Acyclicity Conditions and their Application to Query Answering in Description Logics
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Transcript of Acyclicity Conditions and their Application to Query Answering in Description Logics
![Page 1: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/1.jpg)
ACYCLICITY CONDITIONS AND THEIR
APPLICATION TO QUERY ANSWERING
IN DESCRIPTION LOGICS
Bernardo Cuenca Grau, Ian Horrocks, Markus Krötzsch,Clemens Kupke, Despoina Magka, Boris Motik, Zhe Wang
Department of Computer Science, University of Oxford
June 14, 2012
![Page 2: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/2.jpg)
OUTLINE
1 MOTIVATION
2 MFA AND MSA
3 QUERYING ACYCLIC DL ONTOLOGIES
4 EXPERIMENTAL RESULTS
1
![Page 3: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/3.jpg)
ONTOLOGICAL QUERY ANSWERING
Key reasoning task for DL and rule-based applications
Answering CQs over DLs high computational complexityMaterialisation-based paradigm: chase ABox using TBoxand evaluate Q in the computed ABox
+
ABox A
TBox T
Query Q
T ∪A ⊨ Q ?
ABox A
TBox T
Query Q Chase
T(A)⊨Q ?
MaterialisedABox
Chase
2
![Page 4: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/4.jpg)
ONTOLOGICAL QUERY ANSWERING
Key reasoning task for DL and rule-based applicationsAnswering CQs over DLs high computational complexity
Materialisation-based paradigm: chase ABox using TBoxand evaluate Q in the computed ABox
+
ABox A
TBox T
Query Q
T ∪A ⊨ Q ?
ABox A
TBox T
Query Q Chase
T(A)⊨Q ?
MaterialisedABox
Chase
2
![Page 5: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/5.jpg)
ONTOLOGICAL QUERY ANSWERING
Key reasoning task for DL and rule-based applicationsAnswering CQs over DLs high computational complexityMaterialisation-based paradigm: chase ABox using TBoxand evaluate Q in the computed ABox
+
ABox A
TBox T
Query Q
T ∪A ⊨ Q ?
ABox A
TBox T
Query Q Chase
T(A)⊨Q ?
MaterialisedABox
Chase
2
![Page 6: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/6.jpg)
EXISTENTIAL RULES
Positive, function-free, FOL implications with existentiallyquantified variables in the head
Existential rules fundamental for several KR formalisms:
1 Schema constraints in databases
2 Data transformation rules in data exchange
3 Foundation for Datalog± family of KR languages
4 Ubiquitous in Description Logics
Chase termination is undecidable for existential rules
CQ answering is undecidable for existential rules
3
![Page 7: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/7.jpg)
EXISTENTIAL RULES
Positive, function-free, FOL implications with existentiallyquantified variables in the head
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y) DL-equivalent A v ∃R.B
Existential rules fundamental for several KR formalisms:
1 Schema constraints in databases
2 Data transformation rules in data exchange
3 Foundation for Datalog± family of KR languages
4 Ubiquitous in Description Logics
Chase termination is undecidable for existential rules
CQ answering is undecidable for existential rules
3
![Page 8: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/8.jpg)
EXISTENTIAL RULES
Positive, function-free, FOL implications with existentiallyquantified variables in the head
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y) DL-equivalent A v ∃R.B
Existential rules fundamental for several KR formalisms:
1 Schema constraints in databases
2 Data transformation rules in data exchange
3 Foundation for Datalog± family of KR languages
4 Ubiquitous in Description Logics
Chase termination is undecidable for existential rules
CQ answering is undecidable for existential rules
3
![Page 9: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/9.jpg)
EXISTENTIAL RULES
Positive, function-free, FOL implications with existentiallyquantified variables in the head
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y) DL-equivalent A v ∃R.B
Existential rules fundamental for several KR formalisms:
1 Schema constraints in databases
2 Data transformation rules in data exchange
3 Foundation for Datalog± family of KR languages
4 Ubiquitous in Description Logics
Chase termination is undecidable for existential rules
CQ answering is undecidable for existential rules
3
![Page 10: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/10.jpg)
EXISTENTIAL RULES
Positive, function-free, FOL implications with existentiallyquantified variables in the head
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y) DL-equivalent A v ∃R.B
Existential rules fundamental for several KR formalisms:
1 Schema constraints in databases
2 Data transformation rules in data exchange
3 Foundation for Datalog± family of KR languages
4 Ubiquitous in Description Logics
Chase termination is undecidable for existential rules
CQ answering is undecidable for existential rules
3
![Page 11: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/11.jpg)
EXISTENTIAL RULES
Positive, function-free, FOL implications with existentiallyquantified variables in the head
