Download presentation

Presentation is loading. Please wait.

Published byDeasia Ballinger Modified over 3 years ago

1
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

2
CS848: Topics in Databases: Foundations of Query Optimization Single column QL D ::=THING |C Q ::=D as x |(empty x) |(THING as x minus C as x) |(from Q 1, Q 2 ) |(elim x from x.A = y, elim y from y = x, Q) |(x.Pf 1 = x.Pf 2 ) |(THING as x minus x.Pf 1 = x.Pf 2 ) |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

3
CS848: Topics in Databases: Foundations of Query Optimization Initial analysis The language L 2 consists of all formulae of FOPC with equality and constant functions that use at most two distinct variables. Theorem: The satisfiability problem for L 2 is NEXPTIME-complete. Corollary: The query containment problem for single column QL is decidable for queries that are attribute free.

4
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ?, |(empty x) |(THING as x minus C as x) |(from Q 1, Q 2 ) |(elim x from x.A = y, elim y from y = x, Q) |(x.Pf 1 = x.Pf 2 ) |(THING as x minus x.Pf 1 = x.Pf 2 ) |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

5
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ? | : C, |(THING as x minus C as x) |(from Q 1, Q 2 ) |(elim x from x.A = y, elim y from y = x, Q) |(x.Pf 1 = x.Pf 2 ) |(THING as x minus x.Pf 1 = x.Pf 2 ) |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

6
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ? | : C |C 1 u C 2, |(from Q 1, Q 2 ) |(elim x from x.A = y, elim y from y = x, Q) |(x.Pf 1 = x.Pf 2 ) |(THING as x minus x.Pf 1 = x.Pf 2 ) |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

7
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ? | : C |C 1 u C 2 | 8 A.D, |(elim x from x.A = y, elim y from y = x, Q) |(x.Pf 1 = x.Pf 2 ) |(THING as x minus x.Pf 1 = x.Pf 2 ) |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

8
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ? | : C |C 1 u C 2 | 8 A.D |Pf 1 = Pf 2, |(x.Pf 1 = x.Pf 2 ) |(THING as x minus x.Pf 1 = x.Pf 2 ) |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

9
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ? | : C |C 1 u C 2 | 8 A.D |Pf 1 = Pf 2 |Pf 1 Pf 2, |(THING as x minus x.Pf 1 = x.Pf 2 ) |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

10
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ? | : C |C 1 u C 2 | 8 A.D |Pf 1 = Pf 2 |Pf 1 Pf 2 | 9 R.THING, |(elim x x.R = y) |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

11
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) D ::=THING |C Q ::=D as x | ? | : C |C 1 u C 2 | 8 A.D |Pf 1 = Pf 2 |Pf 1 Pf 2 | 9 R.THING | 8 R.D, |(THING as x minus elim x from x.R = y, elim y from y = x, THING as x minus Q) |

12
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) Q ::=D as x | D ::=THING |C | ? | : C |C 1 u C 2 | 8 A.D |Pf 1 = Pf 2 |Pf 1 Pf 2 | 9 R.THING | 8 R.D |(D)

13
CS848: Topics in Databases: Foundations of Query Optimization New syntax (cont’d) Q ::=D as x | D ::= > |C | ? | : C |C 1 u C 2 | 8 A.D |Pf 1 = Pf 2 |Pf 1 Pf 2 | 9 R. > | 8 R.D |(D)

14
CS848: Topics in Databases: Foundations of Query Optimization 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 D 1 v D 2 and corresponds to the query containment dependency (D 1 as x) v (D 2 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).

15
CS848: Topics in Databases: Foundations of Query Optimization CLASSIC † (our first DL) (syntax) (semantics) D ::= (universal concept) | > (primitive concept) |C (C) I (bottom concept) | ? ; (atomic negation) | : C – (C) I (intersection) |D 1 u D 2 (D 1 ) I Å (D 2 ) I (attribute value restriction) | 8 A.D {e : (A) I (e) 2 (D) I } (path agreement) |Pf 1 = Pf 2 {e : (Pf 1 ) I (e) = (Pf 2 ) I (e)} (path disagreement) |Pf 1 Pf 2 {e : (Pf 1 ) I (e) (Pf 2 ) I (e)} (existential quantification) | 9 R.D {e 1 : 9 e 2 : (e 1, e 2 ) 2 (R) I Æ e 2 2 (D) I } (role value restriction) | 8 R.D {e 1 : 8 (e 1, e 2 ) 2 (R) I : e 2 2 (D) I } |(D) † [Borgida and Patel-Schneider, 1994]

16
CS848: Topics in Databases: Foundations of Query Optimization Concept dependencies (cont’d) The concept inclusion problem for a given DL is to determine if a concept inclusion dependency in the DL, D 1 v D 2, is an axiom; that is, to determine if (D 1 ) I µ (D 2 ) I for any database I. Theorem: The concept inclusion problem for CLASSIC is solvable in low order polynomial time.

