Learning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus

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1 Forml Query Lnguges: Reltionl Alger nd Reltionl Clculus Chpter 4 Lerning Gols Given dtse ( set of tles ) you will e le to express dtse query in Reltionl Alger (RA), involving the sic opertors (selection, projection, cross product, renming, set union, intersection, difference ), join, division nd ssignment rewrite RA queries using suset of the opertors with queries using nother suset show tht two RA queries re/ren t equivlent express query in Domin Reltionl Clculus (DRC) trnslte RA queries to DRC queries nd vice vers CS304, R. Ng CS304, R. Ng Reltionl Query Lnguges Query lnguges: Allow mnipultion nd retrievl of dt from dtse Reltionl model supports simple, powerful QLs: Strong forml foundtion sed on logic Allows for much optimiztion Query Lnguges!= progrmming lnguges QLs not intended to e used for complex clcultions QLs support esy, efficient ccess to lrge dt sets Forml Reltionl Query Lnguges Two mthemticl Query Lnguges form the sis for rel lnguges (e.g., SQL), nd for implementtion: Reltionl Alger (RA): opertionl, need to specify wht you wnt nd lso how to compute it; very useful for representing execution plns Reltionl Clculus (RC): declrtive you only need to specify wht you wnt, ut not how to compute it CS304, R. Ng 3 CS304, R. Ng 4

2 Who cres, i.e. why do we not jump to SQL? RA nd RC form the mthemticl foundtion of querying in the reltionl model Focus on clen core concepts, without worrying out syntx nd implementtion In prctice, RA teches you out query optimiztion nd RC is the foundtion of querying with forms SQL ctully implements fetures from oth forml lnguges Preliminries A query is pplied to reltion instnces, nd the result of query is lso reltion instnce Schems of input reltions for query re fixed The schem for the result of given query is lso fixed! Determined y definition of query lnguge constructs CS304, R. Ng 5 CS304, R. Ng 6 Exmple Instnces Silors nd Reserves reltions for our exmples S (Silors) dustin luer rusty R (Reserves) sid id dy 0 0/0/ //96 S (Silors) 8 yuppy luer guppy rusty CS304, R. Ng 7 Reltionl Alger Bsic opertions: σ π Selection ( ) Selects suset of rows from reltion Projection ( ) Deletes unwnted columns from reltion Cross-product ( ) Allows us to comine two reltions Set-difference ( ) Tuples in reln., ut not in reln. Union ( U ) Tuples in reln. nd in reln. (notice tht some opertions re unry, while others re inry) Additionl opertions: Intersection, join, division, renming: not essentil, ut very useful Since ech opertion returns reltion, opertions cn e composed. (Alger is closed ) CS304, R. Ng 8

3 Projection (π (pi)) Nottion: π A, A,, Ak (r) where A,,Ak re ttriutes (the projection list) nd r is reltion Intention: to discrd columns not listed The result: reltion of the k ttriutes A, A,, AK otined from r y ersing the columns tht re not listed Duplicte rows removed from result (reltions re sets) CS304, R. Ng 9 Projection Exmple Reltion r: A B C π A,C (r) A C A C = CS304, R. Ng 0 Projection Exmples cont. S 8 yuppy luer guppy rusty π ge (S) ge π ( S) snme, rting snme rting yuppy 9 luer 8 guppy 5 rusty 0 CS304, R. Ng Selection (σ (sigm)) Nottion: σ p (r) Intention: to retrieve tuples stisfying given conditions Set of tuples of r tht Defined s: σ p (r) = {t t r nd p(t)} stisfy p Where p is formul in propositionl clculus consisting of predictes connectives : (nd), (or), (not) A predicte is one of: <ttriute> op <ttriute> or <ttriute> op <constnt> where op is one of: =,, >,, <, Result schem is sme s r s schem CS304, R. Ng

4 Selection Exmple Reltion r A B C D Selection Exmple S 8 yuppy rusty σ ( S ) rting >9 σ A=B ^ D > 5 (r) A B C D rusty CS304, R. Ng 3 CS304, R. Ng 4 Selection (cont.) Result reltion cn e the input for nother reltionl lger opertion! (Opertor composition) π 8 yuppy rusty σ S rting >8 ) snme rting yuppy 9 rusty 0 ( σ ( S )) snme, rting rting>8 Which queries would you like to sk? CS304, R. Ng 5 CS304, R. Ng 6

5 Union, Intersection, Set-Difference Union,Int., Diff. Exmples Nottion: r s r s r s Defined s: r s = {t t r or t s} r s ={ t t r nd t s } r s = {t t r nd t s} For these opertions to e well-defined:. r, s must hve the sme rity (sme numer of ttriutes). The ttriute domins must e comptile (e.g., nd column of r hs sme domin of vlues s the nd column of s) Wht is the schem of the result? Reltions r, s: A B r r s: A B r s: 3 A B A B 3 s r s: A B CS304, R. Ng 7 CS304, R. Ng 8 Union,Int., Diff. Exmples S S 8 yuppy dustin luer luer guppy rusty rusty S S S S dustin dustin S S 3 luer rusty luer guppy rusty yuppy CS304, R. Ng 9 Crtesin (or Cross)-Product Nottion: r x s Intention: to form new tuples y conctenting ll possile pirs Defined s: r x s = { t q t r nd q s} Assume tht ttriutes of r(r) nd s(s) re disjoint (i.e., R S = ). If r nd s hve common ttriutes, they must e renmed in the result CS304, R. Ng 0