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y) DL-equivalent A v ∃R.B
Existential rules fundamental for several KR formalisms:
1 Schema constraints in databases
2 Data transformation rules in data exchange
3 Foundation for Datalog± family of KR languages
4 Ubiquitous in Description Logics
Chase termination is undecidable for existential rules
CQ answering is undecidable for existential rules
3
![Page 12: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/12.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 13: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/13.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets
2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 14: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/14.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 15: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/15.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 16: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/16.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 17: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/17.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 18: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/18.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 19: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/19.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 20: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/20.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 21: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/21.jpg)
TACKLING UNDECIDABILITY
1 Identify groups of rules for which query answering isdecidable
Guarded rules, sticky rules, bounded treewidth sets2 Acyclicity conditions: weak acyclicity [Kolaitis et al., ICDT,
2002], super-weak acyclicity [Marnette, PODS, 2009], jointacyclicity [Krötzsch and Rudolph, IJCAI, 2011],. . .
Acyclic set of rules
Guarantees chase termination
Yields finite materialisation
Plus
I No restriction on the shape ofrules (unlike guarded rules)
II Materialised ABoxes can bestored as databases
Minus
I Only sets of rules with modelsof bounded size
II Acyclicity conditions might betoo restrictive
4
![Page 22: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/22.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]
For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL)
2 Apply existential rules in a restricted way
Suggestion: materialise ABoxes only over acyclic TBoxesAlways complete 3Provably terminating 3
5
![Page 23: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/23.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL)
2 Apply existential rules in a restricted way
Suggestion: materialise ABoxes only over acyclic TBoxesAlways complete 3Provably terminating 3
5
![Page 24: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/24.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Risk of non-termination
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL)
2 Apply existential rules in a restricted way
Suggestion: materialise ABoxes only over acyclic TBoxesAlways complete 3Provably terminating 3
5
![Page 25: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/25.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Risk of non-termination
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL)
2 Apply existential rules in a restricted waySuggestion: materialise ABoxes only over acyclic TBoxes
Always complete 3Provably terminating 3
5
![Page 26: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/26.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Risk of non-termination
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL): can miss
answers 8
2 Apply existential rules in a restricted waySuggestion: materialise ABoxes only over acyclic TBoxes
Always complete 3Provably terminating 3
5
![Page 27: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/27.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Risk of non-termination
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL): can miss
answers 82 Apply existential rules in a restricted way
Suggestion: materialise ABoxes only over acyclic TBoxesAlways complete 3Provably terminating 3
5
![Page 28: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/28.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Risk of non-termination
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL): can miss
answers 82 Apply existential rules in a restricted way: can still miss
answers and/or not terminate 8
Suggestion: materialise ABoxes only over acyclic TBoxesAlways complete 3Provably terminating 3
5
![Page 29: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/29.jpg)
MATERIALISATION-BASED REASONING
Answering CQs over expressive DLs is expensive, e.g.EXPTIME-complete for Horn-SHOIQ [Ortiz, Rudolph andSimkus, 2011]For Horn ontologies, consequences can be precomputed,stored and used for query evaluation, e.g. by the RDFrepositories Sesame, Jena, OWLIM, DLE-Jena,. . .
