cs848: topics in databases: foundations of query optimization topics covered introduction to...

CS848: Topics in Databases: Foundations of Query Optimization

Topics covered

Introduction to description logic: Single column QL

The ALC family of dialects

Terminologies

Language extensions


Initial analysis

The language L2 consists of all formulae of FOPC with equality and constant functions that use at most two distinct variables.

Theorem: The satisfiability problem for L2 is NEXPTIME-complete.

Corollary: The query containment problem for single column QL is decidable for queries that are attribute free.



Q ::= D as x |

D ::= THING | C | ? | :C | C1 u C2

| 8A.D | Pf1 = Pf2

| Pf1 Pf2

| 9R.THING | 8R.D | (D)



Q ::= D as x |

D ::= > | C | ? | :C | C1 u C2

| 8A.D | Pf1 = Pf2

| Pf1 Pf2

| 9R.> | 8R.D | (D)


Concept dependencies

On terminology and notation: We call an instance of the language generated by D for a given DL a concept. A concept inclusion dependency C for a given DL is written

D1 v D2

and corresponds to the query containment dependency

(D1 as x) v (D2 as x).

A concept definition C for a given DL is written

C ´ D

and corresponds to the query equivalence dependency

(C as x) ´ (D as x).


CLASSIC† (our first DL) (syntax) (semantics)

D ::= (universal concept) | > (primitive concept) | C (C)I

(bottom concept) | ? ; (atomic negation) | :C – (C)I

(intersection) | D1 u D2 (D1)I Å (D2)I

(attribute value restriction) | 8A.D {e : (A)I(e) 2 (D)I} (path agreement) | Pf1 = Pf2 {e : (Pf1)I(e) = (Pf2)I(e)} (path disagreement) | Pf1 Pf2 {e : (Pf1)I(e) (Pf2)I(e)} (existential quantification) | 9R.D {e1 : 9e2 : (e1, e2) 2 (R)I Æ e2 2 (D)I} (role value restriction) | 8R.D {e1 : 8(e1, e2) 2 (R)I : e2 2 (D)I}

| (D)

†[Borgida and Patel-Schneider, 1994]


Concept dependencies (cont’d)

The concept inclusion problem for a given DL is to determine if a concept inclusion dependency in the DL, D1 v D2, is an axiom; that is, to determine if (D1)I µ (D2)I for any database I.

Theorem: The concept inclusion problem for CLASSIC is solvable in low order polynomial time.


An efficient decision procedure

Theorem: The following procedure decides if C = (D1 v D2) is anaxiom for CLASSIC, and can be implemented in low order polynomialtime.

1. Create a partial database I1 consisting of a single individual e in concept D1. Perform a simple chase of I1 to obtain a partial database I2.

2. Return true if the domain of I2 is empty, or if the tuple

hx : e , cnt : 1i

occurs in «D2 as x¬(I2)†; otherwise return false.

†Use forced semantics for agreements and disagreements.


The simple chase

n : {D1 t D2} [ L n : {D1, D2} [ L

n1 : {8A.D} [ L n2 : {D}n1 : LA

n1 : {9R.D} [ L n2 : {D}n1 : LR


The simple chase (cont’d)

n2 : L2n1 : {8R.D} [ L1

R

n2 : {D} [ L2n1 : L1

R

n : {A1.A2. .Ar = B1.B2. .Bs} [ L

n : L u1 : ; ur : ;A1 ArA2

v1 : ; vs : ;BsB2B1



n : {A1.A2. .Ar B1.B2. .Bs} [ L

n : L u1 : ; ur : ;A1 ArA2

v1 : ; vs : ;BsB2B1

w : L u : L1

A

v : L2

A

w : L u : L1

A

v : L2

A



n1 : L1 n2 : L2 n1 : L1 [ L2 n2 : L1 [ L2

n1 : L1 n2 : L2 n3 : L3

n1 : L1 n2 : L2 n3 : L3

u : L1 v : L3

A

x : L4

Aw : L2

u : L1 v : L3

A

x : L4

Aw : L2



w : L u : L1

A

v : L2

A

w : {?} u : L1

A

v : L2

A

u : L1 v : L3

A

x : L4

Aw : L2

u : L1 v : L3

A

x : L4

Aw : L2



(remove all nodes and incident arcs)n : {?} [ L

or

m : L1 n : L2

n : {C, :C } [ L

or


Evaluating agreements and disagreements

Note that agreements and disagreements can navigate missing attribute values. In such cases, assume a forced semantics. In particular, a node n satisfies an agreement iff the agreement has the form

Pf1.Pf = Pf2.Pf

where (Pf1)I(n) and (Pf2)I(n) are defined and lead to nodes connected by an equality arc; n satisfies a disagreement iff it has the form

Pf1 = Pf2

where (Pf1)I(n) and (Pf2)I(n) are defined and lead to nodes connected by an inequality arc.


