Combining Existential Rules and Description Logics
Antoine Amarilli1,2, Michael Benedikt2
1Télécom ParisTech, Paris, France
2University of Oxford, Oxford, United Kingdom
July 28, 2015
1/8
Open-world query answering (QA)
Open-worldquery answering:
We are given:
RelationalinstanceI(ground facts)
! LogicalconstraintsΣ
? Boolean conjunctivequeryq
We ask:
Consider all possiblecompletionsJ⊇I
Restrict to those that satisfy theconstraintsΣ
→ Isqcertainamong them?
Open-world query answering (QA)
Open-worldquery answering:
We are given:
RelationalinstanceI(ground facts)
! LogicalconstraintsΣ
? Boolean conjunctivequeryq We ask:
Consider all possiblecompletionsJ⊇I
Restrict to those that satisfy theconstraintsΣ
→ Isqcertainamong them?
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Open-world query answering (QA)
Open-worldquery answering: –query entailment orcontainment We are given:
RelationalinstanceI(ground facts) –A-Box
! LogicalconstraintsΣ–T-Box
? Boolean conjunctivequeryq We ask:
Consider all possiblecompletionsJ⊇I
Restrict to those that satisfy theconstraintsΣ
→ Isqcertainamong them?
Decidable constraint languages for QA
Rich description logics (DLs) Frontier-guarded existential rules
Emp⊑CEO⊔(∃Mgr−.Emp) ∀pwv Acpt(p,w,v) →∃fTrip(p,f,v) Arity-twoonly Arbitraryarity
Rich(disjunction, etc.) Poor(conjunction and implication) Functionality asserts
Funct(Mgr−) ! n/a
→ QA isdecidable foreither language
3/8
Decidable constraint languages for QA
Rich description logics (DLs) Frontier-guarded existential rules Emp ⊑CEO⊔(∃Mgr−.Emp) ∀pwv Acpt(p,w,v) →∃f Trip(p,f,v)
Arity-twoonly Arbitraryarity
Rich(disjunction, etc.) Poor(conjunction and implication) Functionality asserts
Funct(Mgr−) ! n/a
→ QA isdecidable foreither language
Decidable constraint languages for QA
Rich description logics (DLs) Frontier-guarded existential rules
Emp ⊑CEO⊔(∃Mgr−.Emp) ∀pwv Acpt(p,w,v) →∃f Trip(p,f,v)
Arity-two only Arbitraryarity
Rich(disjunction, etc.) Poor(conjunction and implication) Functionality asserts
Funct(Mgr−) ! n/a
→ QA isdecidable foreither language
3/8
Decidable constraint languages for QA
Rich description logics (DLs) Frontier-guarded existential rules
Emp ⊑CEO⊔(∃Mgr−.Emp) ∀pwv Acpt(p,w,v) →∃f Trip(p,f,v)
Arity-two only Arbitraryarity
Rich (disjunction, etc.) Poor(conjunction and implication)
Functionality asserts
Funct(Mgr−) ! n/a
→ QA isdecidable foreither language
Decidable constraint languages for QA
Rich description logics (DLs) Frontier-guarded existential rules
Emp ⊑CEO⊔(∃Mgr−.Emp) ∀pwv Acpt(p,w,v) →∃f Trip(p,f,v)
Arity-two only Arbitraryarity
Rich (disjunction, etc.) Poor(conjunction and implication) Functionality asserts
Funct(Mgr−) ! n/a
→ QA isdecidable foreither language
3/8
Decidable constraint languages for QA
Rich description logics (DLs) Frontier-guarded existential rules
Emp ⊑CEO⊔(∃Mgr−.Emp) ∀pwv Acpt(p,w,v) →∃f Trip(p,f,v)
Arity-two only Arbitraryarity
Rich (disjunction, etc.) Poor(conjunction and implication) Functionality asserts
Funct(Mgr−) ! n/a
→ QA isdecidable foreither language
Our problem
Can we have the best of both worlds?
QA is decidable forrich DLs(i.e., expressible in GC2, guarded two-variable first-order logic with counting) QA is decidable forfrontier-guarded existential rules
→ Is QA decidable for rich DLs+ someclasses of rules? We show:
QA isundecidable for richDLsand frontier-guarded rules QA with rich DLs isdecidable for some newrule classes Functional dependencies can be added under someconditions
4/8
Our problem
Can we have the best of both worlds?
QA is decidable forrich DLs(i.e., expressible in GC2, guarded two-variable first-order logic with counting) QA is decidable forfrontier-guarded existential rules
→ Is QA decidable for rich DLs+ someclasses of rules?
