Logic Aspects of Databases teacher: doc. RNDr. Stanislav Kraj ci, - - PowerPoint PPT Presentation
Pavol Jozef Saf arik University in Ko sice Faculty of Science Supportive textbooks in course Logic Aspects of Databases teacher: doc. RNDr. Stanislav Kraj ci, PhD. study programme: Informatics Project 2005/NP1-051 11230100466
Pavol Jozef ˇ Saf´ arik University in Koˇ sice Faculty of Science Supportive textbooks in course
teacher: doc. RNDr. Stanislav Krajˇ ci, PhD. study programme: Informatics Project 2005/NP1-051 11230100466
Stanislav Krajˇ ci Logic Aspects of Databases
◮ assessment: examination ◮ duration: 2 hours per week, 26 hours per study ◮ final examination: written test ◮ course objective: to acquire information connecting logic and
the databases theory
◮ course content: first order logic, database, relational algebra
Stanislav Krajˇ ci Logic Aspects of Databases
◮ S. Abiteboul, R. Hull, V. Vian: Foundations of Databases,
Addison-Wesley Publishing Company, 1995, ISBN 0-201-53771-0
Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ UCnn = {¬
¬ ¬} – unary connectives
◮ BCnn = {∧
∧ ∧,∨ ∨ ∨,→ → →,↔ ↔ ↔,↑ ↑ ↑,↓ ↓ ↓,⊕ ⊕ ⊕} – binary connectives
◮ Cnn = UCnn ∪ BCnn – connectives ◮ Qua = {∀
∀ ∀,∃ ∃ ∃} – quantifiers
◮ Aux = {(
( (,) ) ),, , ,} – auxiliary symbols
◮ BLS = Cnn ∪ Qua ∪ Aux – basic logic symbols
Stanislav Krajˇ ci Logic Aspects of Databases
◮ sets of special symbols
◮ DatL – data types ◮ VarL – variables ◮ ConL – constant symbols ◮ FunL – function symbols ◮ PreL – predicate symbols
◮ all these sets are disjoint and disjoint of BLS
Stanislav Krajˇ ci Logic Aspects of Databases
◮ L = DatL, typVarL, typConL, sigFunL, sigPreL – typed first
◮ typVarL : VarL → DatL – variable types ◮ typConL : ConL → DatL – constant symbol types ◮ sigFunL : FunL →
n∈N+ Datn L × DatL – function symbol
signatures
◮ sigPreL : PreL →
n∈N+ Datn L – predicate symbol signatures
◮ all often written without the index L
Stanislav Krajˇ ci Logic Aspects of Databases
◮ TrmL (or shortly Trm) – the least set fulfilling the following
conditions:
1a VarL ⊆ TrmL and typ(v) = typVar(v) for each v ∈ VarL, 1b ConL ⊆ TrmL and typ(c) = typCon(c) for each c ∈ ConL 2 if f ∈ FunL s.t. sigFun(f ) = D1, . . . , Dn, D and for all i ∈ {1, . . . , n}, ti ∈ TrmL s.t. typ(ti) = Di then t = f ( ( (t1, , , . . ., , ,tn) ) ) ∈ TrmL and typ(t) = D
◮ L-terms (or shortly terms) – elements of TrmL
Stanislav Krajˇ ci Logic Aspects of Databases
◮ FrmL (or shortly Frm) – the least set fulfilling the following
conditions:
1 if p ∈ PreL s.t. sigPre(p) = D1, . . . , Dn and for all i ∈ {1, . . . , n}, ti ∈ TrmL s.t. typ(ti) = Di then p( ( (t1, , , . . ., , ,tn) ) ) ∈ FrmL (so-called atomic formulae or atoms) 2a if ψ ∈ FrmL then ( ( (¬ ¬ ¬ψ) ) ) ∈ FrmL 2b if ψ1, ψ2 ∈ FrmL and @ ∈ BCnn then ( ( (ψ1@ψ2) ) ) ∈ FrmL 2c if ψ ∈ FrmL, # ∈ Qua and v ∈ VarL then ( ( (( ( (#v) ) )ψ) ) ) ∈ FrmL
◮ L-formulae (or shortly formulae) – elements of FrmL
Stanislav Krajˇ ci Logic Aspects of Databases
◮ define function ConTrm : Trm → P(Con) in the following
way:
1a if t = v ∈ Var then ConTrm(t) = ∅ 1b if t = c ∈ Fun then ConTrm(t) = {c} 2 if t = f ( ( (t1, , , . . ., , ,tn) ) ) then ConTrm(t) =
i∈{1,...,n} ConTrm(ti)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ define function VarTrm : Trm → P(Var) in the following
way:
1a if t = v ∈ Var then VarTrm(t) = {v} 1b if t = c ∈ Fun then VarTrm(t) = ∅ 2 if t = f ( ( (t1, , , . . ., , ,tn) ) ) then VarTrm(t) =
i∈{1,...,n} VarTrm(ti)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ define functions FVarFrm, BVarFrm : Frm → P(Var) in the
following way:
1 if ϕ = p( ( (t1, , , . . ., , ,tn) ) ) then FVarFrm(ϕ) =
i∈{1,...,n} VarTrm(ti)
and BVarFrm(ϕ) = ∅ 2a if ϕ = ( ( (¬ ¬ ¬ψ) ) ) then FVarFrm(ϕ) = FVarFrm(ψ) and BVarFrm(ϕ) = BVarFrm(ψ) 2b if ϕ = ( ( (ψ1@ψ2) ) ) then FVarFrm(ϕ) = FVarFrm(ψ1) ∪ FVarFrm(ϕ2) and BVarFrm(ϕ) = BVarFrm(ψ1) ∪ BVarFrm(ϕ2) 2c if ϕ = ( ( (( ( (#v) ) )ψ) ) ) then FVarFrm(ϕ) = FVarFrm(ψ) {v} and BVarFrm(ϕ) = BVarFrm(ψ) ∪ {v}
◮ define VarFrm(ϕ) = FVarFrm(ϕ) ∪ BVarFrm(ϕ)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ data type interpretation:
if D ∈ Dat then IntDat(D) is some non-empty set
◮ constant symbol interpretation:
if c ∈ Con and typCon(c) = D then IntCon(c) ∈ IntDat(D)
◮ function symbol interpretation:
if f ∈ Fun and sigFun(f ) = D1, . . . , Dn, D then IntFun(f ) : IntDat(D1) × · · · × IntDat(Dn) → IntDat(D)
◮ such IntDat, IntCon, IntFun is called a pre-interpretation
Stanislav Krajˇ ci Logic Aspects of Databases
◮ evaluation of variables:
an arbitrary function E s. t. Dom(E) = Var and E(v) ∈ typVar(v)
◮ if E is an evaluation of variables, v is a variable and
m ∈ IntDat(typVar(v)) then define Ev/m in the following way: Ev/m(w) =
if w = v m if w = v
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let E is an evaluation of variables and
IntDat, IntCon, IntFun is a pre-interpretation
◮ define function E with Dom(E) = Trm by induction:
1a if t = v ∈ Var then E(t) = E(v) 1b if t = c ∈ Con then E(t) = IntCon(c) 2 if t = f ( ( (t1, , , . . ., , ,tn) ) ) where f ∈ Fun and t1, . . . , tn ∈ Trm then E(t) = (IntFun(f )) (E(t1), . . . , E(tn))
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let p ∈ Pre and sigPre(p) = D1, . . . , Dn ◮ two possible interpretations:
◮ as a relation:
IntPreRel(p) ⊆ IntDat(D1) × · · · × IntDat(Dn)
◮ (alternatively) as a function to a set TrV of truth values:
IntPreFun(p) : IntDat(D1) × · · · × IntDat(Dn) → TrV
◮ if TrV = {0, 1} both interpretations are clearly in this
correspondence: x1, . . . , xn ∈ IntPreRel(p) iff (IntPreFun(p)) (x1, . . . , xn) = 1
Stanislav Krajˇ ci Logic Aspects of Databases
◮
x y B∨
∨ ∨(x, y)
B∧
∧ ∧(x, y)
B→
→ →(x, y)
B↔
↔ ↔(x, y)
1 1 1 1 1 1 1 1 1 1 1 1
◮
x y B↓
↓ ↓(x, y)
B↑
↑ ↑(x, y)
B⊕
⊕ ⊕(x, y)
1 1 1 1 1 1 1 1 1 1
Stanislav Krajˇ ci Logic Aspects of Databases
◮
x B¬
¬ ¬(x)
1 1
◮
X B∃
∃ ∃(X)
B∀
∀ ∀(X)
∅ 1 {0} {1} 1 1 {0, 1} 1 1
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let E is an evaluation of variables,
IntDat, IntCon, IntFun is a pre-interpretation and IntPreFun is a interpretation of predicate symbols
◮ define function E with Dom(E) = Frm by induction:
1 if ϕ = p( ( (t1, , , . . ., , ,tn) ) ) where f ∈ Fun and t1, . . . , tn ∈ Trm then E(ϕ) = (IntPreFun(p)) (E(t1), . . . , E(tn)) 2a if ϕ = ( ( (¬ ¬ ¬ψ) ) ) where ψ ∈ Frm then E(ϕ) = B¬
¬ ¬(E(ψ))
2b if ϕ = ( ( (ψ1@ψ2) ) ) where ψ1, ψ2 ∈ Frm and @ ∈ BCnn then E(ϕ) = B@(E(ψ1), E(ψ2)) 2c if ϕ = ( ( (( ( (#v) ) )ψ) ) ) where ψ ∈ Frm, # ∈ Qua and v ∈ Var then E(ϕ) = B#({Ev/m(ψ) : m ∈ IntDat(typVar(v))})
Stanislav Krajˇ ci Logic Aspects of Databases
◮ σ : Var → Trm is called a substitution
if for all v ∈ Var, typ(σ(v)) = typVar(v)
◮ a special case – identity Id:
for all v ∈ Var, Id(v) = v
◮ if σ is a substitution, w is a variable and t is a term s. t.
