Finding Explanations of Inconsistency in Multi-Context Systems

Transcription

1 Finding Explanations of Inconsistency in Multi-Context Systems Thomas Eiter Michael Fink Peter Schüller Antonius Weinzierl Principles of Knowledge Representation and Reasoning May 10, 2010 Vienna University of Technology Institute for Information Systems Knowledge-Based Systems Group Supported by the Vienna Science and Technology Fund (WWTF) under grant ICT08-020

2 Motivation Interlinking and Integrating Knowledge Focus on decentralized systems Heterogeneous and nonmonotonic system parts, here called contexts (databases, ontologies, answer set programs,... ) Fixed (small) amount of contexts Fixed topology Example: companies linking their business logics unifying formalism: Multi-Context Systems 1 / 20

3 Motivation Interlinking and Integrating Knowledge Focus on decentralized systems Heterogeneous and nonmonotonic system parts, here called contexts (databases, ontologies, answer set programs,... ) Fixed (small) amount of contexts Fixed topology Example: companies linking their business logics unifying formalism: Multi-Context Systems Inconsistencies arise easily, even if all contexts are consistent: Unforseen effects of information exchange Complexity of application and data We seek to understand and give reasons for inconsistencies. 1 / 20

4 Multi-Context Systems MCSs introduced by [Giunchiglia & Serafini, 1994]: represent inter-contextual information flow express reasoning w.r.t. contextual information allow decentralized, pointwise information exchange Framework extended for integrating heterogeneous non-monotonic logics [Brewka & Eiter, 2007]. 2 / 20

5 Syntax and Semantics of MCSs (1) What is a multi-context system? a collection M = (C 1,..., C n ) of contexts What is a context? Ci = (L i, kb i, br i ) L i : a logic kb i : the context s knowledge base br i : a set of bridge rules 3 / 20

6 Syntax and Semantics of MCSs (1) What is a multi-context system? a collection M = (C 1,..., C n ) of contexts What is a context? Ci = (L i, kb i, br i ) L i : a logic kb i : the context s knowledge base br i : a set of bridge rules What is a logic? L = (KBL, BS L, ACC L ) KBL : set of well-formed knowledge bases BS L : is the set of possible belief sets ACC L : KB L 2 BSL : acceptability function: Which belief sets are accepted by a knowledge base? 3 / 20

7 Syntax and Semantics of MCSs (2) M = (C 1,..., C n ) C i = (L i, kb i, br i ) L = (KB L, BS L, ACC L ) What is a belief state? S i BS Li is a belief set at C i S = (S 1,..., S n ) is a belief state of M 4 / 20

8 Syntax and Semantics of MCSs (2) M = (C 1,..., C n ) C i = (L i, kb i, br i ) L = (KB L, BS L, ACC L ) What is a belief state? S i BS Li is a belief set at C i S = (S 1,..., S n ) is a belief state of M What is a bridge rule? (k : s) (c 1 : p 1 ),..., (c j : p j ), not (c j+1 : p j+1 ),..., not (c m : p m ). Given a bridge rule r, intuitively (c : p) looks at presence of belief p at context C c (belief set S c )... r is applicable if positive p i are present and negative p i are absent... applicable s is added to knowledge base of context k 4 / 20

9 Syntax and Semantics of MCSs (3) Equilibrium semantics: A belief state S = (S 1,..., S n )... makes certain bridge rules applicable,... so we can add their heads to the kb i of the contexts. imported beliefs = bridge rule heads C 1 C 2 accepted beliefs = belief sets 5 / 20

10 Syntax and Semantics of MCSs (3) Equilibrium semantics: A belief state S = (S 1,..., S n )... makes certain bridge rules applicable,... so we can add their heads to the kb i of the contexts. C 1 C 2 5 / 20

11 Syntax and Semantics of MCSs (3) Equilibrium semantics: A belief state S = (S 1,..., S n )... makes certain bridge rules applicable,... so we can add their heads to the kb i of the contexts. S is an equilibrium iff each context plus these heads accepts S i. Equilibrium condition: S i ACC(kb i H i ) for all C i a c kb 1 kb 2 {a, c} 5 / 20

12 Example - Contexts Health care decision support system (wrt. medication and pneumonia): patient history database C 1, blood and X-Ray analysis database C 2, ontology of diseases C 3 (description logic), expert system C 4 (disjunctive logic program). C 1 = {allergy_strong_ab} C 2 = { blood_marker, xray_pneumonia} C 3 = {Pneumonia Marker AtypPneumonia} C 4 = {give_strong give_weak need_ab. give_strong need_strong. give_strong, not allow_strong_ab. give_nothing not need_ab, not need_strong.} 6 / 20