Risk of non-termination
Approaches taken:1 Saturate only non-existential rules (OWL 2 RL): can miss
answers 82 Apply existential rules in a restricted way: can still miss
answers and/or not terminate 8
Suggestion: materialise ABoxes only over acyclic TBoxesAlways complete 3Provably terminating 3
5
![Page 30: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/30.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA
2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 31: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/31.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 32: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/32.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 33: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/33.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditions
Horn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 34: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/34.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hard
Horn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete4 Experimental evaluation on DL ontologies
83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 35: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/35.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 36: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/36.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies
83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 37: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/37.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)
materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 38: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/38.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large
× 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 39: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/39.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 40: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/40.jpg)
RESULTS OVERVIEW
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large
× 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
6
![Page 41: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/41.jpg)
OUTLINE
1 MOTIVATION
2 MFA AND MSA
3 QUERYING ACYCLIC DL ONTOLOGIES
4 EXPERIMENTAL RESULTS
7
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EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ ∃y1.R(u, y1) ∧ B(y1)
r2 : B(v)→ ∃y2.R(v, y2) ∧ C(y2)
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 43: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/43.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ ∃y1.R(u, y1) ∧ B(y1)
r2 : B(v)→ ∃y2.R(v, y2) ∧ C(y2)
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 44: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/44.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ ∃y1.R(u, y1) ∧ B(y1)
r2 : B(v)→ ∃y2.R(v, y2) ∧ C(y2)
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 45: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/45.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ ∃y2.R(v, y2) ∧ C(y2)
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 46: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/46.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ ∃y2.R(v, y2) ∧ C(y2)
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 47: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/47.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 48: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/48.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 49: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/49.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 50: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/50.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 51: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/51.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 52: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/52.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1
, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 53: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/53.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1, R|1
, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 54: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/54.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆
Move(f (u)) = {R|2, B|1, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 55: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/55.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1}
⊆
Move(f (u)) = {R|2, B|1, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
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EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆ Move(f (u)) = {R|2, B|1, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 57: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/57.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆ Move(f (u)) = {R|2, B|1, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 58: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/58.jpg)
EXISTING ACYCLICITY CONDITIONS
EXAMPLE
r1 : A(u)→ R(u, f (u)) ∧ B(f (u))
r2 : B(v)→ R(v, g(v)) ∧ C(g(v))
r3 : R(w, z) ∧ B(z)→ A(w)
A ≡ ∃R.BB v ∃R.C
A,B,C
B C
C
∗
f (∗) g(∗)
g(f (∗))
R R
R
R
f (u)
not in JA
PosB(u) = {A|1} ⊆ Move(f (u)) = {R|2, B|1, R|1, A|1}
Joint acyclicity1 Tracks value generation and propagation to detect cyclic
creation of terms2 Polynomial time to check
May overestimate rule applicability
8
![Page 59: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/59.jpg)
MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u))
∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v))
∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u))
∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v))
∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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MODEL-FAITHFUL ACYCLICITY
Track rule applications more ‘faithfully’
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u)) ∧ S(u, f (u)) ∧ Ff (f (u))
B(v)→ R(v, g(v)) ∧ C(g(v)) ∧ S(v, g(v)) ∧ Fg(g(v))
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Ff (x) ∧ D(x, y) ∧ Ff (y)→ Cycle
Fg(x) ∧ D(x, y) ∧ Fg(y)→ Cycle
A,B,CR
B,Ff C,Fg
C,Fg
∗
f (∗) g(∗)
g(f (∗))
R, S,D
R, S,D
R, S,D
D
For Σ a set of rules, Σ is MFA if I∗Σ ∪MFA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is MFA
9
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COST OF CHECKING MFATesting model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
2EXPTIME-complete (tree with branching factor |~x| andheight the total number of function symbols )
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
2EXPTIME-complete
3 Rules from Horn-SRI
EXPTIME-hard
4 Rules from Horn-SHIQ
PSPACE-complete
Existing acyclicity conditions can be checked in PTIMEIsn’t computational complexity too high?
10
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COST OF CHECKING MFATesting model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
2EXPTIME-complete (tree with branching factor |~x| andheight the total number of function symbols )
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
2EXPTIME-complete
3 Rules from Horn-SRI
EXPTIME-hard
4 Rules from Horn-SHIQ
PSPACE-complete
Existing acyclicity conditions can be checked in PTIMEIsn’t computational complexity too high?
10
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COST OF CHECKING MFATesting model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
2EXPTIME-complete (tree with branching factor |~x| andheight the total number of function symbols )
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
2EXPTIME-complete
3 Rules from Horn-SRI
EXPTIME-hard
4 Rules from Horn-SHIQ
PSPACE-complete
Existing acyclicity conditions can be checked in PTIMEIsn’t computational complexity too high?