typ(t) = typVar(w) then σw/t is a substitution defined by σw/t(v) =
if v = w σ(v), if v = w
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let σ is a substitution ◮ define σ : Trm → Trm:
1a if t = v ∈ Var then σ(t) = σ(v) 1b if t = c ∈ Con then σ(t) = c 2 if t = f ( ( (t1, , , . . ., , ,tn) ) ) where f ∈ Fun and t1, . . . , tn ∈ Trm then σ(t) = f ( ( (σ(t1), , , . . ., , ,σ(tn)) ) )
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let σ is a substitution ◮ define σ : Frm → Frm:
1 if ϕ = p( ( (t1, , , . . ., , ,tn) ) ) where f ∈ Fun and t1, . . . , tn ∈ Trm then σ(ϕ) = p( ( (σ(t1), , , . . ., , ,σ(tn)) ) ) 2a if ϕ = ( ( (¬ ¬ ¬ψ) ) ) where ψ ∈ Frm then σ(ϕ) = ( ( (¬ ¬ ¬σ(ψ)) ) ) 2b if ϕ = ( ( (ψ1@ψ2) ) ) where ψ1, ψ2 ∈ Frm and @ ∈ BCnn then σ(ϕ) = ( ( (σ(ψ1)@σ(ψ2)) ) ) 2c if ϕ = ( ( (( ( (#v) ) )ψ) ) ) where ψ ∈ Frm, # ∈ Qua and v ∈ Var then σ(ϕ) = ( ( (( ( (#) ) )σv/v(ψ)) ) )
Stanislav Krajˇ ci Logic Aspects of Databases
◮ theorem:
Let v and w are variables of the same type and ϕ be a formula s. t. v, w / ∈ BVarFrm(ϕ) and moreover v / ∈ FVarFrm(ϕ). Then for all E E(( ( (( ( (∃ ∃ ∃w) ) )ϕ) ) )) = E(( ( (( ( (∃ ∃ ∃v) ) )Idw/vϕ) ) )).
Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ Abc – the English abc including other special characters ◮ ≤Abc (shortly ≤) – an arbitrary but fixed linear ordering of
the set Abc (e.g. lexicographical)
◮ TbN – table names – an arbitrary but fixed infinite countable
subset of Abc+
◮ AtN – attribute names – an arbitrary but fixed infinite
countable subset of Abc+
◮ att : TbN → Pfin(AtN) – s.t. att−1(U) is infinite for all
U ∈ Pfin(AtN)
◮ if M ∈ TbN then ari(M) = |att(M)|
Stanislav Krajˇ ci Logic Aspects of Databases
◮ All – the universe (or (not very correctly) domain) – an
arbitrary but fixed infinite set
◮ T = M, D – an abstract table (or shortly table)
◮ M ∈ TbN – the T’s metadata (or name),
M = met(T), T is an instantion of M
◮ D ⊆ Allatt(M) – the T’s data,
D = dat(T)
◮ T1 T2 (T1 and T2 are tables) – if met(T1) = met(T2) and
dat(T1) ⊆ dat(T2)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ database – an arbitrary finite subset of tables with different
metadata
◮ sch(B) – database schema of a database B – the set of all
metadata of all tables from B; B is an instantion of sch(B)
◮ tab(B, M) (B is a database and M ∈ sch(B)) – the only table
T of the database B s.t. met(T) = M
◮ B1 B2 (B1 and B2 are databases) – if sch(B1) = sch(B2)
and for each M ∈ sch(B1), tab(B1, M) tab(B2, M)
◮ the active universe (or active domain) of the database B –
act(B) =
T∈B
Stanislav Krajˇ ci Logic Aspects of Databases
◮ table T1 with a name “cinema
cinema cinema”:
“cinema address cinema address cinema address” “cinema name cinema name cinema name” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1” “´ Usmev ´ Usmev ´ Usmev” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8” “Tatra Tatra Tatra” ◮ table T2 with a name “movie
movie movie”:
“country country country” “director director director” “movie name movie name movie name” “year year year” “USA USA USA” “Robert Zemeckis Robert Zemeckis Robert Zemeckis” “Forrest Gump Forrest Gump Forrest Gump” “1994 1994 1994” “USA USA USA” “Frank Darabont Frank Darabont Frank Darabont” “The Shawshank Redemption The Shawshank Redemption The Shawshank Redemption” “1994 1994 1994” “USA USA USA” “David Leane David Leane David Leane” “The Bridge over the River Kwai The Bridge over the River Kwai The Bridge over the River Kwai” “1959 1959 1959” “USA USA USA” “Robert Zemeckis Robert Zemeckis Robert Zemeckis” “Back to the Future Back to the Future Back to the Future” “1985 1985 1985” “Czechia Czechia Czechia” “Oldˇ rich Lipsk´ y Oldˇ rich Lipsk´ y Oldˇ rich Lipsk´ y” “Mareˇ cku, podejte mi pero! Mareˇ cku, podejte mi pero! Mareˇ cku, podejte mi pero!” “1976 1976 1976”
Stanislav Krajˇ ci Logic Aspects of Databases
◮ table T3 with a name “performance
performance performance”:
“cinema name cinema name cinema name” “datetime datetime datetime” “movie name movie name movie name” “Capitol Capitol Capitol” “2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00” “Forrest Gump Forrest Gump Forrest Gump” “´ Usmev ´ Usmev ´ Usmev” “2007-03-26 18:00 2007-03-26 18:00 2007-03-26 18:00” “The Shawshank Redemption The Shawshank Redemption The Shawshank Redemption” “´ Usmev ´ Usmev ´ Usmev” “2007-03-27 16:00 2007-03-27 16:00 2007-03-27 16:00” “The Bridge over the River Kwai The Bridge over the River Kwai The Bridge over the River Kwai” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-28 16:00 2007-03-28 16:00 2007-03-28 16:00” “Back to the Future Back to the Future Back to the Future” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-29 20:00 2007-03-29 20:00 2007-03-29 20:00” “Mareˇ cku, podejte mi pero! Mareˇ cku, podejte mi pero! Mareˇ cku, podejte mi pero!”