We show:
QA isundecidable for richDLsand frontier-guarded rules QA with rich DLs isdecidable for some newrule classes Functional dependencies can be added under someconditions
Our problem
Can we have the best of both worlds?
QA is decidable forrich DLs(i.e., expressible in GC2, guarded two-variable first-order logic with counting) QA is decidable forfrontier-guarded existential rules
→ Is QA decidable for rich DLs+ someclasses of rules?
We show:
QA isundecidable for richDLsand frontier-guarded rules QA with rich DLs isdecidable for some newrule classes Functional dependencies can be added under someconditions
4/8
Our problem
Can we have the best of both worlds?
QA is decidable forrich DLs(i.e., expressible in GC2, guarded two-variable first-order logic with counting) QA is decidable forfrontier-guarded existential rules
→ Is QA decidable for rich DLs+ someclasses of rules?
We show:
QA isundecidable for richDLsand frontier-guarded rules QA with rich DLs isdecidable for some newrule classes Functional dependencies can be added under someconditions
Restricting the language
Theorem
QA isundecidablefor rich DLs and frontier-guarded rules
Problem: inclusion dependencies + Funct = ID/FD implication
→ Frontier-one rules: ∀xyϕ(x,y)→ ∃z ψ(x,z)
Theorem
QA isundecidablefor rich DLs and frontier-one rules Problem: cycles in rules + Funct = grid
→ Non-looping rules: prohibitcycles R(x,y)S(y,z)T(z,x)
R(x,y,z)S(x,y,z)
5/8
Restricting the language
Theorem
QA isundecidablefor rich DLs and frontier-guarded rules Problem: inclusion dependencies + Funct = ID/FD implication
→ Frontier-one rules: ∀xyϕ(x,y)→ ∃z ψ(x,z)
Theorem
QA isundecidablefor rich DLs and frontier-one rules Problem: cycles in rules + Funct = grid
→ Non-looping rules: prohibitcycles R(x,y)S(y,z)T(z,x)
R(x,y,z)S(x,y,z)
Restricting the language
Theorem
QA isundecidablefor rich DLs and frontier-guarded rules
Problem: inclusion dependencies + Funct = ID/FD implication
→ Frontier-one rules: ∀xyϕ(x,y)→ ∃z ψ(x,z)
Theorem
QA isundecidablefor rich DLs and frontier-one rules Problem: cycles in rules + Funct = grid
→ Non-looping rules: prohibitcycles R(x,y)S(y,z)T(z,x)
R(x,y,z)S(x,y,z)
5/8
Restricting the language
Theorem
QA isundecidablefor rich DLs and frontier-guarded rules
Problem: inclusion dependencies + Funct = ID/FD implication
→ Frontier-one rules: ∀xyϕ(x,y)→ ∃z ψ(x,z)
Theorem
QA isundecidablefor rich DLs and frontier-one rules
Problem: cycles in rules + Funct = grid
→ Non-looping rules: prohibitcycles R(x,y)S(y,z)T(z,x)
R(x,y,z)S(x,y,z)
Restricting the language
Theorem
QA isundecidablefor rich DLs and frontier-guarded rules
Problem: inclusion dependencies + Funct = ID/FD implication
→ Frontier-one rules: ∀xyϕ(x,y)→ ∃z ψ(x,z)
Theorem
QA isundecidablefor rich DLs and frontier-one rules Problem: cycles in rules + Funct = grid
→ Non-looping rules: prohibitcycles R(x,y)S(y,z)T(z,x)
R(x,y,z)S(x,y,z)
5/8
Restricting the language
Theorem
QA isundecidablefor rich DLs and frontier-guarded rules
Problem: inclusion dependencies + Funct = ID/FD implication
→ Frontier-one rules: ∀xyϕ(x,y)→ ∃z ψ(x,z)
Theorem
QA isundecidablefor rich DLs and frontier-one rules Problem: cycles in rules + Funct = grid
→ Non-looping rules: prohibitcycles R(x,y)S(y,z)T(z,x)
QA decidability
Non-loopingfrontier-one rules: no cycles in body and in head
→ We can shred them to DL rules Theorem
QA isdecidablefor non-looping frontier-one rules + rich DLs
Head-non-loopingfrontier-one rules: no cycles in head
→ We can treeify the rules, soundness byunravellingthe models Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
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QA decidability
Non-loopingfrontier-one rules: no cycles in body and in head
→ We can shredthem to DL rules Theorem
QA isdecidablefor non-looping frontier-one rules + rich DLs
Head-non-loopingfrontier-one rules: no cycles in head
→ We can treeify the rules, soundness byunravellingthe models Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
QA decidability
Non-loopingfrontier-one rules: no cycles in body and in head
→ We can shredthem to DL rules Theorem
QA isdecidablefor non-looping frontier-one rules + rich DLs
