CS848: Topics in Databases: Foundations of Query Optimization Topics covered Overview of DEMO  Capturing database schema in QL  Differential query optimization  Dialect DLA  Capturing database schema in DLA

CS848: Topics in Databases: Foundations of Query Optimization Capturing database schema in QL  Another view on views  Another view on physical design

CS848: Topics in Databases: Foundations of Query Optimization Tables: current practice  Tables that are base  Tables that are views (defined by queries on base tables) create table EMP ( attribute name on STR; attribute age on INT; attribute loc on STR; primary key (name); ) create view WATEMP ( select name, age from EMP where loc = ‘Waterloo’; ) create view TOREMP ( select name from EMP where loc = ‘Toronto’; )

CS848: Topics in Databases: Foundations of Query Optimization Tables: an alternative  Tables  Constraints (query containment dependencies) ( select name, age from EMP where loc = ‘Waterloo’ ) ´ ( select name, age from WATEMP ) ( select name from EMP where loc = ‘Toronto’ ) ´ ( select name from TOREMP ) create table EMP ( attribute name on STR; attribute age on INT; attribute loc on STR; primary key (name); updatable; ) create table WATEMP ( attribute name on STR; attribute age on INT; ) create table TOREMP ( attribute name on STR; )

CS848: Topics in Databases: Foundations of Query Optimization Table: an alternative (cont’d)  Easy to do much more! ( select name, age from EMP where loc = ‘Waterloo’ ) ´ ( select name, age from WATEMP ) ( select name from EMP where loc = ‘Toronto’ ) ´ ( select name from TOREMP ) ( select name from EMP ) ´ ( select name from WATEMP ) union all ( select name from TOREMP ) create table EMP ( attribute name on STR; attribute age on INT; attribute loc on STR; primary key (name); updatable; ) create table WATEMP ( attribute name on STR; attribute age on INT; ) create table TOREMP ( attribute name on STR; )

CS848: Topics in Databases: Foundations of Query Optimization Physical design: current practice  Records and fields (implicit)  Clustered indices on base tables (a default file created)  Secondary indices on base tables  Materialized views (a default file created)  Secondary indices on materialized views  Etc.  External engines (collections of tables, views and the above) create index on EMP (loc asc, age desc) alter view WATEMP ( materialized; ) create index on WATEMP (name asc)

CS848: Topics in Databases: Foundations of Query Optimization Physical design: an alternative  Explicit named record identifiers and stored attributes  Iterator signatures  Secondary indices are also base relations alter table EMP ( record identifier emp; stored (name, age, loc); iterator; ) create table EMPX ( attribute loc on STR; attribute age on INT; attribute eref on RID; stored (loc, age, eref); iterator (loc asc, age desc); ) ( select loc, age, emp from EMP ) ´ ( select loc, age, eref from EMPX )

CS848: Topics in Databases: Foundations of Query Optimization Physical design: an alternative (cont’d) alter table WATEMP ( record identifier emp; stored (name, age); iterator; ) create table WATEMPX ( attribute name on STR; attribute eref on RID; stored (name, eref); iterator (name asc); ( select name, emp from WATEMP ) ´ ( select name, eref from WATEMPX )

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization select name, age from EMP where loc = ‘Waterloo’ and name = :p (based on simple hill climbing) 1.Expand conjunctive subquery using constraints. 2.Initialize subplan. 3.Select next iterator or disjunction from query for subplan; fail if none available. 4.If disjunction selected, recursively apply steps 1 to 5 on each disjunct; fail if any recursive application fails. 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) 1.Expand conjunctive subquery using constraints. select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age ( select name, age from EMP where loc = ‘Waterloo’ ) ´ ( select name, age from WATEMP ) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) 1.Expand conjunctive subquery using constraints. select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age ( select name, emp from WATEMP ) ´ ( select name, eref from WATEMPX ) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 2.Initialize subplan. select from PV as :p (plan) (query parameter)

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available. select from PV as :p (plan)

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name where name = :p (plan) (nested loops; string copy) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available.

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name where name = :p (plan) 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name where name = :p (expanded plan) 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name where name = :p (plan) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available.

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name, WATEMPX as x where name = :p and x.name = name (plan) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available. (nested loops; index scan)

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name, WATEMPX as x where name = :p and x.name = name (plan) 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name, age from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w, INT as age, EMP as e where name = :p and x.name = name and w.name = x.name and w.emp = x.eref and e.name = w.name and e.age = w.age and e.age = age and e.loc = ‘Waterloo’ (expanded plan) 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name, WATEMPX as x where name = :p and x.name = name (plan) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available.

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w where name = :p and x.name = name and w.emp = x.eref (plan) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available. (nested loops; pointer navigation)

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w where name = :p and x.name = name and w.emp = x.eref (plan) 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name, age from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w, INT as age, EMP as e where name = :p and x.name = name and w.emp = x.eref and w.name = x.name and e.name = w.name and e.age = w.age and e.age = age and e.loc = ‘Waterloo’ (expanded plan) 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w where name = :p and x.name = name and w.emp = x.eref (plan) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available.