Stanislav Krajˇ ci Logic Aspects of Databases
◮ TbN ⊇ {“cinema
cinema cinema”,“movie movie movie”,“performance performance performance”}
◮ AtN ⊇ {“cinema address
cinema address cinema address”,“cinema name cinema name cinema name”,“country country country”, “director director director”,“movie name movie name movie name”,“year year year”,“datetime datetime datetime”}
◮ att(“cinema
cinema cinema”) = {“cinema address cinema address cinema address”,“cinema name cinema name cinema name”}, att(“movie movie movie”) = {“country country country”,“director director director”,“movie name movie name movie name”,“year year year”}, att(“performance performance performance”) = {“cinema name cinema name cinema name”,“datetime datetime datetime”,“movie name movie name movie name”}
◮ ari(“cinema
cinema cinema”) = 2, ari(“movie movie movie”) = 4, ari(“performance performance performance”) = 3
Stanislav Krajˇ ci Logic Aspects of Databases
◮ All ⊇ {“Hronsk´
a 12 Hronsk´ a 12 Hronsk´ a 12”,“Druˇ zba Druˇ zba Druˇ zba”,“Masarykova 2 Masarykova 2 Masarykova 2”, “Capitol Capitol Capitol”, . . . ,“USA USA USA”,“Robert Zemeckis Robert Zemeckis Robert Zemeckis”, “Forrest Gump Forrest Gump Forrest Gump”,“1994 1994 1994”, . . . ,“2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00”, . . . }
◮ e. g. dat(T1) = {f1, f2, f3, f4} where
f1 = {“cinema address cinema address cinema address”,“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12”, “cinema name cinema name cinema name”,“Druˇ zba Druˇ zba Druˇ zba”}, f2 = {“cinema address cinema address cinema address”,“Masarykova 2 Masarykova 2 Masarykova 2”, “cinema name cinema name cinema name”,“Capitol Capitol Capitol”}, f3 = {“cinema address cinema address cinema address”,“Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1”, “cinema name cinema name cinema name”,“´ Usmev ´ Usmev ´ Usmev”}, f4 = {“cinema address cinema address cinema address”,“Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8”, “cinema name cinema name cinema name”,“Tatra Tatra Tatra”}
Stanislav Krajˇ ci Logic Aspects of Databases
◮ met(T1) = “cinema
cinema cinema”, met(T2) = “movie movie movie”, met(T3) = “performance performance performance”
◮ database B = {T1, T2, T3} ◮ sch(B) = {“cinema
cinema cinema”,“movie movie movie”,“performance performance performance”}
◮ tab(B,“cinema
cinema cinema”) = T1, tab(B,“movie movie movie”) = T2, tab(B,“performance performance performance”) = T3
Stanislav Krajˇ ci Logic Aspects of Databases
◮ table T ′ 1 with a name “cinema
cinema cinema”:
“cinema address cinema address cinema address” “cinema name cinema name cinema name” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1” “´ Usmev ´ Usmev ´ Usmev” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8” “Tatra Tatra Tatra” “Hlavn´ a 59 Hlavn´ a 59 Hlavn´ a 59” “Slovan Slovan Slovan” ◮ T1 T ′ 1
because met(T ′
1) = metT1 and dat(T ′ 1) ⊇ datT1 ◮ if B′ = {T ′ 1, T2, T3} then
sch(B′) = sch(B) and B B′
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let n ∈ N and U is an n-element set of attributes,
let U = {u1, . . . , un} where u1 < · · · < un; if f : U → All then tup(f ) = f (u1), . . . , f (un)
◮ in a special case n = 0 clearly f = ∅,
hence tup(f ) =
◮ if f = {“cinema address
cinema address cinema address”,“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12”, “cinema name cinema name cinema name”,“Druˇ zba Druˇ zba Druˇ zba”}, then tup(f ) = “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12”,“Druˇ zba Druˇ zba Druˇ zba” because “cinema address cinema address cinema address” ≤ “cinema name cinema name cinema name”
Stanislav Krajˇ ci Logic Aspects of Databases
◮ (abstract) table relation:
if T is an abstract table then rel(T) = {tup(f ) : f ∈ dat(T)}
◮ clearly T1 T2 iff rel(T1) ⊆ rel(T2) ◮ in our example rel(T1) =
{“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12”,“Druˇ zba Druˇ zba Druˇ zba” “Masarykova 2 Masarykova 2 Masarykova 2”,“Capitol Capitol Capitol” “Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1”,“´ Usmev ´ Usmev ´ Usmev” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8”,“Slovan Slovan Slovan”}
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
If met(T) = M and rel(T) = R then dat(T) = {f : f : att(M) → All ∧ f (u1), . . . , f (un) ∈ R} where {u1, . . . , un} = att(M) and u1 < · · · < un.
◮ proof:
simply using definitions
Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ special symbols
◮ Dat – the only one (in our simplified version) ◮ Var – unspecified ◮ Con – the set All, they are interpreted by themselves ◮ Fun – none ◮ Pre – the set TbN
◮ a new auxiliary symbol ←
← ←
◮ i. e. terms are variables and constant symbols only
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let M1, . . . , Mn ∈ TbN and for all i ∈ {1, . . . , n},
mi = ari(Mi) and ti
1, . . . , ti mi are terms ◮ let K ∈ TbN, k = ari(K) and s1, . . . , sk are terms ◮ if k i=1 VarTrm(si) ⊆ n i=1
mi
j=1 VarTrm(ti j )
then the word K( ( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) ) is called a conjunctive query
◮ its head / body – the part before / after ←
← ←
◮ if S is a database schema, M1, . . . , Mn ∈ S and K /
∈ S then this is conjunctive query over S
Stanislav Krajˇ ci Logic Aspects of Databases
◮
Forrest Gump cinemas Forrest Gump cinemas Forrest Gump cinemas( ( (y y y, , ,x x x) ) )← ← ← cinema cinema cinema( ( (x x x, , ,y y y) ) ), , ,performance performance performance( ( (x x x, , ,z z z, , ,Forrest Gump Forrest Gump Forrest Gump) ) ) is a conjunctive query over S
◮ k = 2, n = 2, m1 = 2, m2 = 3 ◮ K = Forrest Gump cinemas
Forrest Gump cinemas Forrest Gump cinemas / ∈ S, M1 = cinema cinema cinema ∈ S, M2 = performance performance performance ∈ S
◮ s1 = y
y y ∈ Var, s2 = x x x ∈ Var, t1
1 = x
x x ∈ Var, t1
2 = y
y y ∈ Var, t2
1 = x
x x ∈ Var, t2
2 = z
z z ∈ Var, t2
3 = Forrest Gump
Forrest Gump Forrest Gump ∈ Con
◮ k i=1 VarTrm(si) = VarTrm(y
y y) ∪ VarTrm(x x x) = {x x x,y y y}, n
i=1
mi
j=1 VarTrm(ti j ) = (VarTrm(x
x x) ∪ VarTrm(y y y)) ∪ (VarTrm(x x x) ∪ VarTrm(z z z) ∪ VarTrm(Forrest Gump Forrest Gump Forrest Gump)) = {x x x,y y y,z z z}
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let B be a database and q be a conjunctive query over sch(B),
q = K( ( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) )
◮ answer of the database B to a conjunctive query q:
a table T = aC(B, q) s. t. met(T) = K and rel(T) = {E(s1), . . . , E(sk) : E : Var → All ∧ ∧ (∀i ∈ {1, . . . , n})E(ti
1), . . . , E(ti mi) ∈ rel(tab(B, Mi))}
Stanislav Krajˇ ci Logic Aspects of Databases
◮ the answer T of our database B to the previous conjunctive
query:
◮ met(T) = Forrest Gump cinemas
Forrest Gump cinemas Forrest Gump cinemas
◮ rel(T)
= {E(y y y), E(x x x) : E : Var → All ∧ ∧ E(x x x), E(y y y) ∈ rel(tab(B,cinema cinema cinema)) ∧ ∧ E(x x x), E(z z z), E(Forrest Gump Forrest Gump Forrest Gump) ∈ ∈ rel(tab(B,performance performance performance))} = {E(y y y), E(x x x) : E : Var → All ∧ ∧ E(x x x) = Masarykova 2 Masarykova 2 Masarykova 2 ∧ ∧ E(y y y) = Capitol Capitol Capitol ∧ ∧ E(z z z) = “2007-03-23 20:00” “2007-03-23 20:00” “2007-03-23 20:00”} = {Capitol Capitol Capitol,Masarykova 2 Masarykova 2 Masarykova 2}
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let S be a database schema and q be a conjunctive query
S, B1 B2 implies aC(B1, q) aC(B2, q).
◮ proof:
simply using definitions
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let S be a database schema and q be a conjunctive query
◮ proof:
◮ let q be K(
( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) )
◮ let C be the set of all constant symbols in q and a ∈ Con C ◮ for all M ∈ S take its instantiation TM s. t.
rel(TM) = (C ∪ {a})ari(M), and take B = {TM : M ∈ S}
◮ clearly TM = tab(B, M) and sch(B) = S ◮ let E be an evaluation of variables s. t. E(v) = a ◮ then for all i ∈ {1, . . . , n},
E(ti
1), . . . , E(ti mi) ∈ rel(TMi) = rel(tab(B, Mi))}
hence E(s1), . . . , E(sk) ∈ rel(aC(B, q))
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let S be a database schema and q be a conjunctive query
q = K( ( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) ) Then
Rng(f ) ⊆ ⊆
n
Rng(f ) ∪
k
ConTrm(sj).
