1. Turning The Partial-closed World Assumption
Upside Down
Simon Razniewski, Ognjen Savkovic and Werner Nutt Free University of Bozen-Bolzano

2. 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

3. 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)

4. How can we model the partial-closed world assumption?
Incompleteness as default (IAD) Describe complete parts
Completeness as default (CAD) Describe incomplete parts

5. 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?

6. 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

7. 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: ?

8. 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

9. 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

10. 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

11. 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

12. 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”

13. PCWA
IAD
CAD
Questions?