10
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COST OF CHECKING MFATesting model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
2EXPTIME-complete (tree with branching factor |~x| andheight the total number of function symbols )
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
2EXPTIME-complete
3 Rules from Horn-SRI
EXPTIME-hard
4 Rules from Horn-SHIQ
PSPACE-complete
Existing acyclicity conditions can be checked in PTIMEIsn’t computational complexity too high?
10
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COST OF CHECKING MFATesting model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
2EXPTIME-complete (tree with branching factor |~x| andheight the total number of function symbols )
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
2EXPTIME-complete
3 Rules from Horn-SRI
EXPTIME-hard
4 Rules from Horn-SHIQ
PSPACE-complete
Existing acyclicity conditions can be checked in PTIMEIsn’t computational complexity too high?
10
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COST OF CHECKING MFATesting model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
2EXPTIME-complete (tree with branching factor |~x| andheight the total number of function symbols )
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
2EXPTIME-complete
3 Rules from Horn-SRI
EXPTIME-hard
4 Rules from Horn-SHIQ
PSPACE-complete
Existing acyclicity conditions can be checked in PTIME
Isn’t computational complexity too high?
10
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COST OF CHECKING MFATesting model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
2EXPTIME-complete (tree with branching factor |~x| andheight the total number of function symbols )
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
2EXPTIME-complete
3 Rules from Horn-SRI
EXPTIME-hard
4 Rules from Horn-SHIQ
PSPACE-complete
Existing acyclicity conditions can be checked in PTIMEIsn’t computational complexity too high?
10
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, ) ∧ B()
∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, ) ∧ C()
∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u))
∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, g(v)) ∧ C(g(v))
∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, f (u)) ∧ B(f (u))
∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, g(v)) ∧ C(g(v))
∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1)
∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2)
∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1) ∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2) ∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1) ∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2) ∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1) ∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2) ∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1) ∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2) ∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1) ∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2) ∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1) ∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2) ∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= Cycle
Set of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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MODEL-SUMMARISING ACYCLICITY
Track rule applications just ‘faithfully’ enough
EXAMPLE
A(u)→ R(u, c1) ∧ B(c1) ∧ S(u, c1) ∧ Fc1(c1)
B(v)→ R(v, c2) ∧ C(c2) ∧ S(v, c2) ∧ Fc2(c2)
R(w, z) ∧ B(z)→ A(w)
S(x, y)→ D(x, y)
D(x, y) ∧ S(y, z)→ D(x, z)
Fc1(x) ∧ D(x, y) ∧ Fc1(y)→ Cycle
Fc2(x) ∧ D(x, y) ∧ Fc2(y)→ Cycle
d
A,B,CR
B,Fc1 C,Fc2
∗
c1 c2
R, S,D R, S,D
R, S,D
For Σ a set of rules, Σ is MSA if I∗Σ ∪MSA(Σ) 6|= CycleSet of rules that correspond to DL subsumptions{A ≡ ∃R.B,B v ∃R.C} is still MSA
11
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COST OF CHECKING MSA
Testing model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
EXPTIME-complete
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
coNP-complete
3 Rules from Horn-SHIQ
PTIME-complete
Horn-SHIQ TBoxes can be checked in PTIME for MSAbefore potential materialisation-based query answering
12
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COST OF CHECKING MSA
Testing model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
EXPTIME-complete
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
coNP-complete
3 Rules from Horn-SHIQ
PTIME-complete
Horn-SHIQ TBoxes can be checked in PTIME for MSAbefore potential materialisation-based query answering
12