◮ proof:
using definitions
Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ special symbols (almost the same as above)
◮ Dat – the only one (in our simplified version) ◮ Var – unspecified ◮ Con – the set All ◮ Fun – none ◮ Pre – the set TbN, moreover =
= =
◮ i. e. there is a new type of atomic formulae, namely t1 =
= = t2
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let M1, . . . , Mn ∈ TbN and for all i ∈ {1, . . . , n},
mi = ari(Mi) and ti
1, . . . , ti mi are terms ◮ let K ∈ TbN, k = ari(K) and s1, . . . , sk are terms ◮ let u1 1, u1 2, . . . , uℓ 1, uℓ 2 are terms ◮ if k i=1 VarTrm(si) ⊆
n
i=1
mi
j=1 VarTrm(ti j ) ∪ ℓ i=1
2
j=1 VarTrm(ui j)
then the word K( ( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) ), , ,u1
1 =
= = u1
2,
, , . . ., , ,uℓ
1 =
= = uℓ
2
is called an equality-conjunctive query
◮ if S is a database schema, M1, . . . , Mn ∈ S and K /
∈ S then this is equality-conjunctive query over S
◮ clearly each conjunctive query (over S) is an
equality-conjunctive query (over S) (for ℓ = 0)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let B be a database and q be a equality-conjunctive query
q = K( ( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) ), , ,u1
1 =
= = u1
2,
, , . . ., , ,uℓ
1 =
= = uℓ
2 ◮ answer of the database B to an equality-conjunctive query q:
a table T = aEC(B, q) s. t. met(T) = K and rel(T) = {E(s1), . . . , E(sk) : E : Var → All ∧ ∧ (∀i ∈ {1, . . . , n})E(ti
1), . . . , E(ti mi) ∈ rel(tab(B, Mi)) ∧
∧ (∀i ∈ {1, . . . , ℓ})E(ui
1) = E(ui 2)} ◮ for each database S and conjunctive query q over sch(B),
aEC(B, q) = aC(B, q)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let S be a database schema and q be an equality-conjunctive query over S. For all databases B1 a B2 which are instantiations of S, B1 B2 implies aEC(B1, q) aEC(B2, q).
◮ proof:
simply using definitions
Stanislav Krajˇ ci Logic Aspects of Databases
◮ an equality-conjunctive query q over S is called satisfiable
if there is a database B which is an instantiation of S s. t. rel(aEC(B, q)) = ∅
◮ an equality-conjunctive query need not be satisfiable:
K( ( (x) ) )← ← ← M( ( (x) ) ), , ,x = = = a, , ,x = = = b where a and b are different constants
Stanislav Krajˇ ci Logic Aspects of Databases
◮ the set of value-restricted equality-conjunctive query:
1 each conjunctive query is value-restricted 2 if i ∈ {0, . . . , ℓ}, v1 and v2 are terms s. t. v1 or v2 is an element of n
i=1
mi
j=1 VarTrm(ti j ) ∪ ℓ i=1
2
j=1 VarTrm(ui 1)
and K( ( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) ), , , u1
1 =
= = u1
2,
, , . . ., , ,uℓ
1 =
= = uℓ
2
is value-restricted, then K( ( (s1, , , . . ., , ,sk) ) )← ← ← M1( ( (t1
1,
, , . . ., , ,t1
m1)
) ), , , . . ., , ,Mn( ( (tn
1,
, , . . ., , ,tn
mn)
) ), , , u1
1 =
= = u1
2,
, , . . ., , ,ui
1 =
= = ui
2,
, ,v1 = = = v2, , ,ui+1
1
= = = ui+1
2
, , , . . ., , ,uℓ
1 =
= = uℓ
2
is value-restricted
Stanislav Krajˇ ci Logic Aspects of Databases
◮ a value-restricted equality-conjunctive query need not be
satisfiable:
K( ( (x, , ,y) ) )← ← ← M( ( (x) ) ), , ,y = = = z where x, y and z are different variables
◮ theorem:
Let S be a database schema. For all satisfiable value-restricted equality-conjunctive query q over S there exists conjunctive query q′ over S s. t. aEC(B, q) = aC(B, q′).
◮ i. e. these two sets of queries have the same expressive power,
they are equivalent in this sense
◮ proof:
◮ by induction through ℓ Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let S be a database schema;
the set CFrmS of conjunctive S-formulae (or shortly S-formulae):
1 if M ∈ S and t1, . . . , tm are terms then M( ( (t1, , , . . ., , ,tm) ) ) ∈ CFrmS 2a if ψ1, ψ2 ∈ CFrmS then ( ( (ψ1∧ ∧ ∧ψ2) ) ) ∈ CFrmS. 2b if ψ ∈ CFrmS and v ∈ Var then ( ( (( ( (∃ ∃ ∃v) ) )ψ) ) ) ∈ CFrmS.
◮ clearly each S-formula is a formula
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let B be a database s. t. sch(B) = S ◮ set IntPreRel(M) = rel(tab(B, M))
and take corresponding interpretation IntPreFun
◮ denote TV(B, E, ϕ) corresponding E(ϕ) ◮ clearly if E1 ↾ FVarFrm(ϕ) = E2 ↾ FVarFrm(ϕ)
then TV(B, E1, ϕ) = TV(B, E2, ϕ), hence this value depends on valuation of the variables from FVarFrm(ϕ) only
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let S be a database schema ◮ conjunctive calculus query: an expression
{ { {
, , . . ., , ,tn
: :ϕ} } } where t1, . . . , tn are terms, ϕ is a S-formula and
n
VarTrm(ti) = FVarFrm(ϕ).
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let B be a database and d be a conjunctive calculus query,
d = { { {
, , . . ., , ,tn
: :ϕ} } }
◮ answer of the database B to a conjunctive calculus query d:
aCC(B, d) = {E(t1), . . . , E(tn) : E : Var → All ∧ TV(B, E, ϕ) = 1}
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let B be a database, E be a variable valuation and ϕ be a sch(B)-formula. Then If TV(B, E, ϕ) = 1 then Rng(E ↾ FVarFrm(ϕ)) ⊆ act(B).
◮ proof:
◮ by the induction on ϕ
◮ corollary:
Let B be a database and d be a conjunctive calculus query. Then aCC(B, d) = {E(t1), . . . , E(tn) : E : Var → act ∧ TV(B, E, ϕ) = 1}
◮ this means that it is enough to verify the finite number of
valuations of the variables from FVarFrm(ϕ)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ S-formula ϕ is in the normal form
if there is a sequence (ϕ0, . . . , ϕn) s. t. ϕn = ϕ and there are k, ℓ ∈ {0, . . . , n}, k ≤ ℓ s. t.:
1 for each i ∈ {0, . . . , k}, ϕi is an atomic formula 2a for each i ∈ {k + 1, . . . , ℓ} there are h, j < i s. t. ϕi = ( ( (ϕh∧ ∧ ∧ϕj) ) ) 2b for each i ∈ {ℓ + 1, . . . , n} there is j < i and v ∈ Var s. t. ϕi = ( ( (( ( (∃ ∃ ∃v) ) )ϕj) ) )
◮ e. g. (
( (∃ ∃ ∃y y y) ) )( ( (∃ ∃ ∃z z z) ) )( ( (cinema cinema cinema( ( (Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8, , ,x x x) ) )∧ ∧ ∧performance performance performance( ( (x x x, , ,z z z, , ,y y y) ) )) ) )
◮ clearly ϕ is in the normal form
iff ( ( (( ( (∃ ∃ ∃v) ) )ϕ) ) ) is in the normal form
◮ a conjunctive calculus query is in the normal form
iff its formula is in the normal form
◮ {
{ {
x x
: :( ( (∃ ∃ ∃y y y) ) )( ( (∃ ∃ ∃z z z) ) )( ( (cinema cinema cinema( ( (Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8, , ,x x x) ) )∧ ∧ ∧performance performance performance( ( (x x x, , ,z z z, , ,y y y) ) )) ) )} } }
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let Var = {vi : i ∈ N} and all vi’s be different ◮ define function glu gluing two S-formulae in the normal form
into one: glu(( ( (∃ ∃ ∃vi1
1)
) ) . . .( ( (∃ ∃ ∃vi1
n1)
) )ψ1,( ( (∃ ∃ ∃vi2
1)
) ) . . .( ( (∃ ∃ ∃vi2
n2)
) )ψ2) = ( ( (∃ ∃ ∃vj1
1)
) ) . . .( ( (∃ ∃ ∃vj1
n1)
) )( ( (∃ ∃ ∃vj2
1)
) ) . . .( ( (∃ ∃ ∃vj2
n2)
) )( ( (ξ1∧ ∧ ∧ξ2) ) ) where ξh = Idvih
1 /jh 1 (. . . (Idvih nh /jh nh (ψh)) . . . ),
j1
k = i1 k + max({u : vu ∈ VarFrm(ψ1) ∪ VarFrm(ψ2)}),
j2
ℓ = i2 ℓ + max({j1 k : k ∈ {1, . . . , n1}})
Stanislav Krajˇ ci Logic Aspects of Databases
◮ define function NFF from S-formulae to S-formulae in the
normal form:
1 if ϕ is an atomic formula set NFF(ϕ) = ϕ 2a if ϕ = ( ( (ψ1∧ ∧ ∧ψ2) ) ) then NFF(ϕ) = glu(NFF(ψ1), NFF(ψ2)) 2b if ϕ = ( ( (( ( (∃ ∃ ∃v) ) )ψ) ) ) then NFF(ϕ) = ( ( (( ( (∃ ∃ ∃v) ) )NFF(ψ)) ) )
◮ clearly NFF(ϕ) is equivalent to ϕ ◮ corollary:
For each conjunctive query q there exists a conjunctive query in the normal form d′ such that d and d′ are equivalent.
