REWRITING ALCHIQ TO DISJUNCTIVE EXISTENTIAL RULES

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REWRITING ALCH IQ TO DISJUNCTIVE EXISTENTIAL RULES

David Carral Markus Krötzsch Knowledge-Based Systems

TU Dresden

Full paper and video at https://tud.link/h5l5

IJCAI 2020

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Rewriting DLs to Rules

Given a theory T

₁

in a logic L

₁

and a theory T

₂

in a logic L

₂

, T

₂

is a rewriting of T

₁

if,

T

₁

, F | = ϕ iff T

₂

, F | = ϕ

for every set F of ground facts and every

ground fact ϕ over the signature of T

₁

.

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Rules and DLs

Rule languages we encounter:

• Datalog: the “simplest rules conceivable”, e.g., A(x)∧R(x,y)→B(y)

• Datalog^∨: Datalog +∨in heads

• Datalog^∃: Datalog +∃in heads, a.k.a. existential rules

• Datalog^∨∃:Datalog +∨and∃in heads

The DLALCH IQcan be normalised^∗ to rules of nine forms:

A(x)∧B(x)→C(x) AuBvC A(x)→B(x)∨C(x) AvBtC A(x)∧R(x,y)→B(y) Av ∀R.B A(x)→ ∃y.R(x,y)∧B(y) Av ∃R.B R(x,y)∧R(x,z)→y≈z > v61R.> R(x,y)→S(x,y)∨V(x,y) RvStV R(x,y)∧S(x,y)→V(x,y) RuSvV R(y,x)→S(x,y) R⁻vS

A(x)∧R(x,y)∧B(y)→S(x,y) A◦R◦BvS

∗) this is polynomial under unary encoding of numbers

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Rules and DLs

Rule languages we encounter:

• Datalog: the “simplest rules conceivable”, e.g., A(x)∧R(x,y)→B(y)

• Datalog^∨: Datalog +∨in heads

• Datalog^∃: Datalog +∃in heads, a.k.a. existential rules

• Datalog^∨∃:Datalog +∨and∃in heads

The DLALCH IQcan be normalised^∗ to rules of nine forms:

A(x)∧B(x)→C(x) AuBvC A(x)→B(x)∨C(x) AvBtC A(x)∧R(x,y)→B(y) Av ∀R.B A(x)→ ∃y.R(x,y)∧B(y) Av ∃R.B R(x,y)∧R(x,z)→y≈z > v61R.> R(x,y)→S(x,y)∨V(x,y) RvStV R(x,y)∧S(x,y)→V(x,y) RuSvV R(y,x)→S(x,y) R⁻vS

A(x)∧R(x,y)∧B(y)→S(x,y) A◦R◦BvS

∗) this is polynomial under unary encoding of numbers

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Work Source Target Size Rules Hustadt et al. [2007] ALCH IQ Datalog^∨ exp. bounded

Eiter et al. [2012] Horn-SH IQ Datalog exp. bounded Rudolph et al. [2012] SH IQbs Datalog^∨ exp. bounded Bienvenu et al. [2014] SH I Datalog^∨ exp. bounded Carral et al. [2018] Horn-ALCH OIQ Datalog exp. bounded Carral et al. [2019b] Horn-SH IQ Datalog exp. bounded Horn-SRIQ Datalog 2exp. bounded Ortiz et al. [2010] Horn-ALCH OIQ Datalog poly. unbounded Ahmetaj et al. [2016] ALCH IO Datalog^∨ poly. unbounded Krötzsch [2011] EL⁺⁺ Datalog poly. bounded Carral et al. [2019a] Horn-ALC Datalog^∃ poly. bounded

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Work Source Target Size Rules Hustadt et al. [2007] ALCH IQ Datalog^∨ exp. bounded

Eiter et al. [2012] Horn-SH IQ Datalog exp. bounded Rudolph et al. [2012] SH IQbs Datalog^∨ exp. bounded Bienvenu et al. [2014] SH I Datalog^∨ exp. bounded Carral et al. [2018] Horn-ALCH OIQ Datalog exp. bounded Carral et al. [2019b] Horn-SH IQ Datalog exp. bounded Horn-SRIQ Datalog 2exp. bounded Ortiz et al. [2010] Horn-ALCH OIQ Datalog poly. unbounded Ahmetaj et al. [2016] ALCH IO Datalog^∨ poly. unbounded Krötzsch [2011] EL⁺⁺ Datalog poly. bounded Carral et al. [2019a] Horn-ALC Datalog^∃ poly. bounded ALCH IQ Datalog^∨ poly. unbounded ALCH IQ Datalog^∨∃ poly. bounded Horn-ALCH IQ Datalog^∃ poly. bounded

NEW!

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From ALCH IQ to Datalog

^∨

using types

We decomposeALCH IQmodels into structures of bounded size, i.e. “types”:

c C~

d

~D

s1 ~E1

... s` ~E`

~R

~S1

~S_`

A type is given by a fixed number of:

• sets of conceptsC,~ D,~ E~1,. . . ,E~`

• sets of (inverse) relations~R,S~1,. . . ,S~`

• where`is the number ofALCH IQ axioms of formA(x)→ ∃y.R(x,y)∧B(y)

⇒We can represent sets as bit vectors and store types as facts Type(1,0,1,0,1,0,. . .

| {z }

suitably long bit vector

)

AnALCH IQontology is satisfiable iff it admits a consistent set of types.

⇒Datalog^∨encoding: axiomatise required types and consistency conditions

Markus Krötzsch RewritingALCH IQto Disjunctive Existential Rules slide 6 of 9

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From ALCH IQ to Datalog

^∨

using types

c C~

d

~D

s1 ~E1

...

s` ~E`

~R

~S1

~S_`

| {z }

)

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From ALCH IQ to Datalog

^∨

using types

c C~

d

~D

s1 ~E1

...

s` ~E`

~R

~S1

~S_`

| {z }

)

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From ALCH IQ to Datalog

^∨

using types

c C~

d

~D

s1 ~E1

...

s` ~E`

~R

~S1

~S_`

| {z }

)

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

We construct a tableau-like structure:

c A

,A

d B

,B

S

,S

n1

A^¬,

B

S^¬,

R,Succ

n2 A^¬,B Succ,R^¬,S

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

A(x)→A(x) B(x)→B(x) S(x,y)→S(x,y)

n1

A^¬,

B

S^¬,

R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,

B

S^¬,

R,Succ

A(x)→ ∃y.R(x,y)∧B(y)∧Succ(x,y) Av ∃R.B

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

Succ(x,y)→S(x,y)∨S^¬(x,y) Unnamed(x)→A(x)∨A^¬(x)

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬ Succ,R,S^¬

SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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From ALCH IQ to Datalog

^∨∃

by simulating tableau

c A,A

d B,B S,S

n1

A^¬,B S^¬,R,Succ

n3

A,B Succ,R,S

n4

A,B Succ,R,S

n6

A,B^¬

Succ,R,S^¬ SameTyp

Succ,R,S^¬

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Further Results and Outlook

Result Summary: There are polynomial time, fact-preserving rewritings from

• ALCHIQto Datalog^∨

• ALCHIQto Datalog^∨∃

• Horn-ALCH IQto Datalog^∃ (not shown here)

where all translations with∃use rules of bounded size on which the (disjunctive) restricted chase will terminate when prioritising rules without∃

Open Challenges

• Can a chase-based system be worst-case optimal for non-Horn logics?

• Rewritings for more DLs (ALCH OIQanyone?)

• Further exploitation in implementations