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name, age from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w, (INT u QV) as age where name = :p and x.name = name and w.emp = x.eref and age = w.age (plan) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age, WATEMP as w, WATEMPX as x where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age and e.name = w.name and e.age = w.age and w.name = x.name and w.emp = x.eref 3.Select next iterator or disjunction from query for subplan; fail if none available. (nested loops; field extraction)

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name, age from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w, (INT u QV) as age where name = :p and x.name = name and w.emp = x.eref and age = w.age (plan) (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3.

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name, age from PV as :p, (STR u QV) as name, WATEMPX as x, WATEMP as w, (INT u QV) as age, EMP as e where name = :p and x.name = name and w.emp = x.eref and age = w.age and w.name = x.name and e.name = w.name and e.age = w.age and e.age = age and e.loc = ‘Waterloo’ (expanded plan) 5.If expand of subplan contains subquery, return subplan; otherwise repeat from 3. (expanded query) select name, age from EMP as e, PV as :p, (STR u QV) as name, (INT u QV) as age where e.loc = ‘Waterloo’ and e.name = :p and e.name = name and e.age = age

CS848: Topics in Databases: Foundations of Query Optimization Differential query optimization (cont’d) select name, age from PV as :p, // input parameter (STR u QV) as name, // string copy WATEMPX as x, // index scan WATEMP as w, // pointer navigation (INT u QV) as age // field extraction where name = :p and x.name = name and w.emp = x.eref and age = x.age (final plan)

CS848: Topics in Databases: Foundations of Query Optimization Dialect DLA D ::= (universal concept) | >  (primitive concept) |C (C) I (bottom concept) | ? ; (negation) | : D  – (D) I (intersection) |D 1 u D 2 (D 1 ) I Å (D 2 ) I (union) |D 1 t D 2 (D 1 ) I [ (D 2 ) I (attribute value restriction) | 8 A.D {e : (A) I (e) 2 (D) I } (quantified attribute inverse) |( > n A) {e 1 : |{e 2 : (A) I (e 2 ) = e 1 }| ¸ n} (quantified attribute inverse) |( 6 n A) {e 1 : n ¸ |{e 2 : (A) I (e 2 ) = e 1 }|} (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)} (path functional dependency) |C : L 1 ! L 2 (next slide) (an individual) | ¤ 2  Pf ::=id | A. Pf L ::=id | A | L. L | L, L | { L }

CS848: Topics in Databases: Foundations of Query Optimization Semantics of PFDs (C : L 1 ! L 2 ) I ´ f e 1 : 8 e 2 2 (C) I : Æ (Pf ) I (e 1 ) = (Pf ) I (e 2 ) ! Æ (Pf ) I (e 1 ) = (Pf ) I (e 2 ) g Pf 2 L (L 1 ) Pf 2 L (L 2 ) where L (id) ´ {id} L (A) ´ {A. id} L (L 1. L 2 ) ´ {Pf 1 ± Pf 2 : Pf 1 2 L (L 1 ) Æ Pf 2 2 L (L 2 )} L (L 1, L 2 ) ´ L (L 1 ) [ L (L 2 ) L ({L}) ´ L (L) where id ± Pf ´ Pf (A. Pf 1 ) ± Pf 2 ´ A. (Pf 1 ± Pf 2 )

CS848: Topics in Databases: Foundations of Query Optimization Sample partial database EMP emp (RID) name (STR) Waterloo Toronto Waterloo age (INT) Mary Ann Fred @ EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age

CS848: Topics in Databases: Foundations of Query Optimization DLA : primitive concepts EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR

CS848: Topics in Databases: Foundations of Query Optimization DLA : primitive concepts (cont’d) EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR  EMP

CS848: Topics in Databases: Foundations of Query Optimization DLA : attribute value restriction EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR  EMP  8 loc.STR

CS848: Topics in Databases: Foundations of Query Optimization EMP DLA : path functional dependencies EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR  EMP  8 loc.STR  EMP:{loc} ! {name}

CS848: Topics in Databases: Foundations of Query Optimization EMP DLA : quantified attribute inverse EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR  EMP  8 loc.STR  EMP:{loc} ! {name}  ( > 1 loc)

CS848: Topics in Databases: Foundations of Query Optimization EMP DLA : path agreement EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR  EMP  8 loc.STR  EMP:{loc} ! {name}  ( > 1 loc)  name = loc

CS848: Topics in Databases: Foundations of Query Optimization EMP DLA : intersection EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR  EMP  8 loc.STR  EMP:{loc} ! {name}  ( > 1 loc)  name = loc  RID u ( > 1 emp)

CS848: Topics in Databases: Foundations of Query Optimization EMP DLA : negation EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age  STR  EMP  8 loc.STR  EMP:{loc} ! {name}  ( > 1 loc)  name = loc  RID u ( > 1 emp)  : ( > 1 emp)