Example

Observation: The chase decision procedure for CLASSIC can be implemented in O(n log n) time, where n is the length of the component descriptions.

select e from EMP as ewhere e = e.b.b.b and e = e.b.b.b.b.b

(from (EMP as x), (from (x = x.b.b.b), (x = x.b.b.b.b.b)))

´ EMP u (id = b.b.b) u (id = b.b.b.b.b) as x

EMP u (id = b.b.b) u (id = b.b.b.b.b)

´ EMP u (id = id.b)

EMP u (id = b) as x)

select e from EMP as e where e = e.b


The ALC family of DLs

(syntax) (semantics)

D ::= (primitive concept) | C (C)I

(universal concept) | > (bottom concept) | ? ; (atomic negation) | :C – (C)I

(intersection) | D1 u D2 (D1)I Å (D2)I

(role value restriction) | 8R.D {e1 : 8(e1, e2) 2 (R)I : e2 2 (D)I} (limited existential quantification) | 9R.> {e1 : 9e2 : (e1, e2) 2 (R)I Æ e2 2 (D)I} (union) | D1 t D2 (D1)I [ (D2)I

(full existential quantification) | 9R.D {e1 : 9e2 : (e1, e2) 2 (R)I Æ e2 2 (D)I} (quantified number restriction) | (> n R) {e1 : |{e2 : (e1, e2) 2 (R)I}| ¸ n} (quantified number restriction) | (6 n R) {e1 : n ¸ |{e2 : (e1, e2) 2 (R)I}|} (full negation) | :D – (D)I


The ALC family of DLs (cont’d)

FL0 FL– AL ALN

D ::= C p p p p | > p p p | ? p p p | :C p p | D1 u D2 p p p p | 8R.D p p p p | 9R.> p p p | D1 t D2

| 9R.D | (> n R) p | (6 n R) p | :D


Some complexity results

Theorem: The concept inclusion problems for ALC and ALCN are PSPACE-complete.

A consistency problem for a given set of concepts is to determine if there exists a database that interprets a given member of the set as nonempty.

Observation: The consistency problem for ALC (resp. ALCN ) coincides with the concept inclusion problem for ALC (resp. ALCN ). In particular,

D1 v D2

is an axiom iff the concept(D1 u :D2)

is not consistent.


Testing consistency in ALC

Theorem: The following procedure decides if a given concept D in ALCis consistent.

1. Create a singleton set S1 = {I} of partial databases in which I consists of a single individual e in concept D. Perform a union generalized chase of S1 to obtain a set of partial databases S2 = {I1, … , In}.

2. Return true if the domain of any database in S2 is nonempty; otherwise return false.


Union generalized chase

Repeatedly do the following to a given set of partial databases S until nochanges occur.

1. Apply the simple chase augmented with the negation rule to a member of S.

2. If S contains a partial database I that in turn contains a node n with the form on the left below, then replace I with two partial databases I1 and I2 in S in which the labeling of node n is revised to the forms on the right below.

e : {D1t D2} [ L e : {D1} [ L e : {D2} [ L

(old node n in I) (new node n in I2)(new node n in I1)


The negation rule

Exhaustively apply the following rewrites to the concept labeling for any given node:†

:> ) ?:? ) >::D ) D:(D1 u D2) ) (:D1) t (:D2):8A.D ) 8A.:D:8R. D ) 9R.:D:9R. D ) 8R.:D:(D1 t D2) ) (:D1) u (:D2)

†Obtains negation normal form for concept descriptions.


A general membership problem

A database schema T that consists of concept dependencies in which no primitive concept occurs more than once on the left-hand-side of a concept definition is called a terminology.

The membership problem for a DL dialect is to determine, given a set

{C1, … , Cn, C} of concept dependencies in the DL, if {C1, … , Cn} ² C; that is, if every database I that models each Ci also models C.

Theorem: The membership problem for CLASSIC is undecidable.

Theorem: The membership problem for ALCN is DEXPTIME-complete.


Varieties of terminologies

A terminology T with only concept definitions is definitional.

For each C1 ´ D occurring in a terminology T and each primitive concept C2 occurring in D, C1 has a direct use of C2. The use relation is the transitive closure of direct use.

T is cyclic iff there exists an atomic concept in T that has a use of itself.

T is acyclic iff it is definitional and is not cyclic.


An acyclic terminology in ALC

WOMAN ´ PERSON u FEMALE

MAN ´ PERSON u :WOMAN

MOTHER ´ WOMAN u 9hasChild.PERSON

FATHER ´ MAN u 9hasChild.PERSON

PARENT ´ FATHER t MOTHER

GRANDMOTHER ´ MOTHER u 9hasChild.PARENT

MOTHERWITHMANYCHILDREN ´ MOTHER u > 3 hasChild

MOTHERWITHOUTDAUGHTER ´ MOTHER u 8hasChild.:WOMAN

WIFE ´ WOMAN u 9hasHusband.MAN


More complexity results

Theorem: The membership problem for FL0 with acyclic terminologies is CoNP-complete.

Theorem: The membership problem for ALC with acyclic terminologies is PSPACE-complete.

The DL ALCF extends ALC with agreements and disagreements of path functions.

Theorem: The concept inclusion problem for ALCF is PSPACE-complete.

Theorem: The membership problem for ALCF with acyclic terminologies is NEXPTIME-complete.


Blocking

Theorem: The membership problem for ALCN is DEXPTIME-complete.

The membership problem for ALCN can be solved by a refinement of theconsistency checking algorithm for concepts in ALC. There are twoimportant tricks to note.

1. Each concept dependency occurring in the terminology, e.g. D1 v D2, is internalized to each new node by adding a corresponding concept, e.g. (:D1 t D2), to the node’s label.

2. To ensure termination, no chasing is performed on blocked nodes. A node is blocked if its concepts are included in an older node.


Language extensions

Role constructors

Role value maps

Uniqueness constraints

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