1. Topics covered (class assignment)
SHIQ (from [Horrocks, Sattler and Tobies])
CS848: Topics in Databases: Foundations of Query Optimization

2. QL dependencies to DLA dependencies
(class assignment)
( select name, age
from EMP
where loc = ‘Waterloo’ )
´ ( select name, age from WATEMP )
( select name
where loc = ‘Toronto’ )
´ ( select name from TOREMP )
( select name from EMP )
´ ( select name from WATEMP )
union all
( select name from TOREMP )
CS848: Topics in Databases: Foundations of Query Optimization

3. Dependency translation (cont’d)
WEMP ´ EMP u (8loc.WATERLOO)
WEMP v (> 1 eref)
(8eref.WEMP) v C1
C1 v (8eref.WEMP) u (8wref.WATEMP)
C1 v (C1:eref ! id) u (C1:wref ! id)
C1 v (eref.name = wref.name) u (eref.age = wref.age)
WATEMP v (> 1 wref)
(8wref.WATEMP) v C1
TEMP ´ EMP u (8loc.TORONTO)
TEMP v (> 1 eref)
(8eref.TEMP) v C2
C2 v (8eref.TEMP) u (8tref.TOREMP)
C2 v (C2:eref ! id) u (C2:tref ! id)
C2 v (eref.name = tref.name)
TOREMP v (> 1 tref)
(8tref.TOREMP) v C2
CS848: Topics in Databases: Foundations of Query Optimization

4. Dependency translation (cont’d)
EMP v (WEMP t TEMP)
WATERLOO v STR u (:TORONTO) u (WATERLOO:! id)
TORONTO v STR u (:WATERLOO) u (TORONTO:! id)
¤ v 8waterloo.WATERLOO
¤ v 8toronto.TORONTO
CS848: Topics in Databases: Foundations of Query Optimization

5. Another approach WEMP ´ EMP u (8loc.WATERLOO)
WEMP v (8wref.WATEMP) u (WEMP:wref ! id)
WATEMP v (8eref.WEMP) u (WATEMP:eref ! id)
WATEMP v (name = eref.name) u (age = eref.age)
TEMP ´ EMP u (8loc.TORONTO)
TEMP v (8tref.TOREMP) u (TEMP:tref ! id)
TOREMP v (8eref.TEMP) u (TOREMP:eref ! id)
TOREMP v (name = eref.name)
EMP v (WEMP t TEMP)
WATERLOO v STR u (:TORONTO) u (WATERLOO:! id)
TORONTO v STR u (:WATERLOO) u (TORONTO:! id)
¤ v 8waterloo.WATERLOO
¤ v 8toronto.TORONTO
CS848: Topics in Databases: Foundations of Query Optimization

6. Dialect SHIQ The S part stands for ALC plus transitive roles R+ µ R.
Also introduces inverse roles.
SI roles consist of (R [ {R- : R 2 R})
Inv(R) = R-; Inv(R-) = R.
Trans(R) = true iff R 2 R+ or Inv(R) 2 R+
The H part means that we allows role inclusion axioms R1 v R2. Given a set R of role inclusion axioms, define the role hierarchy
R+ ´ (R [ {Inv(R1) v Inv(R2) : R1 v R2 2 R}, v*)
where v* is the reflexive transitive closure of v over
R [ {Inv(R1) v Inv(R2) : R1 v R2 2 R}.
CS848: Topics in Databases: Foundations of Query Optimization

7. Dialect SHIQ (cont’d) (union) D1 t D2 (D1)I [ (D2)I
(universal concept) > D
(primitive concept) C (C)I
(transitive role) R 2 R (R)I = ((R)I )+
(negation) :D D – (D)I
(intersection) D1 u D (D1)I Å (D2)I S
(union) D1 t D (D1)I [ (D2)I
(role value restriction) 8R.D {e1 : 8e2: (e1, e2) 2 (R)I ! e2 2 (D)I}
(role existence) 9R.D {e1 : 9e2: (e1, e2) 2 (R)I Æ e2 2 (D)I}
(role hierarchy) R1 v R (R1)I µ (R2)I H
(inverse role) R {(e2, e1) : (e1, e2) 2 (R)I} I
(number restriction) (> n R) {e1 : |{e2 : (e1, e2) 2 (R)I}| ¸ n} N
(number restriction) (6 n R) {e1 : n ¸ |{e2 : (e1, e2) 2 (R)I}|}
(qualified number restriction) (> n R D) {e1 : |{e2 : (e1, e2) 2 (R)I Æ e2 2 (D)I}| ¸ n} Q
(qualified number restriction) (6 n R D) {e1 : n ¸ |{e2 : (e1, e2) 2 (R)I Æ e2 2 (D)I}|}
CS848: Topics in Databases: Foundations of Query Optimization

8. Dialect SHIQ (cont’d)
An interpretation I satisfies a role hierarchy R+ iff (R1)I µ (R2)I for each R1 v* R2 2 R+.
Concept D is satisfiable with respect to role hierarchy R+ iff there exists I that satisfies R+ and for which (D)I ¹ ;.
CS848: Topics in Databases: Foundations of Query Optimization

9. Internalization in SHIQ
Given terminology T, role inclusion axioms R, SHIQ concept D and transitive role U not occurring in T, D or R, define
DT ´ u {:D1 t D2 : D1 v D2 2 T }
and
RU ´ R [ {R v U, Inv(R) v U : R occurs in T, D or R}.
Then D is satisfiable with respect to T and R+ iff
D u DT u 8U.DT
is satisfiable with respect to (RU)+.
CS848: Topics in Databases: Foundations of Query Optimization