Stanislav Krajˇ ci Logic Aspects of Databases
◮ theorem:
Let S be a database schema.
1 For each query of conjunctive calculus d in the normal form there exists a conjunctive query q s. t. for all databases B of the schema S aCC(B, d) = rel(aC(B, q)). 2 For each conjunctive query q there exists a query of conjunctive calculus d in the normal form s. t. for all databases B of the schema S aCC(B, d) = rel(aC(B, q)).
◮ i. e. conjunctive calculus (wlog in the normal form) and
conjunctive queries have the same expressive power
Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ selection
◮ σn
k,ℓ : P(Alln) → P(Alln) where n ∈ N and k, ℓ ∈ {1, . . . , n}:
σn
k,ℓ(R) = {x1, . . . , xn ∈ R : xk = xℓ}
◮ projection
◮ πn
j1,...,jk : P(Alln) → P(Alln) where if n ∈ N and
j1, . . . , jk ∈ {1, . . . , n}: πn
j1,...,jk(R) = {xj1, . . . , xjk :
(∃y1) . . . (∃yn)(y1, . . . , yn ∈ R ∧ (∀i ∈ {1, . . . , k})yji = xji)}
◮ cross-product (isomorphic (but not equal) to Cartesian one)
◮ κn,m : P(Alln) × P(Allm) → P(Alln+m) where if n, m ∈ N:
κn,m(R, S) = {x1, . . . , xn, y1, . . . , ym : x1, . . . , xn ∈ R ∧ y1, . . . , ym ∈ S}
◮ shortly (and not very correctly) R × S Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let n ∈ N, k1, k2, ℓ1, ℓ2 ∈ {1, . . . , n} and R ⊆ Alln. Then σn
k1,ℓ1(σn k2,ℓ2(R)) = σn k2,ℓ2(σn k1,ℓ1(R)). ◮ proof:
◮ σn
k1,ℓ1(σn k2,ℓ2(R))
= σn
k1,ℓ1({x1, . . . , xn ∈ R : xk2 = xℓ2})
= {y1, . . . , yn ∈ {x1, . . . , xn ∈ R : xk2 = xℓ2} : yk1 = yℓ1}) = {y1, . . . , yn ∈ R : yk2 = yℓ2 ∧ yk1 = yℓ1} = {y1, . . . , yn ∈ R : yk1 = yℓ1 ∧ yk2 = yℓ2} = {y1, . . . , yn ∈ {x1, . . . , xn ∈ R : xk1 = xℓ1} : yk2 = yℓ2}) = σn
k2,ℓ2({x1, . . . , xn ∈ R : xk1 = xℓ1})
= σn
k2,ℓ2(σn k1,ℓ1(R))
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let n, m, k ∈ N, j1, . . . , jk ∈ {1, . . . , n} i1, . . . , in ∈ {1, . . . , m} and R ⊆ Allm. Then πn
j1,...,jk(πm i1,...,in(R)) = πm ij1,...,ijk (R). ◮ proof:
◮ πn
j1,...,jk(πm i1,...,in(R))
= πn
j1,...,jk({xi1, . . . , xin : (∃y1, . . . , ym)(y1, . . . , ym ∈ R∧
∧ (∀p ∈ {1, . . . , n})yip = xip)}) = {zj1, . . . , zjk : (∃w1, . . . , wn)(w1, . . . , wn ∈ {xi1, . . . , xin : (∃y1, . . . , ym)(y1, . . . , ym ∈ R∧(∀p ∈ {1, . . . , n})yip = xip)} ∧ (∀q ∈ {1, . . . , k})wjq = zjq)} = {zj1, . . . , zjk : (∃w1, . . . , wn)(∃y1, . . . , ym)(y1, . . . , ym ∈ R ∧ (∀p ∈ {1, . . . , n})yip = wp ∧ (∀q ∈ {1, . . . , k})wjq = zjq)} = {zj1, . . . , zjk : (∃y1, . . . , ym)(y1, . . . , ym ∈ R ∧ (∀q ∈ {1, . . . , k})yijq = zjq)} = πm
ij1,...,ijk (R)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let R, S and T are relations. Then (R × S) × T = R × (S × T).