Head-non-loopingfrontier-one rules: no cycles in head
→ We can treeify the rules, soundness byunravellingthe models Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
6/8
QA decidability
Non-loopingfrontier-one rules: no cycles in body and in head
→ We can shredthem to DL rules Theorem
QA isdecidablefor non-looping frontier-one rules + rich DLs
Head-non-loopingfrontier-one rules: no cycles in head
→ We can treeify the rules, soundness byunravellingthe models Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
Adding functional dependencies
Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
We want to add:
Functional dependencies(FDs) onarbitrary predicates: Talk[speaker,session]determines Talk[title]
FDs plus single-head frontier-onerules already undecidable
→ Must impose thenon-conflicting condition
Theorem
Decidable QAfor: Rich DLconstraints
Single-head frontier-one rules Non-conflicting FDs
7/8
Adding functional dependencies
Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
We want to add:
Functional dependencies(FDs) onarbitrary predicates:
Talk[speaker,session]determines Talk[title]
FDs plus single-head frontier-onerules already undecidable
→ Must impose thenon-conflicting condition
Theorem
Decidable QAfor: Rich DLconstraints
Single-head frontier-one rules Non-conflicting FDs
Adding functional dependencies
Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
We want to add:
Functional dependencies(FDs) onarbitrary predicates:
Talk[speaker,session]determines Talk[title]
FDs plus single-head frontier-onerules already undecidable
→ Must impose thenon-conflicting condition
Theorem
Decidable QAfor: Rich DLconstraints
Single-head frontier-one rules Non-conflicting FDs
7/8
Adding functional dependencies
Theorem
QA isdecidablefor head-non-looping frontier-one rules + rich DLs
We want to add:
Functional dependencies(FDs) onarbitrary predicates:
Talk[speaker,session]determines Talk[title]
FDs plus single-head frontier-onerules already undecidable
→ Must impose thenon-conflicting condition
Theorem
Decidable QAfor:
Rich DLconstraints
Summary of results
Combining Existential Rules and Description Logics
Open-worldquery answering(QA) under: RichDLconstraints
Existential rules
For which rule classes is QA decidablewith rich DLs?
→ Must restrict tofrontier-one rules
→ Must prohibit cyclesin rule heads
→ QA isdecidable for head-non-looping frontier-one + rich DLs
→ Can add non-conflicting FDs
Thanks for your attention! More details: see poster 76
8/8
Summary of results
Combining Existential Rules and Description Logics
Open-worldquery answering(QA) under:
RichDLconstraints Existential rules
For which rule classes is QA decidablewith rich DLs?
→ Must restrict tofrontier-one rules
→ Must prohibit cyclesin rule heads
→ QA isdecidable for head-non-looping frontier-one + rich DLs
→ Can add non-conflicting FDs
Thanks for your attention! More details: see poster 76
Summary of results
Combining Existential Rules and Description Logics
Open-worldquery answering(QA) under:
RichDLconstraints Existential rules
For which rule classes is QA decidablewith rich DLs?
→ Must restrict tofrontier-one rules
→ Must prohibit cyclesin rule heads
→ QA isdecidable for head-non-looping frontier-one + rich DLs
→ Can add non-conflicting FDs
Thanks for your attention! More details: see poster 76
8/8
Summary of results
Combining Existential Rules and Description Logics
Open-worldquery answering(QA) under:
RichDLconstraints Existential rules
For which rule classes is QA decidablewith rich DLs?
→ Must restrict tofrontier-one rules
→ Must prohibit cyclesin rule heads
→ QA isdecidable for head-non-looping frontier-one + rich DLs
→ Can add non-conflicting FDs
Thanks for your attention! More details: see poster 76
Summary of results
Combining Existential Rules and Description Logics
Open-worldquery answering(QA) under:
RichDLconstraints Existential rules
For which rule classes is QA decidablewith rich DLs?
→ Must restrict tofrontier-one rules
→ Must prohibit cyclesin rule heads
→ QA isdecidable for head-non-looping frontier-one + rich DLs
→ Can add non-conflicting FDs
Thanks for your attention!
More details: see poster 76
8/8