10. Reasoning in SHIQ
Based on an extension of the union generalized chase for ALC.
To accommodate role inclusion hierarchies, allow arcs in a partial database to be labeled with sets of role names and role inverses. Write L(x, y) to denote the set of (possibly inverse) roles that label an arc from x to y.
Write »D to denote the negation normal form of a SHIQ concept :D, and clos(D) to denote the smallest set of concepts in negation normal form that contains »D and is closed under subconcepts and ».
Node y is an R2-successor of node x if an arc from x to y exists such that R1 2 L(x, y) and R1 v* R2. Node y is an R-neighbour of x iff y is an
R-successor of x, or if x is an Inv(R)-successor of y.
CS848: Topics in Databases: Foundations of Query Optimization

11. :C u 9(R1)-.(C u (6 1 R1)) u 8(R2)-.(9(R1)-.(C u (6 1 R1)))
Blocking in SHIQ
As with ALC with arbitrary terminologies, SHIN does not have the finite model property; e.g., when R2 is a transitive super-role of R1,
:C u 9(R1)-.(C u (6 1 R1)) u 8(R2)-.(9(R1)-.(C u (6 1 R1)))
requires the existence of an infinite (R1)- path. Therefore need blocking in chases to ensure termination.
CS848: Topics in Databases: Foundations of Query Optimization

12. Blocking in SHIQ (cont’d)
Node y2 is directly blocked in a partial database if it has ancestors y1, x2 and x1 such that
1. arcs exists from y1 to y2 and from x1 to x2,
2. L(x1, x2) = L(y1, y2),
3. L(x1) = L(y1) and
4. L(x2) = L(y2).
In such cases, x2 is said to block y2.
A node is indirectly blocked iff one of its ancestors is blocked or it has an incoming arc with an empty label.
A node is blocked iff it is directly or indirectly blocked.
CS848: Topics in Databases: Foundations of Query Optimization

13. The SHIQ chase u–rule: if 1. D1 u D2 2 L(x),
2. x is not indirectly blocked, and
3. {D1, D2} * L(x)
then L(x) Ã L(x) [ {D1, D2}
t–rule:
if D1 t D2 2 L(x),
2. x is not indirectly blocked and
3. {D1, D2} Å L(x) = ;
then L(x) Ã L(x) [ {D1} or L(x) Ã L(x) [ {D2}.
CS848: Topics in Databases: Foundations of Query Optimization

14. The SHIQ chase (cont’d)
9–rule:
if R.D 2 L(x),
2. x is not blocked and
3. x has no R-neighbour y with D 2 L(y)
then create a new node y and arc (x, y) with L(x, y) Ã {R}
and L(y) Ã {D}.
8–rule:
if R.D 2 L(x),
2. x is nor indirectly blocked and
3. there is an R-neighbour y of x with D Ï L(y)
then L(y) Ã L(y) [ {D}.
CS848: Topics in Databases: Foundations of Query Optimization

15. The SHIQ chase (cont’d)
8+ rule:
if R2.D 2 L(x),
2. x is not indirectly blocked,
3. there is some R1 such that Trans(R1) and R1 v* R2,
4. there is an R1–neighbour y of x such that 8R1.D Ï L(y)
then L(y) Ã L(y) [ {8R1.D}.
choose-rule:
if (6 n R D) 2 L(x) or (> n R D) 2 L(x),
3. there is an R-neighbour y of x such that {D, »D} Å L(y) = ;
then L(y) Ã L(y) [ {D} or L(y) Ã L(y) [ {»D}.
CS848: Topics in Databases: Foundations of Query Optimization

16. The SHIQ chase (cont’d)
>–rule:
if (> n R D) 2 L(x),
2. x is not blocked,
3. there are not n R-neighbours y1, … , yn of x with D 2 L(yi)
and yi ¹ yj for 1  i < j  n
then create n new nodes y1, … , yn and arcs (x, yi) with L(x, yi) Ã {R},
L(yi) Ã {D} and yi ¹ yj for 1  i < j  n.
CS848: Topics in Databases: Foundations of Query Optimization

17. The SHIQ chase (cont’d)
6–rule:
if (6 n R D) 2 L(x),
2. x is not indirectly blocked,
3. there are n+1 R-neighbours y1, … , yn+1 of x such that
D 2 L(yi), y2 is not an ancestor of x and not y1 ¹ y2
then do the following:
1. L(y1) Ã L(y1) [ L(y2);
2. L(y1, x) Ã L(y1, x) [ Inv(L(x, y2)) if y1 is an ancestor of x;
3. L(x, y1) Ã L(x, y1) [ L(x, y2) if y1 is not an ancestor of x;
4. L(x, y2) Ã ; and
5. set z ¹ y1 for all z such that z ¹ y2.
CS848: Topics in Databases: Foundations of Query Optimization

18. Complexity of reasoning
A role is simple if it is neither transitive nor has transitive subroles.
Theorem: The concept membership problem for SHIQ in which only simple roles occur within the scope of number restrictions is DEXPTIME-complete.
Theorem: The concept membership problem for SI is PSPACE-complete.
Theorem: The concept inclusion problem for SHIN in which arbitrary roles can occur within the scope of number restrictions is undecidable.
CS848: Topics in Databases: Foundations of Query Optimization