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COST OF CHECKING MSA
Testing model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
EXPTIME-complete
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
coNP-complete
3 Rules from Horn-SHIQ
PTIME-complete
Horn-SHIQ TBoxes can be checked in PTIME for MSAbefore potential materialisation-based query answering
12
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COST OF CHECKING MSA
Testing model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
EXPTIME-complete
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
coNP-complete
3 Rules from Horn-SHIQ
PTIME-complete
Horn-SHIQ TBoxes can be checked in PTIME for MSAbefore potential materialisation-based query answering
12
![Page 92: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/92.jpg)
COST OF CHECKING MSA
Testing model-faithful acyclicity for a set of rules Σ
1 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) (no restriction)
EXPTIME-complete
2 Rules of the form ϕ(~x,~z)→ ∃~y.ψ(~x,~y) with predicates ofbounded arity
coNP-complete
3 Rules from Horn-SHIQ
PTIME-complete
Horn-SHIQ TBoxes can be checked in PTIME for MSAbefore potential materialisation-based query answering
12
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ACYCLICITY CONDITIONS (PARTIAL) TAXONOMY
Our contributions:
1 MSA strictly subsumes SWA
2 MFA strictly subsumes MSA
JA ( SWA MSA MFA
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y)
B(x)→ ∃y.S(x, y) ∧ T(y, x)
A(z) ∧ S(z, x)→ C(x)
C(z) ∧ T(z, x)→ A(x) MFA but not MSA
MSA and MFA coincide in experimental evaluation of DLontologies
13
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ACYCLICITY CONDITIONS (PARTIAL) TAXONOMY
Our contributions:
1 MSA strictly subsumes SWA
2 MFA strictly subsumes MSA
JA ( SWA MSA MFA
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y)
B(x)→ ∃y.S(x, y) ∧ T(y, x)
A(z) ∧ S(z, x)→ C(x)
C(z) ∧ T(z, x)→ A(x) MFA but not MSA
MSA and MFA coincide in experimental evaluation of DLontologies
13
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ACYCLICITY CONDITIONS (PARTIAL) TAXONOMY
Our contributions:
1 MSA strictly subsumes SWA
2 MFA strictly subsumes MSA
JA ( SWA ( MSA MFA
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y)
B(x)→ ∃y.S(x, y) ∧ T(y, x)
A(z) ∧ S(z, x)→ C(x)
C(z) ∧ T(z, x)→ A(x) MFA but not MSA
MSA and MFA coincide in experimental evaluation of DLontologies
13
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ACYCLICITY CONDITIONS (PARTIAL) TAXONOMY
Our contributions:
1 MSA strictly subsumes SWA
2 MFA strictly subsumes MSA
JA ( SWA ( MSA ( MFA
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y)
B(x)→ ∃y.S(x, y) ∧ T(y, x)
A(z) ∧ S(z, x)→ C(x)
C(z) ∧ T(z, x)→ A(x) MFA but not MSA
MSA and MFA coincide in experimental evaluation of DLontologies
13
![Page 97: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/97.jpg)
ACYCLICITY CONDITIONS (PARTIAL) TAXONOMY
Our contributions:
1 MSA strictly subsumes SWA
2 MFA strictly subsumes MSA
JA ( SWA ( MSA ( MFA
EXAMPLE
A(x)→ ∃y.R(x, y) ∧ B(y)
B(x)→ ∃y.S(x, y) ∧ T(y, x)
A(z) ∧ S(z, x)→ C(x)
C(z) ∧ T(z, x)→ A(x) MFA but not MSA
MSA and MFA coincide in experimental evaluation of DLontologies
13
![Page 98: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/98.jpg)
OUTLINE
1 MOTIVATION
2 MFA AND MSA
3 QUERYING ACYCLIC DL ONTOLOGIES
4 EXPERIMENTAL RESULTS
14
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TRANSLATING DLS INTO RULES
Axioms of normalised Horn-SRIQ ontologies can beconverted to (existential) rules
A v ∃R.B A(x) → ∃y.R(x, y) ∧ B(y)A v ≤ 1 R.B A(z) ∧ R(z, x1) ∧ B(x1) ∧ R(z, x2)
∧ B(x2) → x1≈x2A u B v C A(x) ∧ B(x) → C(x)
A v ∀R.B A(z) ∧ R(z, x) → B(x)R v S R(x1, x2) → S(x1, x2)
R ◦ S v T R(x1, z) ∧ S(z, x2) → T(x1, x2)
Equality is handled with a modification of the singularisation[Marnette, PODS, 2009] technique
15
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TRANSLATING DLS INTO RULES
Axioms of normalised Horn-SRIQ ontologies can beconverted to (existential) rules
A v ∃R.B A(x) → ∃y.R(x, y) ∧ B(y)A v ≤ 1 R.B A(z) ∧ R(z, x1) ∧ B(x1) ∧ R(z, x2)
∧ B(x2) → x1≈x2A u B v C A(x) ∧ B(x) → C(x)
A v ∀R.B A(z) ∧ R(z, x) → B(x)R v S R(x1, x2) → S(x1, x2)
R ◦ S v T R(x1, z) ∧ S(z, x2) → T(x1, x2)
Equality is handled with a modification of the singularisation[Marnette, PODS, 2009] technique
15
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DL QUERY ANSWERING UNDER ACYCLICITY
Answering conjunctive queries for the DL Horn-SHIQ isEXPTIME-complete [Eiter et al., 2008]
Does acyclicity affect complexity for DL Query Answering?
