Relational Calculus Dr Paolo Guagliardo University of Edinburgh Fall 2016 First-order logic term t := x (variable) j c (constant) j f(t1; : : : ; tn) (function application) formula ' := P (t1; : : : ; tn) j t1 op t2 with op 2 f=; 6=; >; <; >; 6g j '1 ^ '2 j '1 _ '2 j :' j '1 ! '2 j 9x ' j 8x ' if x 2 free(') free(') = f variables that are not in the scope of any quantifier g Notation: we write 9x19x2 · · · 9xn ' as 9x1; : : : ; xn ' 1 / 24 Relational calculus A relational calculus query is an expression of the form fx¯ j 'g where the set of variables in x¯ is free(') Examples I Q = x; y j 9z R(x; z) ^ S(z; y) I Q = y; x j 9z R(x; z) ^ S(z; y) I Q = fx; x j 8y R(x; y)g Queries without free variables are called Boolean queries Examples I Q = f() j 8x R(x; x)g I Q = f() j 8x 9y R(x; y)g 2 / 24 Q2: Name and age of customers having an account in London y; z j 9x Customer(x; y; z) ^ 9w Account(w; `London'; x) Q3: ID of customers who have an account in every branch x j 9y; z Customer(x; y; z) ^ 8u; w; v Account(u; w; v) ! 9u0 Account(u0; w; x) Examples Customer : ID, Name, Age Account : Number, Branch, CustID Q1: Name of customers younger than 33 or older than 50 y j 9x; z Customer(x; y; z) ^ (z < 33 _ z > 50) 3 / 24 Q1: Name of customers younger than 33 or older than 50 y j 9x; z Customer(x; y; z) ^ (z < 33 _ z > 50) Q3: ID of customers who have an account in every branch x j 9y; z Customer(x; y; z) ^ 8u; w; v Account(u; w; v) ! 9u0 Account(u0; w; x) Examples Customer : ID, Name, Age Account : Number, Branch, CustID Q2: Name and age of customers having an account in London y; z j 9x Customer(x; y; z) ^ 9w Account(w; `London'; x) 3 / 24 Q1: Name of customers younger than 33 or older than 50 y j 9x; z Customer(x; y; z) ^ (z < 33 _ z > 50) Q2: Name and age of customers having an account in London y; z j 9x Customer(x; y; z) ^ 9w Account(w; `London'; x) Examples Customer : ID, Name, Age Account : Number, Branch, CustID Q3: ID of customers who have an account in every branch x j 9y; z Customer(x; y; z) ^ 8u; w; v Account(u; w; v) ! 9u0 Account(u0; w; x) 3 / 24 Standard Name Assumption (SNA) Every constant is intepreted as itself: cI = c Interpretations First-order structure I ∆ non-empty domain of objects (universe) ·I gives meaning to constant/function/relation symbols cI 2 ∆ f I : ∆n ! ∆ RI ⊆ ∆n 4 / 24 Interpretations First-order structure I ∆ non-empty domain of objects (universe) ·I gives meaning to constant/function/relation symbols cI 2 ∆ f I : ∆n ! ∆ RI ⊆ ∆n Standard Name Assumption (SNA) Every constant is intepreted as itself: cI = c 4 / 24 The answer to a Boolean query is either f()g (true) or ? (false) Answers to queries I Fix an underlying domain ∆ under SNA I Fix an underlying interpretation of function symbols =) \interpretations" are just databases Recall: an assignment α maps variables to objects in ∆ The answer to a query Q = fx¯ j 'g on a database D is Q(D) = α(¯x) j α: free(') ! ∆ such that D; α j= ' 5 / 24 Answers to queries I Fix an underlying domain ∆ under SNA I Fix an underlying interpretation of function symbols =) \interpretations" are just databases Recall: an assignment α maps variables to objects in ∆ The answer to a query Q = fx¯ j 'g on a database D is Q(D) = α(¯x) j α: free(') ! ∆ such that D; α j= ' The answer to a Boolean query is either f()g (true) or ? (false) 5 / 24 Bad news Whether a relational calculus query is safe is undecidable Safety A query is safe if it gives a finite answer on all databases Examples of unsafe queries: I x j :R(x) I x; y j R(x) _ R(y) I x; y j x = y Question: Are Boolean queries safe? 6 / 24 Safety A query is safe if it gives a finite answer on all databases Examples of unsafe queries: I x j :R(x) I x; y j R(x) _ R(y) I x; y j x = y Question: Are Boolean queries safe? Bad news Whether a relational calculus query is safe is undecidable 6 / 24 Active domain Adom(R) = f all constants occuring in R g Example 0 R AB 1 Adom@ a1 b1 A = a1; b1; b2 a1 b2 The active domain of a database D is [ Adom(D) = Adom(R) R2D 7 / 24 Active domain and safety For a safe query Q, we have that AdomQ(D) ⊆ Adom(D) Active domain semantics Evaluate queries within Adom(D)= ) safe relational calculus Q(D) = f α(¯x) j α: free(') ! Adom(D) s.t. D; α j= ' For each α: free(') ! Adom(D) (there are finitely many) output