6 Crtesin-Product Exmple r x s: r A A B B C γ γ D E s CS304, R. Ng C γ D E σ A=C (r x s) A B C D E Crtesin-Product Exmple S dustin luer rusty R sid id dy 0 0/0/ //96 S x R S. R.sid id dy dustin /0/96 dustin //96 3 luer /0/96 3 luer //96 58 rusty /0/96 58 rusty //96 conflicting nmes CS304, R. Ng Renme (ρ (rho)) Cn we perform self product, i.e., S X S? How cn we find pirs of silors (s, s) such tht s is younger thn s ut hs higher rting? Nottion: ρ (X, E) returns the expression E under the nme X If E hs rity n, then ρ(r(a, A,, A n ), E) returns the result of expression E under the reltion nme R, nd with the ttriutes renmed to A, A,., A n ρ(r(b A,, B k A k ), E) is s efore, ut it only renmes ttriutes B,,B k of E to A,,A k Additionl Opertions They cn e defined in terms of the primitive opertions They re dded for convenience They re: Join (Condition, Equi-, Nturl) ( ) Division (/) Assignment ( ) CS304, R. Ng 3 CS304, R. Ng 4

7 Joins ( ) Condition Join: R >< c S =σ c( R S) Result schem sme s tht of cross-product Fewer tuples thn cross-product might e le to compute more efficiently Sometimes clled thet-join Condition Join Exmple S dustin luer rusty S>< R S. sid < R. sid R sid id S. R.sid id dy dy 0 0/0/ //96 dustin //96 3 luer //96 CS304, R. Ng 5 CS304, R. Ng 6 Equi-Join & Nturl Join Equi-Join: A specil cse of condition join where the condition c contins only equlities Result schem: similr to cross-product, ut contins only one copy of fields for which equlity is specified Nturl Join: Equijoin on ll common ttriutes Result schem: similr to cross-product, ut hs only one copy of ech common ttriute No need to show the condition Wht if reltions hve no common ttriutes? CS304, R. Ng 7 Equi & Nturl Join Exmples sid id dy R 0 0/0/ //96 dustin luer rusty S>< R S. sid = R. sid id dy dustin /0/96 58 rusty //96 S > < R id dy dustin /0/96 58 rusty //96 CS304, R. Ng 8 S

8 Division Nottion: r / s or r s Useful for expressing queries tht include for ll or for every phrse, e.g., Find silors who hve reserved ll ots Let r nd s e reltions on schems R nd S respectively where r = (A,, A m, B,, B n ) s = (B,, B n ) Then r / s is reltion on schem r / s = (A,, A m ) defined s r / s = { t t r-s (r) u s ( tu r ) } i.e., A/B contins ll x tuples (silors) such tht for every y tuple (ot) in B, there is n x,y tuple in A CS304, R. Ng 9 Exmples of Division A/B A B B B3 pno pno s p s s p3 s s p s s3 s4 s4 A/B s s s3 s4 pno A/B s s4 pno p A/B3 s CS304, R. Ng 30 Expressing r/s Using Bsic Opertors Like join, cn e computed from sic opertors Ide: let X the set of ttriutes of r tht re not in s () compute the X-projection of r () compute ll X-projection vlues of r tht re disqulified y some vlue in s. vlue x is disqulified if y ttching y vlue from s, we otin n xy tuple tht is not in r. result is ()-() So, Disqulified x vlues: π (( π ( r) s) r) r/s is π ( r) π (( π ( r) s) r) X X X CS304, R. Ng 3 X X A=R Exmple of Division A/B Revisited pno s p s s p3 s s p s s3 s4 s4 B = S pno A/B s s4 s s s3 s4 π X (R) - π X (π X (R) x S R) π X (R) π X (R) x S π X (R) x S -R CS304, R. Ng 3 s s s3 s4 s s s3 s4 pno s s3 pno

9 Which queries would you like to sk ( revisited)? Find nmes of silors who ve reserved ot #03 Solution : π snme(( σ Re serves) >< Silors) id =03 Solution : π snme( σ (Re serves>< Silors)) id =03 CS304, R. Ng 33 CS304, R. Ng 34 Find nmes of silors who ve reserved red ot Informtion out ot color only ville in Bots; so need n extr join: π snme (( σ Bots serves Silors color ' red ' ) >< Re >< ) = A more efficient solution: π snme ( π π σ sid (( Bots s Silors id color ' red ' ) >< Re ) >< ) = A query optimizer cn find this given the first solution! CS304, R. Ng 35 Find silors who ve reserved red or green ot Cn identify ll red or green ots, then find silors who ve reserved one of these ots: ρ ( Tempots,( σ )) color = ' red ' color = ' green' Bots π snme ( Tempots>< Re serves>< Silors) Cn lso define Tempots using union! (How?) Wht hppens if is replced y in this query? CS304, R. Ng 36

