (R₁)

order1:Pizza

order2

:Pizza

WeeklyOrder

SameDeliverer

n3

:Pizza

. . .

unhappyOrder:Pizza

. . .

λ¹₁,λ¹₂,λ¹₃,λ¹₄, . . .[,λ²₁,λ²₂, . . .]

This is atransfinite chase sequenceon arestrained partitioning. Observation:ρ₁applications could be done earlier: λ¹₁,λ¹₂,λ²₁,λ¹₃,λ¹₄,λ²₂, . . .

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Avoiding Alternative Matches when Chasing

order1:Pizza

SameDeliverer

n3

:Pizza

. . .

unhappyOrder:Pizza

. . .

λ¹₁

,λ¹₂,λ¹₃,λ¹₄, . . .[,λ²₁,λ²₂, . . .]

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Avoiding Alternative Matches when Chasing

order1:Pizza

SameDeliverer

n3

:Pizza

WeeklyOrder

SameDeliverer

. . .

unhappyOrder:Pizza

. . .

λ¹₁,λ¹₂

,λ¹₃,λ¹₄, . . .[,λ²₁,λ²₂, . . .]

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Avoiding Alternative Matches when Chasing

order1:Pizza

SameDeliverer

. . .

unhappyOrder:Pizza

. . .

λ¹₁,λ¹₂,λ¹₃

,λ¹₄, . . .[,λ²₁,λ²₂, . . .]

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Avoiding Alternative Matches when Chasing

order1:Pizza

SameDeliverer

. . . unhappyOrder:Pizza

. . .

λ¹₁,λ¹₂,λ¹₃,λ¹₄, . . .

[,λ²₁,λ²₂, . . .]

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Avoiding Alternative Matches when Chasing

order1:Pizza

SameDeliverer

. . .

λ¹₁,λ¹₂,λ¹₃,λ¹₄, . . .[,λ²₁,λ²₂, . . .]

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Avoiding Alternative Matches when Chasing

order1:Pizza

SameDeliverer

. . .

λ¹₁,λ¹₂,λ¹₃,λ¹₄, . . .[,λ²₁,λ²₂, . . .]

This is atransfinite chase sequenceon arestrained partitioning.

Observation:ρ₁applications could be done earlier: λ¹₁,λ¹₂,λ²₁,λ¹₃,λ¹₄,λ²₂, . . .

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Avoiding Alternative Matches when Chasing

order1:Pizza

SameDeliverer

. . .

λ¹₁,λ¹₂,λ¹₃,λ¹₄, . . .[,λ²₁,λ²₂, . . .]

This is atransfinite chase sequenceon arestrained partitioning.

Observation:ρ₁applications could be done earlier: λ¹₁,λ¹₂,λ²₁,λ¹₃,λ¹₄,λ²₂, . . .

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Restricted and Core Chase coincide

Theorem

For a rule set R∈AM∀∃, we have R∈CT^res_∀∃iff R∈CT^core_∀ .

Corollary

A transfinite chase sequence on a restrained partitioning terminates (yielding a finite universal core model) iff a finite universal (core) model exists.

Theorem

For a rule set R∈AM∀∀, we have R∈CT^res_∀∀iff R∈CT^res_∀∃iff R∈CT^core_∀ . (CT^res_∀∀is decidable for single-head guarded existential rules.)

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Restricted and Core Chase coincide

Theorem

Corollary

Theorem

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Restricted and Core Chase coincide

Theorem

Corollary

Theorem

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

For a rule set of only single-head rules, if there exists an unfair non-terminating restricted chase sequence, then there exists a fair non-terminating restricted chase sequence.

The problem of the Fairness Theorem [Gogacz et al., 2020]: S(a,b,b)

ρ₁:=S(x,y,y)→ ∃z.S(x,z,y)∧S(z,y,y) ρ₂:=S(x,y,z)→S(z,z,z)

Byρ₁, we obtain:

S(a,n1,b),S(n1,b,b),S(n1,n2,b),S(n2,b,b), . . .

Any application ofρ₂yieldsS(b,b,b)and blocks all (further) applications ofρ₁. ρ₂restrainsρ₁and no infinite fair sequence exists.

However:There are also single-head rules that restrain each other.

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

The problem of the Fairness Theorem [Gogacz et al., 2020]:

S(a,b,b)

Byρ₁, we obtain:

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

S(a,n1,b),S(n1,b,b)

,S(n1,n2,b),S(n2,b,b), . . .

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

S(a,n1,b),S(n1,b,b),S(n1,n2,b),S(n2,b,b)

, . . .

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

Any application ofρ₂yieldsS(b,b,b)and blocks all (further) applications ofρ₁.

ρ₂restrainsρ₁and no infinite fair sequence exists.

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

S(a,b,b)

Byρ₁, we obtain:

Any application ofρ₂yieldsS(b,b,b)and blocks all (further) applications ofρ₁. ρ₂strongly restrainsρ₁and no infinite fair sequence exists.