◮ proof:
◮ (R × S) × T
= {x1, . . . , xn, y1, . . . , ym:x1, . . . , xn∈R ∧ y1, . . . , ym∈S} × ×T = {x1, . . . , xn, y1, . . . , ym, z1, . . . , zk : (x1, . . . , xn ∈ R ∧ ∧ y1, . . . , ym ∈ S) ∧ z1, . . . , zk ∈ T} = {x1, . . . , xn, y1, . . . , ym, z1, . . . , zk : x1, . . . , xn ∈ R ∧ ∧ (y1, . . . , ym ∈ S ∧ z1, . . . , zk ∈ T)} = R × × {y1, . . . , ym, z1, . . . , zk:y1, . . . , ym∈S ∧ z1, . . . , zk∈T} = R × (S × T)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let m,n∈N, k,ℓ∈{1, . . . , m}, j1,. . . ,jk ∈{1, . . . , n}, R ⊆Alln. σm
k,ℓ(πn j1,...,jm(R)) = πn j1,...,jm(σn jk,jℓ(R)). ◮ proof:
◮ σm
k,ℓ(πn j1,...,jm(R))
= σm
k,ℓ(
{ xj1, . . . , xjk:(∃y1, . . . , yn∈R)(∀i∈{1, . . . , k})yji=xji) } ) = {z1, . . . , zm ∈ {xj1, . . . , xjk : (∃y1, . . . , yn) (y1, . . . , yn ∈ R ∧ (∀i ∈ {1, . . . , k})yji = xji)} : zk = zℓ} = {xj1, . . . , xjk : (∃y1, . . . , yn)(y1, . . . , yn ∈ R ∧ ∧ (∀i ∈ {1, . . . , k})yji = xji) ∧ xjk = xjℓ} = {xj1, . . . , xjk : (∃y1, . . . , yn)(y1, . . . , yn ∈ R ∧ ∧ (∀i ∈ {1, . . . , k})yji = xji ∧ yjk = yjℓ)} = {xj1, . . . , xjm : (∃y1, . . . , yn)(y1, . . . , yn ∈ {w1, . . . , wn ∈ R : wjk = wjℓ}∧(∀i ∈ {1, . . . , k})yji = xji)} = πm
j1,...,jm({w1, . . . , wn ∈ R : wjk = wjℓ})
= πm
j1,...,jm(σn jk,jℓ(R))
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let k, m, n ∈ N, j1, . . . , jk ∈ {1, . . . , n} and R ⊆ Allm, S ⊆ Alln. Then R × πn
j1,...,jk(S) = πm+n 1,...,m,m+j1,...,m+jk(R × S). ◮ proof:
◮ R × πn
j1,...,jk(S)
= R × {xj1, . . . , xjk : (∃y1) . . . (∃yn) (y1, . . . , yn ∈ S ∧ (∀i ∈ {1, . . . , k})yji = xji)} = {z1, . . . , zm, xj1, . . . , xjk : z1, . . . , zm ∈ R ∧ (∃y1) . . . (∃yn) (y1, . . . , yn ∈ S ∧ (∀i ∈ {1, . . . , k})yji = xji)} = {z1, . . . , zm, xj1, . . . , xjk : (∃y1, . . . , ym+n)(w1, . . . , ym+n ∈ R × S ∧ ∧(∀i ∈ {1, . . . , m})yi = zi ∧(∀i ∈ {1, . . . , k})ym+ji = xji)} = πm+n
1,...,m,m+j1,...,m+jk(R × S)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let k, m, n ∈ N, j1, . . . , jk ∈ {1, . . . , n} and R ⊆ Allm, S ⊆ Alln. Then πm
j1,...,jk(R) × S = πm+n j1,...,jk,m+1,...,m+n(R × S). ◮ proof:
◮ πm
j1,...,jk(R) × S
= {xj1, . . . , xjk : (∃y1) . . . (∃yn) (y1, . . . , ym ∈ R ∧ (∀i ∈ {1, . . . , k})yji = xji)} × S = {xj1, . . . , xjk, z1, . . . , zn : z1, . . . , zn ∈ S ∧ (∃y1) . . . (∃yn) (y1, . . . , ym ∈ R ∧ (∀i ∈ {1, . . . , k})yji = xji)} = {xj1, . . . , xjk, z1, . . . , zn : z1, . . . , zn ∈ S ∧ (∃y1, . . . , yn+m) (y1, . . . , yn+m ∈ R × S ∧ (∀i ∈ {1, . . . , k})yji = xji ∧ (∀i ∈ {1, . . . , n})yi+m = zi)} = πm+n
j1,...,jk,m+1,...,m+n(R × S)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let m, n ∈ N, k, ℓ ∈ {1, . . . , n} and R ⊆ Allm, S ⊆ Alln. Then R × σn
k,ℓ(S) = σm+n m+k,m+ℓ(R × S). ◮ proof:
◮ R × σn
k,ℓ(S)
= R × {x1, . . . , xn ∈ S : xk = xℓ} = {y1, . . . , ym, x1, . . . , xn ∈ R × S : xk = xℓ} = {z1, . . . , zm, zm+1, . . . , zm+n ∈ R × S : zm+k = zm+ℓ} = σm+n
m+k,m+ℓ(R × S)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let m, n ∈ N, k, ℓ ∈ {1, . . . , m} and R ⊆ Allm, S ⊆ Alln. Then σm
k,ℓ(R) × S = σm+n k,ℓ (R × S). ◮ proof:
◮ σm
k,ℓ(R) × S
= {x1, . . . , xm ∈ R : xk = xℓ} × S = {x1, . . . , xm, y1, . . . , yn ∈ R × S : xk = xℓ} × S = {z1, . . . , zm, zm+1, . . . , zm+n ∈ R × S : zk = zℓ} × S = σm+n
k,ℓ (R × S)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let m, n ∈ N, and R ⊆ Allm, S ⊆ Alln. Then S × R = πm+n
m+1,...,m+n,1,...,m(R × S). ◮ proof:
◮ S × R
= {z1, . . . , zm, zm+1, . . . , zm+n ∈ R × S} = {xm+1, . . . , xm+n, x1, . . . , xm : (∃y1) . . . (∃yn)(y1, . . . , yn ∈ R × S ∧ (∀i ∈ {1, . . . , m})yi = xn+i ∧ (∀i ∈ {1, . . . , n})ym+i = xn} = πm+n
m+1,...,m+n,1,...,m(R × S)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let S is a given database schema ◮ special symbols of the language for the SPC algebra)
◮ Dat = N ◮ Var = S;
if M ∈ Var then typVar(M) = ari(M)
◮ Con = {I
I Ia : a ∈ All}; typCon(I I Ia) = 1
◮ Pre = ∅ ◮ Fun – to be continued Stanislav Krajˇ ci Logic Aspects of Databases
◮ special function symbols
◮ Fun contains: ◮ unary σ
σ σn
k, , ,ℓ for all n ∈ N and k, ℓ ∈ {1, . . . , n};
sigFun(σ σ σn
k, , ,ℓ) = n, n ◮ unary π
π πn
j1, , ,..., , ,jk for all n ∈ N and j1, . . . , jk ∈ {1, . . . , n};
sigFun(π π πn
j1, , ,..., , ,jk ) = n, k ◮ binary κ
κ κn,
, ,m for all n, m ∈ N;
sigFun(κ κ κn,
, ,m) = n, m, n + m;
κ κ κn,
, ,m (
( (t1, , ,t2) ) ) will be abbreviated to t1 × × × t2
Stanislav Krajˇ ci Logic Aspects of Databases
◮ (pre-)interpretation
◮ IntDat(n) = P(Alln) ◮ IntCon(I
I Ia) = {a}
◮ IntFun(σ
σ σn
k, , ,ℓ) = σn k,ℓ
◮ IntFun(π
π πn
j1, , ,..., , ,jk) = πn j1,...,jk
◮ IntFun(κ
κ κn,
, ,m) = κn,m
◮ datatypes correspond to arities of engaged relations
Stanislav Krajˇ ci Logic Aspects of Databases
◮ an arbitrary term (in the sense of this language) is called a
SPC query
◮ if sch(B) = S
then EB(M) = rel(tab(B, M)) is the corresponding evaluation
and induced EB is the corresponding evaluation of variables
◮ if t is a SPC query then
aSPC(B, t) = EB(t) is called the answer of database B to a SPC query t
Stanislav Krajˇ ci Logic Aspects of Databases
◮ we want names and adresses all working cinemas ◮ a suitable term: t = π
π π5
5 5 2 2 2, , ,1 1 1(
( (σ σ σ5
5 5 2 2 2, , ,3 3 3(
( (cinema cinema cinema× × ×performance performance performance) ) )) ) )
◮ the answer to it:
aSPC(B, t) = = EB(t) = EB(π π π5
5 5 2 2 2, , ,1 1 1(
( (σ σ σ5
5 5 2 2 2, , ,3 3 3(
( (cinema cinema cinema× × ×performance performance performance) ) )) ) )) = π5
2,1(EB(σ
σ σ5
5 5 2 2 2, , ,3 3 3(
( (cinema cinema cinema× × ×performance performance performance) ) ))) = π5
2,1(σ5 2,3(EB(cinema
cinema cinema× × ×performance performance performance))) = π5
2,1(σ5 2,3(EB(cinema
cinema cinema) × EB(performance performance performance))) = π5
2,1(σ5 2,3(EB(cinema
cinema cinema) × EB(performance performance performance))) = π5
2,1(σ5 2,3(rel(tab(B,cinema
cinema cinema)) × rel(tab(B,performance performance performance))))
Stanislav Krajˇ ci Logic Aspects of Databases
◮ rel(tab(B,cinema
cinema cinema)) × rel(tab(B,performance performance performance)):