17
CS848: Topics in Databases: Foundations of Query Optimization An efficient decision procedure Theorem: The following procedure decides if C = (D 1 v D 2 ) is an axiom for CLASSIC, and can be implemented in low order polynomial time. 1.Create a partial database I 1 consisting of a single individual e in concept D 1. Perform a simple chase of I 1 to obtain a partial database I 2. 2.Return true if the domain of I 2 is empty, or if the tuple h x : e, cnt : 1 i occurs in « D 2 as x ¬ ( I 2 ) † ; otherwise return false. † Use forced semantics for agreements and disagreements.

18
CS848: Topics in Databases: Foundations of Query Optimization The simple chase n : {D 1 t D 2 } [ L n : {D 1, D 2 } [ L n 1 : { 8 A.D} [ L n 2 : {D} n 1 : L A n 1 : { 9 R.D} [ L n 2 : {D} n 1 : L R

19
CS848: Topics in Databases: Foundations of Query Optimization The simple chase (cont’d) n 2 : L 2 n 1 : { 8 R.D} [ L 1 R n 2 : {D} [ L 2 n 1 : L 1 R n : {A 1.A 2. .A r = B 1.B 2. .B s } [ L n : L u 1 : ; u r : ; A1A1 ArAr A2A2 v 1 : ; v s : ; BsBs B2B2 B1B1

20
CS848: Topics in Databases: Foundations of Query Optimization The simple chase (cont’d) n : {A 1.A 2. .A r B 1.B 2. .B s } [ L n : L u 1 : ; u r : ; A1A1 ArAr A2A2 v 1 : ; v s : ; BsBs B2B2 B1B1 w : L u : L 1 A v : L 2 A w : L u : L 1 A v : L 2 A

21
CS848: Topics in Databases: Foundations of Query Optimization The simple chase (cont’d) n 1 : L 1 n 2 : L 2 n 1 : L 1 [ L 2 n 2 : L 1 [ L 2 n 1 : L 1 n 2 : L 2 n 3 : L 3 n 1 : L 1 n 2 : L 2 n 3 : L 3 u : L 1 v : L 3 A x : L 4 A w : L 2 u : L 1 v : L 3 A x : L 4 A w : L 2

22
CS848: Topics in Databases: Foundations of Query Optimization The simple chase (cont’d) w : L u : L 1 A v : L 2 A w : { ? }u : L 1 A v : L 2 A u : L 1 v : L 3 A x : L 4 A w : L 2 u : L 1 v : L 3 A x : L 4 A w : L 2

23
CS848: Topics in Databases: Foundations of Query Optimization The simple chase (cont’d) (remove all nodes and incident arcs) n : { ? } [ L or m : L 1 n : L 2 n : {C, : C } [ L or

24
CS848: Topics in Databases: Foundations of Query Optimization 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 Pf 1.Pf = Pf 2.Pf where (Pf 1 ) I (n) and (Pf 2 ) I (n) are defined and lead to nodes connected by an equality arc; n satisfies a disagreement iff it has the form Pf 1 = Pf 2 where (Pf 1 ) I (n) and (Pf 2 ) I (n) are defined and lead to nodes connected by an inequality arc.

25
CS848: Topics in Databases: Foundations of Query Optimization 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 e where 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

26
CS848: Topics in Databases: Foundations of Query Optimization The ALC family of DLs (syntax) (semantics) D ::= (primitive concept) |C (C) I (universal concept) | > (bottom concept) | ? ; (atomic negation) | : C – (C) I (intersection) |D 1 u D 2 (D 1 ) I Å (D 2 ) I (role value restriction) | 8 R.D {e 1 : 8 (e 1, e 2 ) 2 (R) I : e 2 2 (D) I } (limited existential quantification) | 9 R. > {e 1 : 9 e 2 : (e 1, e 2 ) 2 (R) I Æ e 2 2 (D) I } (union) |D 1 t D 2 (D 1 ) I [ (D 2 ) I (full existential quantification) | 9 R.D {e 1 : 9 e 2 : (e 1, e 2 ) 2 (R) I Æ e 2 2 (D) I } (quantified number restriction) |( > n R) {e 1 : |{e 2 : (e 1, e 2 ) 2 (R) I }| ¸ n} (quantified number restriction) |( 6 n R) {e 1 : n ¸ |{e 2 : (e 1, e 2 ) 2 (R) I }|} (full negation) | : D – (D) I

27
CS848: Topics in Databases: Foundations of Query Optimization The ALC family of DLs (cont’d) FL 0 FL – AL ALN D ::=C p p p p | > p p p | ? p p p | : C p p |D 1 u D 2 p p p p | 8 R.D p p p p | 9 R. > p p p |D 1 t D 2 | 9 R.D |( > n R) p |( 6 n R) p | : D