1 Horn-SHIQ TBox T and ABox AT is MFAQ Boolean conjunctive query
Deciding T ∪ A |= Q is PSPACE-complete2 Horn-SRI TBox T and ABox AT is weakly acyclicF set of facts
Deciding T ∪ A |= F is EXPTIME-hard
16
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DL QUERY ANSWERING UNDER ACYCLICITY
Answering conjunctive queries for the DL Horn-SHIQ isEXPTIME-complete [Eiter et al., 2008]
Does acyclicity affect complexity for DL Query Answering?
1 Horn-SHIQ TBox T and ABox AT is MFAQ Boolean conjunctive query
Deciding T ∪ A |= Q is PSPACE-complete2 Horn-SRI TBox T and ABox AT is weakly acyclicF set of facts
Deciding T ∪ A |= F is EXPTIME-hard
16
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DL QUERY ANSWERING UNDER ACYCLICITY
Answering conjunctive queries for the DL Horn-SHIQ isEXPTIME-complete [Eiter et al., 2008]
Does acyclicity affect complexity for DL Query Answering?
1 Horn-SHIQ TBox T and ABox AT is MFAQ Boolean conjunctive query
Deciding T ∪ A |= Q is PSPACE-complete
2 Horn-SRI TBox T and ABox AT is weakly acyclicF set of facts
Deciding T ∪ A |= F is EXPTIME-hard
16
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DL QUERY ANSWERING UNDER ACYCLICITY
Answering conjunctive queries for the DL Horn-SHIQ isEXPTIME-complete [Eiter et al., 2008]
Does acyclicity affect complexity for DL Query Answering?
1 Horn-SHIQ TBox T and ABox AT is MFAQ Boolean conjunctive query
Deciding T ∪ A |= Q is PSPACE-complete2 Horn-SRI TBox T and ABox AT is weakly acyclicF set of facts
Deciding T ∪ A |= F is EXPTIME-hard
16
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OUTLINE
1 MOTIVATION
2 MFA AND MSA
3 QUERYING ACYCLIC DL ONTOLOGIES
4 EXPERIMENTAL RESULTS
17
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ACYCLICITY TESTS
Checked 149 DL ontologies for WA, JA, MSA, MFA
Existential rules Total MSA JA WA< 100 70 64 64 64
100–1K 33 30 30 231K–5K 20 14 14 12
5K–12K 14 11 6 612K–160K 12 5 3 3All sizes 149 124 117 108
MSA and MFA coincide w.r.t. the test ontologies
83% were found MSA
7 large and expressive OBO ontologies MSA but not JA(only two of them were ELHr and DL-Lite)
18
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ACYCLICITY TESTS
Checked 149 DL ontologies for WA, JA, MSA, MFA
Existential rules Total MSA JA WA< 100 70 64 64 64
100–1K 33 30 30 231K–5K 20 14 14 12
5K–12K 14 11 6 612K–160K 12 5 3 3All sizes 149 124 117 108
MSA and MFA coincide w.r.t. the test ontologies
83% were found MSA
7 large and expressive OBO ontologies MSA but not JA(only two of them were ELHr and DL-Lite)
18
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ACYCLICITY TESTS
Checked 149 DL ontologies for WA, JA, MSA, MFA
Existential rules Total MSA JA WA< 100 70 64 64 64
100–1K 33 30 30 231K–5K 20 14 14 12
5K–12K 14 11 6 612K–160K 12 5 3 3All sizes 149 124 117 108
MSA and MFA coincide w.r.t. the test ontologies
83% were found MSA
7 large and expressive OBO ontologies MSA but not JA(only two of them were ELHr and DL-Lite)
18
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ACYCLICITY TESTS
Checked 149 DL ontologies for WA, JA, MSA, MFA
Existential rules Total MSA JA WA< 100 70 64 64 64
100–1K 33 30 30 231K–5K 20 14 14 12
5K–12K 14 11 6 612K–160K 12 5 3 3All sizes 149 124 117 108
MSA and MFA coincide w.r.t. the test ontologies
83% were found MSA
7 large and expressive OBO ontologies MSA but not JA(only two of them were ELHr and DL-Lite)
18
![Page 110: Acyclicity Conditions and their Application to Query Answering in Description Logics](https://reader033.fdocuments.in/reader033/viewer/2022051412/5495f899ac7959ff2d8b4fce/html5/thumbnails/110.jpg)
ACYCLICITY TESTS