13 Example - Bridge Rules r 1 = (3 : Pneumonia(p)) (2 : xray_pneumonia). r 2 = (3 : Marker(p)) (2 : blood_marker). r 3 = (4 : need_ab) (3 : Pneumonia(p)). r 4 = (4 : need_strong) (3 : AtypPneumonia(p)). r 5 = (4 : allow_strong_ab) not (1 : allergy_strong_ab). C 1 History r 5 r 3 r 2 Laboratory r 1 C 2 Expert System Disease Ontology C 4 r 4 S = ({allergy_strong_ab}, { blood_marker, xray_pneumonia}, C 3 7 / 20

14 Example - Bridge Rules r 1 = (3 : Pneumonia(p)) (2 : xray_pneumonia). r 2 = (3 : Marker(p)) (2 : blood_marker). r 3 = (4 : need_ab) (3 : Pneumonia(p)). r 4 = (4 : need_strong) (3 : AtypPneumonia(p)). r 5 = (4 : allow_strong_ab) not (1 : allergy_strong_ab). C 1 History r 5 r 3 r 2 Laboratory r 1 C 2 C 4 Expert System r 4 Disease Ontology S = ({allergy_strong_ab}, { blood_marker, xray_pneumonia}, {Pneumonia(p)}, C 3 7 / 20

15 Example - Bridge Rules r 1 = (3 : Pneumonia(p)) (2 : xray_pneumonia). r 2 = (3 : Marker(p)) (2 : blood_marker). r 3 = (4 : need_ab) (3 : Pneumonia(p)). r 4 = (4 : need_strong) (3 : AtypPneumonia(p)). r 5 = (4 : allow_strong_ab) not (1 : allergy_strong_ab). C 1 History r 5 r 3 r 2 Laboratory r 1 C 2 C 4 Expert System r 4 Disease Ontology S = ({allergy_strong_ab}, { blood_marker, xray_pneumonia}, {Pneumonia(p)}, {need_ab, C 3 7 / 20

16 Example - Bridge Rules r 1 = (3 : Pneumonia(p)) (2 : xray_pneumonia). r 2 = (3 : Marker(p)) (2 : blood_marker). r 3 = (4 : need_ab) (3 : Pneumonia(p)). r 4 = (4 : need_strong) (3 : AtypPneumonia(p)). r 5 = (4 : allow_strong_ab) not (1 : allergy_strong_ab). C 1 History r 5 r 3 r 2 Laboratory r 1 C 2 C 4 Expert System r 4 Disease Ontology S = ({allergy_strong_ab}, { blood_marker, xray_pneumonia}, {Pneumonia(p)}, {need_ab,give_weak}) is an equilibrium. C 3 7 / 20

17 Inconsistency Analysis Inconsistency is the lack of an equilibrium. We seek to understand and give reasons for inconsistencies. We use ideas from model-based diagnosis [Reiter 1987] Assumptions: Contexts without input are consistent Bridge rules characterize reasons for inconsistency 8 / 20

18 Inconsistency Analysis Inconsistency is the lack of an equilibrium. We seek to understand and give reasons for inconsistencies. We use ideas from model-based diagnosis [Reiter 1987] Assumptions: Rationale: Contexts without input are consistent Bridge rules characterize reasons for inconsistency Context internals are abstracted away not our business Information flow can have unforeseen effects. Knowledge integration between companies: changing company knowledge bases (often) impossible 8 / 20

19 Diagnoses and Explanations Explaining inconsistency: Consistency-based Diagnosis : Which bridge rules need to be changed to get an equilibrium? changed by removing the rule, or changed by adding the rule in its unconditional form identifies some rules as "faulty" (causing inconsistency) provides possible repairs 9 / 20

20 Diagnoses and Explanations Explaining inconsistency: Consistency-based Diagnosis : Which bridge rules need to be changed to get an equilibrium? changed by removing the rule, or changed by adding the rule in its unconditional form identifies some rules as "faulty" (causing inconsistency) provides possible repairs Entailment-based Inconsistency Explanation : Which bridge rules are required for inconsistency? required, assuming all other rules are removed from the MCS finds groups of rules which together cause inconsistency allows to separate inconsistencies (if there are several of them) 9 / 20

