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Published byPekka Jääskeläinen Modified over 5 years ago
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Turning The Partial-closed World Assumption
Upside Down Simon Razniewski, Ognjen Savkovic and Werner Nutt Free University of Bozen-Bolzano
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Overview Partial closed world assumption Translating CAD to IAD
Incompleteness as default (IAD) versus completeness as default (CAD) Translating CAD to IAD Query completeness reasoning under CAD
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Data semantics CWA OWA Partial-closed world assumption (PCWA)
Data space Database is complete in some parts, in others it is potentially incomplete Database is complete Database is potentially incomplete Partial-closed world assumption (PCWA)
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How can we model the partial-closed world assumption?
Incompleteness as default (IAD) Describe complete parts Completeness as default (CAD) Describe incomplete parts
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IAD is not well suited for DBs that are mostly complete
IAD: Lists 9 complete parts CAD: Lists 3 potentially incomplete parts Questions: How can we describe databases under CAD? How can we translate from CAD to IAD? Does using IAD instead of CAD make a difference?
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1. How can we describe databases under CAD?
Database schema: student(name, degree) lecturer(name, faculty) takes(name, course) Formalism Inspired by .. from IAD Example Full table statements Closed predicates in description logics PotInc(takes) Pattern statements Pattern completeness statements [Razniewski et al., SIGMOD 2015] PotInc(takes(_, DB) ) Query statements Query completeness statements [Motro, TODS 1989] PotInc( Q(x):-takes(x, y), student(x, CS).) Local statements Local completeness statements [Levy, VLDB 1996] PotInc(takes(x, y); student(x, CS) ) More expressive
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2. Translating from CAD to IAD
Database schema student(name, degree) lecturer(name, faculty) takes(name, course) CAD: takes is potentially incomplete Other tables are complete by default = IAD: Complete(lecturer) and Complete(student) CAD: takes is potentially incomplete for records of CS students Other tables and rest of takes are complete by default = IAD: ?
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2. The cost of translation
Result: CAD settings can be translated to IAD settings: For full table statements For pattern statements, if attribute domains are finite, or using disequality in statements For local and query statements, using additionally negation in statements
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3. Query completeness reasoning: IAD instead of CAD, what’s the difference?
Consider QLogics(n) :- student(n, c), takes(n, Logics) “Students that take logics” PotInc(takes(n, d); lecturer(n, f)) “Takes records of lecturers” Lecturers currently missing from the database might take Logics QLogics is not guaranteed to be complete. A query is complete if its certain answers are the same as its possible answers Completeness reasoning has been studied extensively in the IAD setting
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3. Variants of completeness reasoning
Input: Query Q Set of potential incompleteness statements C 1. Instance versus schema reasoning Instance reasoning Q is complete wrt. C over database instance I iff Q(I)=Q(I’) for all C-valid extensions I’ of I Schema Reasoning: Q is compl wrt. C iff Q is complete wrt. C over I for all database instances I 2. Query evaluation under bag or set semantics
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3. How complex is query completeness reasoning in the CAD setting?
Set semantics Bag semantics IAD CAD Schema Reasoning Instance Reasoning Full-table statements PTIME П 2 𝑃 -complete coNP-complete Pattern statments NP-complete coNP-hard, in П 2 𝑃 Local statements coNP-hard П 2 𝑃 -hard in П 2 𝑃 Query statements ? Set semantics Bag semantics IAD CAD Schema Reasoning Instance Reasoning Full-table statements PTIME П 2 𝑃 -complete coNP-complete Pattern statments NP-complete coNP-hard, in П 2 𝑃 Local statements coNP-hard П 2 𝑃 -hard in П 2 𝑃 Query statements ? Set semantics Bag semantics IAD CAD Schema Reasoning Instance Reasoning Full-table statements PTIME П 2 𝑃 -complete coNP-complete Pattern statments NP-complete Local statements in П 2 𝑃 Query statements ? Set semantics Bag semantics IAD CAD Schema Reasoning Instance Reasoning Full-table statements PTIME П 2 𝑃 -complete coNP-complete Pattern statments NP-complete coNP-hard, in П 2 𝑃 Local statements coNP-hard П 2 𝑃 -hard in П 2 𝑃 Query statements ? Straightforward new results for IAD Existing results for IAD New tight results for CAD New bounds for CAD Observations Set is hard: Instance reasoning already П 2 𝑃 -complete for full and pattern statements, both under CAD and IAD CAD bag schema is easier than IAD bag schema (PTIME vs. NP-complete for table and query completeness statements) Bag instance reasoning for local and query statements: Decidability open, requires termination of a chase-like procedure
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Open questions Which variants of reasoning are decidable/ what is their complexity? Can we raise the expressiveness without increasing the complexity? How can we reason with definite incompleteness? “Some students are definitely missing” vs. “students may be missing”
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PCWA IAD CAD Questions?
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