“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Capitol Capitol Capitol” “2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00” “Forrest Gump Forrest Gump Forrest Gump” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “´ Usmev ´ Usmev ´ Usmev” “2007-03-26 18:00 2007-03-26 18:00 2007-03-26 18:00” “The Shawshank The Shawshank The Shawshank. . . ” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “´ Usmev ´ Usmev ´ Usmev” “2007-03-27 16:00 2007-03-27 16:00 2007-03-27 16:00” “The Bridge over The Bridge over The Bridge over. . . ” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-28 16:00 2007-03-28 16:00 2007-03-28 16:00” “Back to the Future Back to the Future Back to the Future” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-29 20:00 2007-03-29 20:00 2007-03-29 20:00” “Mareˇ cku, podejte Mareˇ cku, podejte Mareˇ cku, podejte. . . ” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “Capitol Capitol Capitol” “2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00” “Forrest Gump Forrest Gump Forrest Gump” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “´ Usmev ´ Usmev ´ Usmev” “2007-03-26 18:00 2007-03-26 18:00 2007-03-26 18:00” “The Shawshank The Shawshank The Shawshank. . . ” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “´ Usmev ´ Usmev ´ Usmev” “2007-03-27 16:00 2007-03-27 16:00 2007-03-27 16:00” “The Bridge over The Bridge over The Bridge over. . . ” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-28 16:00 2007-03-28 16:00 2007-03-28 16:00” “Back to the Future Back to the Future Back to the Future” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-29 20:00 2007-03-29 20:00 2007-03-29 20:00” “Mareˇ cku, podejte Mareˇ cku, podejte Mareˇ cku, podejte. . . ” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “Capitol Capitol Capitol” “2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00” “Forrest Gump Forrest Gump Forrest Gump” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “´ Usmev ´ Usmev ´ Usmev” “2007-03-26 18:00 2007-03-26 18:00 2007-03-26 18:00” “The Shawshank The Shawshank The Shawshank. . . ” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “´ Usmev ´ Usmev ´ Usmev” “2007-03-27 16:00 2007-03-27 16:00 2007-03-27 16:00” “The Bridge over The Bridge over The Bridge over. . . ” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-28 16:00 2007-03-28 16:00 2007-03-28 16:00” “Back to the Future Back to the Future Back to the Future” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-29 20:00 2007-03-29 20:00 2007-03-29 20:00” “Mareˇ cku, podejte Mareˇ cku, podejte Mareˇ cku, podejte. . . ” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8” “Tatra Tatra Tatra” “Capitol Capitol Capitol” “2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00” “Forrest Gump Forrest Gump Forrest Gump” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8” “Tatra Tatra Tatra” “´ Usmev ´ Usmev ´ Usmev” “2007-03-26 18:00 2007-03-26 18:00 2007-03-26 18:00” “The Shawshank The Shawshank The Shawshank. . . ” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8” “Tatra Tatra Tatra” “´ Usmev ´ Usmev ´ Usmev” “2007-03-27 16:00 2007-03-27 16:00 2007-03-27 16:00” “The Bridge over The Bridge over The Bridge over. . . ” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8” “Tatra Tatra Tatra” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-28 16:00 2007-03-28 16:00 2007-03-28 16:00” “Back to the Future Back to the Future Back to the Future” “Hlavn´ a 8 Hlavn´ a 8 Hlavn´ a 8” “Tatra Tatra Tatra” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-29 20:00 2007-03-29 20:00 2007-03-29 20:00” “Mareˇ cku, podejte Mareˇ cku, podejte Mareˇ cku, podejte. . . ” Stanislav Krajˇ ci Logic Aspects of Databases
◮ σ5 2,3(rel(tab(B,cinema
cinema cinema)) × rel(tab(B,performance performance performance))):
“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-28 16:00 2007-03-28 16:00 2007-03-28 16:00” “Back to the Future Back to the Future Back to the Future” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-29 20:00 2007-03-29 20:00 2007-03-29 20:00” “Mareˇ cku, podejte Mareˇ cku, podejte Mareˇ cku, podejte. . . ” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “Capitol Capitol Capitol” “2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00” “Forrest Gump Forrest Gump Forrest Gump” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “´ Usmev ´ Usmev ´ Usmev” “2007-03-26 18:00 2007-03-26 18:00 2007-03-26 18:00” “The Shawshank The Shawshank The Shawshank. . . ” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “´ Usmev ´ Usmev ´ Usmev” “2007-03-27 16:00 2007-03-27 16:00 2007-03-27 16:00” “The Bridge over The Bridge over The Bridge over. . . ”
◮ i. e.
“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-28 16:00 2007-03-28 16:00 2007-03-28 16:00” “Back to the Future Back to the Future Back to the Future” “Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba” “Druˇ zba Druˇ zba Druˇ zba” “2007-03-29 20:00 2007-03-29 20:00 2007-03-29 20:00” “Mareˇ cku, podejte Mareˇ cku, podejte Mareˇ cku, podejte. . . ” “Masarykova 2 Masarykova 2 Masarykova 2” “Capitol Capitol Capitol” “Capitol Capitol Capitol” “2007-03-23 20:00 2007-03-23 20:00 2007-03-23 20:00” “Forrest Gump Forrest Gump Forrest Gump” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “´ Usmev ´ Usmev ´ Usmev” “2007-03-26 18:00 2007-03-26 18:00 2007-03-26 18:00” “The Shawshank The Shawshank The Shawshank. . . ” “Kas´ arensk´ e Kas´ arensk´ e Kas´ arensk´
“´ Usmev ´ Usmev ´ Usmev” “´ Usmev ´ Usmev ´ Usmev” “2007-03-27 16:00 2007-03-27 16:00 2007-03-27 16:00” “The Bridge over The Bridge over The Bridge over. . . ” Stanislav Krajˇ ci Logic Aspects of Databases
◮ π5 2,1(σ5 2,3(rel(tab(B,cinema
cinema cinema)) × rel(tab(B,performance performance performance)))):
“Druˇ zba Druˇ zba Druˇ zba”“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Druˇ zba Druˇ zba Druˇ zba”“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Capitol Capitol Capitol” “Masarykova 2 Masarykova 2 Masarykova 2” “´ Usmev ´ Usmev ´ Usmev” “Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1” “´ Usmev ´ Usmev ´ Usmev” “Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1”
◮ i. e.:
“Druˇ zba Druˇ zba Druˇ zba”“Hronsk´ a 12 Hronsk´ a 12 Hronsk´ a 12” “Capitol Capitol Capitol” “Masarykova 2 Masarykova 2 Masarykova 2” “´ Usmev ´ Usmev ´ Usmev” “Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1 Kas´ arensk´ e n´ amestie 1”
Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let n ∈ N, j ∈ {1, . . . , n} and a ∈ All ◮ then define a function ϕn j,a : P(Alln) → P(Alln)
ϕn
j,a(R) = {x1, . . . , xn ∈ R : xj = a} ◮ each such function is called fixing selection
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let n ∈ N, j1, j2 ∈ {1, . . . , n} and a1, a2 ∈ All. Then ϕn
j1,a1(ϕn j2,a2(R)) = ϕn j2,a2(ϕn j1,a1(R)). ◮ proof:
◮ ϕn
j1,a1(ϕn j2,a2(R))
= ϕn
j1,a1({x1, . . . , xn ∈ R : xj2 = a2})
= {x1, . . . , xn ∈ R : xj1 = a1 ∧ xj2 = a2} = {x1, . . . , xn ∈ R : xj2 = a2 ∧ xj1 = a1} = ϕn
j2,a2({x1, . . . , xn ∈ R : xj1 = a1})
= ϕn