CS848: Topics in Databases: Foundations of Query Optimization DLA : dependencies EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age Models  EMP v 8 loc.STR

CS848: Topics in Databases: Foundations of Query Optimization DLA : dependencies (cont’d) EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age Models  EMP v 8 loc.STR  ( > 1 loc) v : INT

CS848: Topics in Databases: Foundations of Query Optimization DLA : dependencies (cont’d) EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age Models  EMP v 8 loc.STR  ( > 1 loc) v : INT  EMP v EMP:{Name} ! { id }

CS848: Topics in Databases: Foundations of Query Optimization DLA : dependencies (cont’d) EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age Does not model  EMP v STR Models  EMP v 8 loc.STR  ( > 1 loc) v : INT  EMP v EMP:{Name} ! { id }

CS848: Topics in Databases: Foundations of Query Optimization DLA : dependencies (cont’d) EMP STR Mary Waterloo STR STR Fred Toronto STRSTR Ann 28 INT 37 INT 33 INT ageemp nameloc empage loc name emp name age Does not model  EMP v STR  EMP v EMP:{Loc} ! {Name} Models  EMP v 8 loc.STR  ( > 1 loc) v : INT  EMP v EMP:{Name} ! { id }

CS848: Topics in Databases: Foundations of Query Optimization Capturing database schema in DLA (recall alternative view on views)  Tables  QL dependencies ( select name, age from EMP where loc = ‘Waterloo’ ) ´ ( select name, age from WATEMP ) ( select name from EMP where loc = ‘Toronto’ ) ´ ( select name from TOREMP ) ( select name from EMP ) ´ ( select name from WATEMP ) union all ( select name from TOREMP ) create table EMP ( attribute name on STR; attribute age on INT; attribute loc on STR; primary key (name); updatable; ) create table WATEMP ( attribute name on STR; attribute age on INT; ) create table TOREMP ( attribute name on STR; )

CS848: Topics in Databases: Foundations of Query Optimization DLA schema: capturing tables EMP v ( 8 name.STR ) u ( 8 age.INT ) u ( 8 loc.STR ) WATEMP v ( 8 name.STR ) u ( 8 age.INT ) TOREMP v 8 name.STR (signatures; typing) create table EMP ( attribute name on STR; attribute age on INT; attribute loc on STR; primary key (name); updatable; ) create table WATEMP ( attribute name on STR; attribute age on INT; ) create table TOREMP ( attribute name on STR; )

CS848: Topics in Databases: Foundations of Query Optimization DLA schema: capturing tables EMP v ( 8 name.STR ) u ( 8 age.INT ) u ( 8 loc.STR ) WATEMP v ( 8 name.STR ) u ( 8 age.INT ) TOREMP v 8 name.STR EMP v EMP:{name} ! { id } (primary keys) create table EMP ( attribute name on STR; attribute age on INT; attribute loc on STR; primary key (name); updatable; ) create table WATEMP ( attribute name on STR; attribute age on INT; ) create table TOREMP ( attribute name on STR; )

CS848: Topics in Databases: Foundations of Query Optimization DLA schema: capturing tables EMP v ( 8 name.STR ) u ( 8 age.INT ) u ( 8 loc.STR ) WATEMP v ( 8 name.STR ) u ( 8 age.INT ) TOREMP v 8 name.STR EMP v EMP:{name} ! { id } CLASS v OBJECT u : DOMAIN DOMAIN v OBJECT u : CLASS EMP v CLASS u UPDATABLE u : WATEMP u : TOREMP WATEMP v CLASS u : EMP u : TOREMP TOREMP v OBJECT u : EMP u : WATEMP INT v DOMAIN u : STR STR v DOMAIN u : INT (generalization hierarchies) create table EMP ( attribute name on STR; attribute age on INT; attribute loc on STR; primary key (name); updatable; ) create table WATEMP ( attribute name on STR; attribute age on INT; ) create table TOREMP ( attribute name on STR; )

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

CS848: Topics in Databases: Foundations of Query Optimization Physical design to DLA dependencies (homework assignment)

CS848: Topics in Databases: Foundations of Query Optimization Interfacing to a DLA theorem prover Observation: A partial database I encodes both a query Q( I ) in conjunctive QL and a concept description D( I ) that characterizes any tuple in the evaluation of Q( I ). Example normal form: select name, age, :p from(QV u IV) as name, (QV u IV) as age, (PV) as :p, (IV u EMP) as e, ( elim name, age, :p from(EV u WATERLOO) as c wherename = :p, name = e.name, age = e.age, e.loc = c, ( fromname = name, age = age, :p = :p )) select name, age from EMP where loc = ‘Waterloo’ and name = :p

CS848: Topics in Databases: Foundations of Query Optimization Query expansion and the simple chase Theorem: Let I 1 denote a partial database and I 2 the result of a simple chase of I 1 relative to terminology T. Then T ² Q( I 1 ) ´ Q( I 2 ).