28
CS848: Topics in Databases: Foundations of Query Optimization The ALC family of DLs (cont’d) ALU ALE ALUEALC ALCN D ::=C p p p p p | > p p p p p | ? p p p p p | : C p p p p p |D 1 u D 2 p p p p p | 8 R.D p p p p p | 9 R. > p p p p p |D 1 t D 2 p p ± p | 9 R.D p p ± p |( > n R) p |( 6 n R) p | : D ± p p

29
CS848: Topics in Databases: Foundations of Query Optimization 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, D 1 v D 2 is an axiom iff the concept (D 1 u : D 2 ) is not consistent.

30
CS848: Topics in Databases: Foundations of Query Optimization Testing consistency in ALC Theorem: The following procedure decides if a given concept D in ALC is consistent. 1.Create a singleton set S 1 = { I } of partial databases in which I consists of a single individual e in concept D. Perform a union generalized chase of S 1 to obtain a set of partial databases S 2 = { I 1, …, I n }. 2.Return true if the domain of any database in S 2 is nonempty; otherwise return false.

31
CS848: Topics in Databases: Foundations of Query Optimization Union generalized chase Repeatedly do the following to a given set of partial databases S until no changes 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 I 1 and I 2 in S in which the labeling of node n is revised to the forms on the right below. e : {D 1 t D 2 } [ L e : {D 1 } [ L e : {D 2 } [ L (old node n in I )(new node n in I 2 )(new node n in I 1 )

32
CS848: Topics in Databases: Foundations of Query Optimization The negation rule Exhaustively apply the following rewrites to the concept labeling for any given node: † :>) ? :?) > :: D ) D : (D 1 u D 2 ) ) ( : D 1 ) t ( : D 2 ) :8 A.D ) 8 A. : D :8 R. D ) 9 R. : D :9 R. D ) 8 R. : D : (D 1 t D 2 ) ) ( : D 1 ) u ( : D 2 ) † Obtains negation normal form for concept descriptions.

33
CS848: Topics in Databases: Foundations of Query Optimization 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 { C 1, …, C n, C } of concept dependencies in the DL, if { C 1, …, C n } ² C ; that is, if every database I that models each C i also models C. Theorem: The membership problem for CLASSIC is undecidable. Theorem: The membership problem for ALCN is DEXPTIME-complete.

34
CS848: Topics in Databases: Foundations of Query Optimization Varieties of terminologies A terminology T with only concept definitions is definitional. For each C 1 ´ D occurring in a terminology T and each primitive concept C 2 occurring in D, C 1 has a direct use of C 2. 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.

35
CS848: Topics in Databases: Foundations of Query Optimization An acyclic terminology in ALC WOMAN ´ PERSON u FEMALE MAN ´ PERSON u : WOMAN MOTHER ´ WOMAN u 9 hasChild.PERSON FATHER ´ MAN u 9 hasChild.PERSON PARENT ´ FATHER t MOTHER GRANDMOTHER ´ MOTHER u 9 hasChild.PARENT MOTHERWITHMANYCHILDREN ´ MOTHER u > 3 hasChild MOTHERWITHOUTDAUGHTER ´ MOTHER u 8 hasChild. : WOMAN WIFE ´ WOMAN u 9 hasHusband.MAN

36
CS848: Topics in Databases: Foundations of Query Optimization More complexity results Theorem: The membership problem for FL 0 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.

37
CS848: Topics in Databases: Foundations of Query Optimization Blocking Theorem: The membership problem for ALCN is DEXPTIME-complete. The membership problem for ALCN can be solved by a refinement of the consistency checking algorithm for concepts in ALC. There are two important tricks to note. 1.Each concept dependency occurring in the terminology, e.g. D 1 v D 2, is internalized to each new node by adding a corresponding concept, e.g. ( : D 1 t D 2 ), 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.

38
CS848: Topics in Databases: Foundations of Query Optimization Language extensions Role constructors Role value maps Uniqueness constraints

Similar presentations

OK

Presented by Kyumars Sheykh Esmaili Description Logics for Data Bases (DLHB,Chapter 16) Semantic Web Seminar.

Presented by Kyumars Sheykh Esmaili Description Logics for Data Bases (DLHB,Chapter 16) Semantic Web Seminar.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on masculine and feminine gender for grade 3 Burn ppt on dvd Ppt on synthesis and degradation of purines and pyrimidines structure Ppt on tamper resistant packaging Ppt on power system stability using facts Ppt on power grid failure 2015 Ppt on primary data collection methods Ppt on management by objectives peter Ppt on viruses and bacteria worksheets Free ppt on types of houses