j2,a2(ϕn j1,a1(R))
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let n ∈ N, j, k, ℓ ∈ {1, . . . , n} and a ∈ All. Then ϕn
j,a(σn k,ℓ(R)) = σn k,ℓ(ϕn j,a(R)). ◮ proof:
◮ ϕn
j,a(σn k,ℓ(R))
= ϕn
j,a({x1, . . . , xn ∈ R : xk = xℓ})
= {x1, . . . , xn ∈ R : xj = a ∧ xk = xℓ} = {x1, . . . , xn ∈ R : xk = xℓ ∧ xj = a} = σn
k,ℓ({x1, . . . , xn ∈ R : xj = a})
= σn
k,ℓ(ϕn j,a(R))
Stanislav Krajˇ ci Logic Aspects of Databases
◮ lemma:
Let n ∈ N, j ∈ {1, . . . , n} and a ∈ All. Then ϕn
j,a(R) = πn+1 1,...,n
j,n+1(R × {a})
◮ proof:
◮ πn+1
1,...,n(σn+1 j,n+1(R × {a}))
πn+1
1,...,n({x1, . . . , xn+1 ∈ R × {a} : xj = xn+1})
= πn+1
1,...,n({x1, . . . , xn, a ∈ R × {a} : xj = a})
= {x1, . . . , xn ∈ R : xj = a} = ϕn
j,a(R)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ define for fixed n ∈ N functions µn F,G : P(Alln) → P(Alln)
where F = j1, a1, . . . , ju, au, G = k1, ℓ1, . . . , kv, ℓv, a1, . . . , au ∈ All, j1, . . . , ju, k1, . . . , kv, ℓ1, . . . , ℓv ∈ {1, . . . , n}
1 for all R ⊆ Alln µn
,(R) = R
2a if j ∈ {1, . . . , n} and a ∈ All then for all R ⊆ Alln µn
F ||j,a,G(R) = ϕn j,a
F,G(R)
µn
F,G ||k,ℓ(R) = σn k,ℓ
F,G(R)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ let F = j1, a1, . . . , ju, au, G = k1, ℓ1, . . . , kv, ℓv ◮ let p is a permutation of {1, . . . , u}
and q is a permutation of {1, . . . , v}
◮ let Fp = jp(1), ap(1), . . . , jp(u), ap(u)
and Gq = kq(1), ℓq(1), . . . , kq(v), ℓq(v)
◮ clearly
µn
F,G = µn Fp,Gq
(because functions σn
k,ℓ and ϕn j,a commute) ◮ clearly
µn
F1,G1(µn F2,G2(R)) = µn F2,G2(µn F1,G1(R))
(again because functions σn
k,ℓ and ϕn j,a commute)
Stanislav Krajˇ ci Logic Aspects of Databases
◮ define for fixed n, m ∈ N, u ∈ N, k1, . . . , ku ∈ {1, . . . , n}
ℓ1, . . . , ℓu ∈ {1, . . . , m} and H = k1, ℓ1, . . . , ku, ℓu function ⊲ ⊳m,n
H : P(Alln) → P(Alln):
⊲ ⊳m,n
H
(R, S) = {x1, . . . , xn, y1, . . . , ym : x1, . . . , xn ∈ R ∧ y1, . . . , ym ∈ S ∧(∀i ∈ {1, . . . , u})xki = yℓi}
◮ each such function is called a join
Stanislav Krajˇ ci Logic Aspects of Databases
◮ if H = k1, ℓ1, . . . , ku, ℓu and a, b ∈ N
then set Ha,b = {a + k1, b + ℓ1, . . . , a + ku, b + ℓu}
◮ lemma:
Let m, n, u ∈ N, j ∈ {1, . . . , n}, k1, . . . , ku ∈ {1, . . . , m}, ℓ1, . . . , ℓu ∈ {1, . . . , n}, H = k1, ℓ1, . . . , ku, ℓu and R ⊆ Allm, S ⊆ Alln. Then ⊲ ⊳m,n
H
(R, S) = µm+n
,H0,n(R × S). ◮ proof:
◮ by induction on m Stanislav Krajˇ ci Logic Aspects of Databases
◮ as for SPC algebra but: ◮ Fun contains:
◮ unary µ
µ µn
, ,a1
, ,..., , ,
, ,au
, ,
, ,ℓ1
, ,..., , ,
, ,ℓv
j1, . . . , ju, k1, . . . , kv, ℓ1, . . . , ℓv ∈ {1, . . . , n}, a1, . . . , au ∈ All; sigFun(µ µ µn
, ,a1
, ,..., , ,
, ,au
, ,
, ,ℓ1
, ,..., , ,
, ,ℓv
◮ unary π
π πn
j1, , ,..., , ,jk as before
◮ binary κ
κ κn,
, ,m as before
◮ binary ⊲
⊳ ⊲ ⊳ ⊲ ⊳n,
, ,m
, ,ℓ1
, ,..., , ,
, ,ℓu
k1, . . . , ku ∈ {1, . . . , n} ℓ1, . . . , ℓu ∈ {1, . . . , m}; sigFun(⊲ ⊳ ⊲ ⊳ ⊲ ⊳n,
, ,m
, ,ℓ1
, ,..., , ,
, ,ℓu
◮ expected interpretation of the new symbols
◮ IntFun(µ
µ µn
, ,a1
, ,..., , ,
, ,au
, ,
, ,ℓ1
, ,..., , ,
, ,ℓv
µn
j1,a1,...,ju,au,k1,ℓ1,...,kv,ℓv
◮ IntFun(⊲
⊳ ⊲ ⊳ ⊲ ⊳n,
, ,m
, ,ℓ1
, ,..., , ,
, ,ℓu
⊳n,m
k1,ℓ1,...,ku,ℓu
Stanislav Krajˇ ci Logic Aspects of Databases
◮ an arbitrary term (in the sense of this language) is called a
SPCJ query
◮ if sch(B) = S
then EB(M) = rel(tab(B, M)) is the corresponding evaluation
and induced EB is the corresponding evaluation of variables
◮ if t is a SPCJ query then
aSPCJ(B, t) = EB(t) is called the answer of database B to a SPCJ query t
Stanislav Krajˇ ci Logic Aspects of Databases
◮ an SPCJ-term t is in the normal form
if there is a sequence (t0, . . . , tp) s. t. tp = t and there are k > 0 and ℓ s. t. 2k + 1 + 2ℓ + 1 = n and:
◮ for each i ∈ {0, . . . , k}, ti ∈ Var ◮ for each i ∈ {k + 1, . . . , 2k}, ti = (
( (ti−2 × × × ti−1) ) )
◮ t2k+1 = µ
µ µn
, ,a1
, ,..., , ,
, ,au
, ,
, ,ℓ1
, ,..., , ,
, ,ℓv
( (t2k) ) ) for some suitable parameters
◮ for each i ∈ {2k + 2, . . . , 2k + 1 + ℓ}, ti ∈ Con ◮ for each i ∈ {2k + 2 + ℓ, . . . , 2k + 1 + 2ℓ}, ti = (
( (ti−2 × × × ti−1) ) )
◮ tp = π
π πn
j1, , ,..., , ,jk(
( (tp−1) ) ) for some suitable parameters
◮ i. e. a normal form is simply
π π π...
...(
( (µ µ µ...
...(
( (M1 × × × . . .× × × Mk) ) )× × ×I I Ia1 × × × . . .× × ×I I Iaℓ) ) ) where M1, . . . , Mk ∈ S and a1, . . . , aℓ ∈ All
Stanislav Krajˇ ci Logic Aspects of Databases
◮ a normal form
π π π...
...(
( (µ µ µ...
...(
( (M1 × × × . . .× × × Mk) ) )× × ×I I Ia1 × × × . . .× × ×I I Iaℓ) ) ) corresponds to an SQL query SELECT columns given by down-indices of π π π (including ai’s) FROM M1, . . . , Mk WHERE conditions given by down-indices of µ µ µ
Stanislav Krajˇ ci Logic Aspects of Databases
◮ theorem:
Let S be a database schema. For each SPCJ query d there exists a SPCJ query e in the normal form s. t. for all databases B of the schema S aSPCJ(B, d) = aSPCJ(B, e).
◮ proof:
◮ rewrite each ⊲
⊳...
... by means of µ... ...’s and ×’s
◮ rewrite each µ...
... by means of σ... ...’s and ϕ... ...’s
◮ use changing lemmas in order to obtain a form
π...
...(. . . (π... ...((X × Y ))) . . . ) where Y = {a1} × · · · × {a...},
X = ϕ...
...(. . . (ϕ... ...(Z)) . . . ), Z = σ... ...(. . . (σ... ...(W )) . . . ) and
W = reltab(B, M1) × reltab(B, Mk)
◮ rewrite π...
...(. . . (π... ...((X × Y ))) . . . ) as one π... ...(X × Y )
◮ rewrite X as µ...
...,(Z)
◮ rewrite Z as µ...
,...(W )
◮ group this two consequent µ...
... to one
Stanislav Krajˇ ci Logic Aspects of Databases
◮ theorem:
Let S be a database schema.
1 For each satisfiable SPCJ query d in the normal form there exists a conjunctive query q s. t. for all databases B of the schema S aSPCJ(B, d) = rel(aC(B, q)). 2 For each conjunctive query q there exists a satisfiable SPCJ query d in the normal form s. t. for all databases B of the schema S aSPCJ(B, d) = rel(aC(B, q)).
◮ i. e. SPC(J) algebra (wlog in the normal form) and
conjunctive queries have the same expressive power
Stanislav Krajˇ ci Logic Aspects of Databases
Stanislav Krajˇ ci Logic Aspects of Databases
◮ above-mentioned models of databases:
◮ conjunctive queries ◮ equality-conjunctive queries ◮ queries of conjunctive calculus ◮ queries of conjunctive calculus in the normal form ◮ queries of the relational SPC algebra ◮ queries of the relational SPCJ algebra ◮ queries of the relational SPCJ algebra in the normal form
◮ roughly speaking, all of them have the same expressive power
Stanislav Krajˇ ci Logic Aspects of Databases
◮ adding of negation ◮ adding of recursion ◮ . . .
Stanislav Krajˇ ci Logic Aspects of Databases