21 Diagnosis Diagnosis: remove rules, or add them unconditionally, to get consistency Definition A diagnosis is a pair (D 1, D 2 ), D 1, D 2 br M, such that M[br M \ D 1 heads(d 2 )] = Notation: br M bridge rules of MCS M M[R] MCS M with bridge rules R instead of br M M = MCS M is inconsistent heads(r) rules in R in unconditional form (α for α β) 10 / 20

22 Diagnosis Diagnosis: remove rules, or add them unconditionally, to get consistency Definition A diagnosis is a pair (D 1, D 2 ), D 1, D 2 br M, such that M[br M \ D 1 heads(d 2 )] = Notation: br M bridge rules of MCS M M[R] MCS M with bridge rules R instead of br M M = MCS M is inconsistent heads(r) rules in R in unconditional form (α for α β) D ± (M): set of diagnoses of M D ± m(m) D ± (M): set of pointwise -minimal diagnoses of M 10 / 20

23 Example - Diagnoses Assume C 2 = {blood_marker, xray_pneumonia} No equilibrium 11 / 20

24 Example - Diagnoses Assume C 2 = {blood_marker, xray_pneumonia} No equilibrium Minimal diagnoses: ({r 1 }, ), remove r 1 : (3 : Pneumonia(p)) (2 : xray_pneumonia). S 3 = {Marker(p)}, S 4 = {give_nothing} 11 / 20

25 Example - Diagnoses Assume C 2 = {blood_marker, xray_pneumonia} No equilibrium Minimal diagnoses: ({r 1 }, ), ({r 2 }, ), remove r 1 : (3 : Pneumonia(p)) (2 : xray_pneumonia). S 3 = {Marker(p)}, S 4 = {give_nothing} remove r 2 : (3 : Marker(p)) (2 : blood_marker). S 3 = {Pneumonia(p)}, S 4 = {need_ab, give_weak} 11 / 20

26 Example - Diagnoses Assume C 2 = {blood_marker, xray_pneumonia} No equilibrium Minimal diagnoses: ({r 1 }, ), ({r 2 }, ), ({r 4 }, ), remove r 1 : (3 : Pneumonia(p)) (2 : xray_pneumonia). S 3 = {Marker(p)}, S 4 = {give_nothing} remove r 2 : (3 : Marker(p)) (2 : blood_marker). S 3 = {Pneumonia(p)}, S 4 = {need_ab, give_weak} remove r 4 : (4 : need_strong) (3 : AtypPneumonia(p)). S 3 = {Pneumonia(p), Marker(p), AtypPneumonia(p)} S 4 = {need_ab, give_weak} 11 / 20

27 Example - Diagnoses Assume C 2 = {blood_marker, xray_pneumonia} No equilibrium Minimal diagnoses: ({r 1 }, ), ({r 2 }, ), ({r 4 }, ), and (, {r 5 }). remove r 1 : (3 : Pneumonia(p)) (2 : xray_pneumonia). S 3 = {Marker(p)}, S 4 = {give_nothing} remove r 2 : (3 : Marker(p)) (2 : blood_marker). S 3 = {Pneumonia(p)}, S 4 = {need_ab, give_weak} remove r 4 : (4 : need_strong) (3 : AtypPneumonia(p)). S 3 = {Pneumonia(p), Marker(p), AtypPneumonia(p)} S 4 = {need_ab, give_weak} add r 5 : (4 : allow_strong_ab) not (1 : allergy_strong_ab). S 3 = {Pneumonia(p), Marker(p), AtypPneumonia(p)} S 4 = {need_ab, need_strong, allow_strong_ab, give_strong} 11 / 20

28 Inconsistency Explanation Inconsistency Explanation: rules (heads) that must be present (absent) for inconsistency Definition An inconsistency explanation is a pair (E 1, E 2 ), E 1, E 2 br M, such that for each pair (R 1, R 2 ), E 1 R 1 br M, R 2 br M \ E 2 M[R 1 heads(r 2 )] = E ± (M) (E ± m (M)): sets of ( -minimal) inconsistency explanations in M 12 / 20