10 Find silors who ve reserved red nd green ot Find the nmes of silors who ve reserved ll ots Previous pproch won t work! Must identify silors who ve reserved red ots, silors who ve reserved green ots, then find the intersection (note tht sid is key for Silors): ρ ( Tempred, π (( σ Bots) >< Re serves)) sid color = ' red ' ρ ( Tempgreen, π (( σ Bots) >< Re serves)) sid color = ' green' π snme (( Tempred Tempgreen) >< Silors) CS304, R. Ng 37 Uses division; schems of the input reltions must e crefully chosen: ρ ( Tempsids,( π Re serves) / ( π )) sid, id id Bots π snme ( Tempsids>< Silors) To find silors who ve reserved ll Interlke ots:... / π ( σ ) id nme= ' Interlke' Bots CS304, R. Ng 38 Summry: RA The reltionl model hs rigorously defined query lnguges tht re simple nd powerful RA is more opertionl; useful s internl representtion for query evlution plns Severl wys of expressing given query; query optimizer should choose the most efficient version Reltionl Clculus two flvours: Tuple reltionl clculus (TRC) nd Domin reltionl clculus (DRC) Clculus hs vriles, constnts, comprison ops, logicl connectives nd quntifiers TRC: Vriles rnge over (i.e., get ound to) tuples DRC: Vriles rnge over domin/field vlues Both re simple susets of first-order logic Predicte Clculus = toms FOL = predicte clculus + (forll) + (there exists) Expressions in the clculus re clled formuls. An nswer tuple is essentilly n ssignment of constnts to vriles tht mke the formul evlute to true CS304, R. Ng 39 CS304, R. Ng 40

11 Domin Reltionl Clculus Query hs the form: { <x, x,, x n > p(<x, x,, x n > ) } Answer includes ll tuples <x, x,, x n > tht mke the formul p(<x, x,, x n >) true The formul is recursively defined: strt with simple tomic formuls (tuples from reltions or comprisons of vlues), uild igger nd etter formuls using logicl connectives. DRC Formuls Atomic formul: <x, x,, x n > Rnme, or X op Y, or X op constnt, where op is one of <,>,=,,, Formul: n tomic formul, or pp, qp, q, where p nd q re formuls, or X ( p( X)), where vrile X is free in p(x), or X ( p( X)), where vrile X is free in p(x) The use of quntifiers X nd X is sid to ind X. A vrile tht is not ound is free CS304, R. Ng 4 CS304, R. Ng 4 Find ll silors with rting ove 7 INTA,,, INTA,,, Silors T> 7 The condition I, N, T, A Silors ensures tht the domin vriles I, N, T nd A re ound to fields of the sme Silors tuple The term I, N, T, A to the left of ` (which should e red s such tht) sys tht every tuple I, N, T, A tht stisfies T>7 is in the nswer Modify this query to nswer: Find silors who re older thn 8 or hve rting under 9, nd re clled Joe. CS304, R. Ng 43 Find silors rted > 7 who ve reserved ot #03 I, N, T, A I, N, T, A Silors T > 7 IrBrD,, IrBrD,, Reserves Ir= I Br= 03 ( ) We hve used Ir, Br, D... for Ir Br D... ( ( ( ))) s shorthnd Note the use of to find tuple in Reserves tht `joins with the Silors tuple under considertion. CS304, R. Ng 44

12 Declrtive vs Procedurl Find silors rted > 7 who ve reserved red ot Find nmes of silors who ve reserved ot #03 Reltionl lger π snme ( σ (Reserves) > < ) id = 03 Silors DRC: INTA,,, INTA,,, Silors IrBrD,, IrBrD,, Reserves Ir= I Br= 03 CS304, R. Ng 45 I, N, T, A I, N, T, A Silors T > 7 IrBrD,, IrBrD,, Reserves Ir= I BBNC,, BBNC,, Bots B= Br C= ' red' Oserve how the prentheses control the scope of ech quntifier s inding; scoping is importnt CS304, R. Ng 46 Find silors who ve reserved ll ots I, N, T, A I, N, T, A Silors BBNC,, Bots Ir, Br, D Reserves I = Ir Br = B The use of the for-ll quntifier mkes division queries esy to expressed in DRC CS304, R. Ng 47 Unsfe Queries, Expressive Power It is possile to write syntcticlly correct clculus queries tht hve n infinite numer of nswers! Such queries re clled unsfe e.g., S S Silors It is known tht every query tht cn e expressed in reltionl lger cn e expressed s sfe query in DRC / TRC; the converse is lso true Reltionl Completeness: Query lnguge (e.g., SQL) cn express every query tht is expressile in reltionl lger/clculus CS304, R. Ng 48

13 Summry RC is non-opertionl, nd users define queries in terms of wht they wnt, not in terms of how to compute it (Declrtive) RA nd DRC hve sme expressive power, leding to the notion of reltionl completeness CS304, R. Ng 49