Checked 149 DL ontologies for WA, JA, MSA, MFA
Existential rules Total MSA JA WA< 100 70 64 64 64
100–1K 33 30 30 231K–5K 20 14 14 12
5K–12K 14 11 6 612K–160K 12 5 3 3All sizes 149 124 117 108
MSA and MFA coincide w.r.t. the test ontologies
83% were found MSA
7 large and expressive OBO ontologies MSA but not JA(only two of them were ELHr and DL-Lite)
18
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MATERIALISATION TESTS
Computed materialisation of acyclic TBoxes
Depth # generated size materialisation sizemax avg max avg
< 5 82 27 2 35 55–9 13 37 11 41 13
10–80 14 281 51 283 53
Depth = length of function symbol nesting
generated size =# facts generated by existential rules
# facts in initial ABox
materialisation size =# facts in materialisation
# facts in initial ABox
For ontologies with small depths materialisation seemspractically feasible
19
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MATERIALISATION TESTS
Computed materialisation of acyclic TBoxes
Depth # generated size materialisation sizemax avg max avg
< 5 82 27 2 35 55–9 13 37 11 41 13
10–80 14 281 51 283 53
Depth = length of function symbol nesting
generated size =# facts generated by existential rules
# facts in initial ABox
materialisation size =# facts in materialisation
# facts in initial ABox
For ontologies with small depths materialisation seemspractically feasible
19
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MATERIALISATION TESTS
Computed materialisation of acyclic TBoxes
Depth # generated size materialisation sizemax avg max avg
< 5 82 27 2 35 55–9 13 37 11 41 13
10–80 14 281 51 283 53
Depth = length of function symbol nesting
generated size =# facts generated by existential rules
# facts in initial ABox
materialisation size =# facts in materialisation
# facts in initial ABox
For ontologies with small depths materialisation seemspractically feasible
19
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MATERIALISATION TESTS
Computed materialisation of acyclic TBoxes
Depth # generated size materialisation sizemax avg max avg
< 5 82 27 2 35 55–9 13 37 11 41 13
10–80 14 281 51 283 53
Depth = length of function symbol nesting
generated size =# facts generated by existential rules
# facts in initial ABox
materialisation size =# facts in materialisation
# facts in initial ABox
For ontologies with small depths materialisation seemspractically feasible
19
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SUMMARY OF THE RESULTS
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large × 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
Thank you! Questions?!?
20
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SUMMARY OF THE RESULTS
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large
× 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
Thank you! Questions?!?
20
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SUMMARY OF THE RESULTS
1 More general acyclicity conditions: MSA and MFA2 Complexity analysis for checking MSA and MFA
Horn-SHIQ bounded arity no restrictionMSA PTime-complete coNP-complete ExpTime-completeMFA PSpace-complete 2ExpTime-complete 2ExpTime-complete
3 DL query answering under acyclicity conditionsHorn-SRI T in WA: T ∪ A |= F is ExpTime-hardHorn-SHIQ T in MFA: T ∪ A |= Q is PSpace-complete
4 Experimental evaluation on DL ontologies83% ontologies found acyclic (78% JA)materialised ABoxes not too large
× 5 bigger on averagefor ontologies with depth < 5 (= most ontologies)
Materialisation-based reasoning beyond OWL 2 RLmight be practically feasible
Thank you! Questions?!?
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