29 Inconsistency Explanation Inconsistency Explanation: rules (heads) that must be present (absent) for inconsistency Definition An inconsistency explanation is a pair (E 1, E 2 ), E 1, E 2 br M, such that for each pair (R 1, R 2 ), E 1 R 1 br M, R 2 br M \ E 2 M[R 1 heads(r 2 )] = E ± (M) (E ± m (M)): sets of ( -minimal) inconsistency explanations in M Intuition: rules in E 1 create inconsistency all supersets inconsistent inconsistency is relevant in M adding rules from E 2 unconditionally is necessary to restore consistency related to minimal inconsistent sets 12 / 20

30 Example - Inconsistency Explanations Assume C 2 = {blood_marker, xray_pneumonia} (as before) Minimal inconsistency explanation: ({r 1, r 2, r 4 }, {r 5 }). 13 / 20

31 Example - Inconsistency Explanations Assume C 2 = {blood_marker, xray_pneumonia} (as before) Minimal inconsistency explanation: ({r 1, r 2, r 4 }, {r 5 }). Minimal diagnoses: ({r 1 }, ), ({r 2 }, ), ({r 4 }, ), and (, {r 5 }). 13 / 20

32 Duality Assume C 2 = {blood_marker, xray_pneumonia} (as before) Minimal inconsistency explanation: ({r 1, r 2, r 4 }, {r 5 }). Minimal diagnoses: ({r 1 }, ), ({r 2 }, ), ({r 4 }, ), and (, {r 5 }). Theorem For an inconsistent MCS, the unions of all minimal diagnoses D ± m minimal inconsistency explanations E m ± coincide: D ± m (M) = E m ± (M) and all Notation: X = ( {A (A, B) X}, {B (A, B) X} ) for X a set of (A, B) Diagnoses and explanations identify the same bridge rules 13 / 20

33 Context Complexity recall: equilibrium condition S i ACC(kb i H i ) for all C i Output beliefs OUT i : beliefs in bridge rule body literals 14 / 20

34 Context Complexity recall: equilibrium condition S i ACC(kb i H i ) for all C i Output beliefs OUT i : beliefs in bridge rule body literals bridge rules depend on output projected belief sets S i = S i OUT i Context complexity = equilibrium existence condition: S i ACC i(kb i H i ) OUTi 14 / 20

35 Complexity: Inconsistency Analysis Problem: recognition of diagnosis/explanation Input: candidate (D 1, D 2 ) resp. (E 1, E 2 ) and M 15 / 20

36 Complexity: Inconsistency Analysis Problem: recognition of diagnosis/explanation Input: candidate (D 1, D 2 ) resp. (E 1, E 2 ) and M Complexity Results (Completeness): context (D 1, D 2 )? (E 1, E 2 )? complexity D ± (M) D ± m(m) E ± (M) E m ± (M) P NP D P conp D P NP NP D P conp D P Σ P 2 Σ P 2 D P 2 Π P 2 D P 2 PSPACE EXPTIME PSPACE EXPTIME D P : solve both an NP and an independent conp problem 15 / 20

37 D ± Computation using HEX-programs HEX = ASP + Higher order features + external atoms Guess diagnosis Guess output belief state a i atoms Evaluate bridge rules b i atoms Check if output belief state is an output projected equilibrium: equilibrium condition: S i ACC i(kb i H i ) OUTi HEX constraint: not &con_out i [a i, b i ](). Open source implementation is available: 16 / 20

38 Special Cases and Properties Special Cases: s-diagnoses: Which rules must be removed to restore consistency? s-inconsistency Explanations: Which rules must be present to get inconsistency? duality holds Splitting Sets on MCS contexts modularity properties Preference orders which are different from subset-minimality: duality for certain Ceteris Paribus preference orderings 17 / 20

39 Explaining Inconsistency Conclusions We analyze inconsistencies to know "what s going on". Our approach... uses inconsistency to gain information provides possible repairs via diagnoses allows to separate sources of inconsistency via explanations 18 / 20

40 Explaining Inconsistency Conclusions We analyze inconsistencies to know "what s going on". Our approach... uses inconsistency to gain information provides possible repairs via diagnoses allows to separate sources of inconsistency via explanations We aim at configurable inconsistency management: automatic repair may be dangerous (see our example) automatic repair may be useful in other cases diagnoses and explanations form a basis for inconsistency management 18 / 20

41 Current and Future Work Current and Future work aims at... query answers on inconsistent MCSs e.g., defining partial equilibria e.g., defining brave and cautious query answers a local point of view to evaluation distributed algorithms approaches to compare diagnoses/explanations quantitative approaches inconsistency measures qualitative approaches using world knowledge implementations and benchmarks relevant application scenarios distributed implementation 19 / 20