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

Theorem

For a rule set without strong restraining relations, if there exists an unfair non-terminating restricted chase sequence, then there exists a fair non-terminating restricted chase sequence.

Conjecture

Consider a guarded rule setRwithout strong restraining relations. It is decidable if R∈CT^res_∀∀.

(We obtain decidability for CT^core_∀ for guarded rule sets without restraining relations.)

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

Theorem

Conjecture

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Decidability of CT ^res _∀∀

Proposition (Fairness Theorem [Gogacz et al., 2020])

Theorem

Conjecture

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Computing Cores for Non-Core-Stratified Rule Sets

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Computing Cores directly with Alternative Matches

order1:Pizza n1:Pizza SameDeliverer

n₂:Pizza SameDeliverer

. . .

SameDeliverer

. . . n₄:Pizza

SameDeliverer . . .

SameDeliverer

Problem: Alternative Matches do not always yield an endomorphism over the fact set.

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Computing Cores directly with Alternative Matches

order1:Pizza n1:Pizza SameDeliverer

n₂:Pizza SameDeliverer

. . .

SameDeliverer

. . . n₄:Pizza

SameDeliverer . . .

SameDeliverer

Problem: Alternative Matches do not always yield an endomorphism over the fact set.

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Compute Cores directly with Alternative Matches

→ ∃z.P(z) P(x)→ ∃z.Q(x,z)

Q(x,y)→ ∃z.Q(z,y)∧Q(z,c)∧P(z)∧S(z,y) Q(x,y)∧S(x,z)→S(x,y)

c

n₁:P

n2

Q n₃:P

Q S Q

S

n₄:P Q

S

Problem:After remappings of nulls, other remappings may be necessary that are not captured by alternative matches.

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Compute Cores directly with Alternative Matches

→ ∃z.P(z) P(x)→ ∃z.Q(x,z)

c n₁:P

n2

Q n₃:P

Q S Q

S

n₄:P Q

S

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Compute Cores directly with Alternative Matches

→ ∃z.P(z) P(x)→ ∃z.Q(x,z)

c n₁:P

n2

Q

n₃:P Q

S Q

S

n₄:P Q

S

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Compute Cores directly with Alternative Matches

→ ∃z.P(z) P(x)→ ∃z.Q(x,z)

c n₁:P

n2

Q n₃:P

Q S Q

S

n₄:P Q

S

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Compute Cores directly with Alternative Matches

→ ∃z.P(z) P(x)→ ∃z.Q(x,z)

c n₁:P

n2

Q n₃:P

Q S Q

S

n₄:P Q

S

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Compute Cores directly with Alternative Matches

→ ∃z.P(z) P(x)→ ∃z.Q(x,z)

c n₁:P

n2

Q n₃:P

Q S Q

S

n₄:P Q

S

Problem:After remappings of nulls, other remappings may be necessary

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

Q(x,y)→ ∃z.Q(z,y)∧Q(z,c)∧P(z)∧S(z,y) (R2)

Q(x,y)∧S(x,z)→S(x,y) (R₁)

Thehybrid chaseon arelaxed restrained partitioningis defined like the transfinite chase but uses the core chase in the last sequence.

c a:P

S

n2

Q S

n3:P Q

S Q

S

n4:P Q

S

Nulls that are introduced before the last sequence can be treated as constants.

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

c a:P

S

n2

Q

S

n3:P Q

S Q

S

n4:P Q

S

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

c a:P

S

n2

Q S

n3:P Q

S Q

S

n4:P Q

S

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

c a:P

S

n2

Q S

n3:P Q

S Q

S

n4:P Q

S

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

c a:P

S

n2

Q S

n3:P Q

S Q

S

n4:P Q

S

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

c a:P

S

n2

Q S

n3:P Q

S Q

S

n4:P Q

S

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

c a:P

S

n2

Q S

n3:P Q

S Q

S

n4:P Q

S

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The Hybrid Chase

P(x)→ ∃z.Q(x,z) (R₁)

c a:P

S

n2

Q S

n3:P Q

S Q

S

n4:P Q

S

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Summary

Results:

• Restricted and core chase coincide for core-stratified rule sets.

• Conjecture: Slightly larger fragment of guarded rules for which CT^res_∀∀ is decidable.

• Ideas for more efficient computation of universal core models for arbitrary rule sets.

Open Questions / Future Work:

• Is AM∀∃decidable for (single-head) guarded existential rules?

• Is CT^res_∀∃ decidable for (single-head) guarded existential rules?

• Verify decidability of CT^res_∀∀ for guarded rules without strong restraining relations.

• Implement/Evaluate/Improve core computation heuristic and hybrid chase.

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Summary

Results:

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Summary

Results:

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Chase-Based Computation of Cores for Existential Rules