α(¯x) whenever D; α j= ' 8 / 24 Algebra ≡ Safe calculus Fundamental theorem of database theory: Relational algebra and Safe relational calculus equally expressive I For every query in safe relational calculus there exists an equivalent query in relational algebra I For every query in relational algebra there exists an equivalent query in safe relational calculus 9 / 24 From algebra to calculus Translate each RA expression E into a FOL formula ' Assumption: the attributes of a relation are ordered (R over A; B; C means the 1st column is A, the 2nd is B, the 3rd is C) Environment η Maps each attribute A in the schema to a variable xA 10 / 24 Example If R is a base relation over A; B η = f A 7! xA;B 7! xB;::: g then R is translated to R(xA; xB) From algebra to calculus Base relation R over A1;:::;An is translated to R η(A1); : : : ; η(An) 11 / 24 From algebra to calculus Base relation R over A1;:::;An is translated to R η(A1); : : : ; η(An) Example If R is a base relation over A; B η = f A 7! xA;B 7! xB;::: g then R is translated to R(xA; xB) 11 / 24 Example If R is a base relation over A; B then ρA!B ρB!C (R) is translated to R(xB; xC ) From algebra to calculus Renaming ρA!B(E) 1. Translate E to ' 2. If there is no mapping for B in η, add fB 7! xBg 3. Replace every occurrence of η(A) in ' by η(B) 12 / 24 From algebra to calculus Renaming ρA!B(E) 1. Translate E to ' 2. If there is no mapping for B in η, add fB 7! xBg 3. Replace every occurrence of η(A) in ' by η(B) Example If R is a base relation over A; B then ρA!B ρB!C (R) is translated to R(xB; xC ) 12 / 24 From algebra to calculus Projection πα(E) is translated to 9X' where I ' is the translation of E I X = free(') − η(α) (attributes that are not projected become quantified) Example If R is a base relation over A; B then πA(R) is translated into 9xB R(xA; xB) 13 / 24 From algebra to calculus Selection σθ(E) is translated to ' ^ η(θ) where I ' is the translation of E I η(θ) is obtained from θ by replacing each attribute A by η(A) Example If R is a base relation over A; B then σA=B(R) is translated into R(xA; xB) ^ xA = xB 14 / 24 From algebra to calculus Cartesian Product, Union, Difference Product E1 × E2 is translated to '1 ^ '2 Union E1 [ E2 is translated to '1 _ '2 Difference E1 − E2 is translated to '1 ^ :'2 where I '1 is the translation of E1 I '2 is the translation of E2 15 / 24 Blackboard time! 9x4 Customer(x1; x2) ^ Account(x3; x4) ^ x1 = x4 Example Customer : CustID, Name Account : Number, CustID Environment η = f CustID 7! x1; Name 7! x2; Number 7! x3 g How do we translate Customer ./ Account? 16 / 24 9x4 Customer(x1; x2) ^ Account(x3; x4) ^ x1 = x4 Example Customer : CustID, Name Account : Number, CustID Environment η = f CustID 7! x1; Name 7! x2; Number 7! x3 g How do we translate Customer ./ Account? Blackboard time! 16 / 24 Example Customer : CustID, Name Account : Number, CustID Environment η = f CustID 7! x1; Name 7! x2; Number 7! x3 g How do we translate Customer ./ Account? Blackboard time! 9x4 Customer(x1; x2) ^ Account(x3; x4) ^ x1 = x4 16 / 24 Active domain in relational algebra For R over attributes A1;:::;An Adom(R) = ρA1!A πA1 (R) [···[ ρAn!A πAn (R) S Adom(D) = R2D Adom(R) AdomA is the relation Adom(D) over attribute A 17 / 24 From calculus to algebra Translate each FOL formula ' into an RA expression E Assumptions (without loss of generality) I No universal quantifiers, implications, double negations I No distinct pair of quantifiers binds the same variable I No variable occurs both free and bound I No variable is repeated within a predicate I No constants in predicates I No atoms of the form x op x or c1 op c2 Environment η Maps each variable x to an attribute Ax 18 / 24 From calculus to algebra Let R be over attributes A1;:::;An Predicate R(x1; : : : ; xn) is translated to ρA1!η(x1);:::;An!η(xn)(R) Example For R over attributes A; B; C, R(x; y; z) is translated into ρA!Ax;B!Ay;C!Az (R) 19 / 24 From calculus to algebra Existential quantification 9x ' is translated to πη(X−{xg)(E) where I E is the translation of ' I X = free(') Example For ' with free variables x; y; z and translation E, 9y ' is translated to πAx;Az (E) 20 / 24 From calculus to algebra Comparisons x op y is translated to ση(x) op η(y) Adomη(x) × Adomη(y) x op c is translated to ση(x) op c Adomη(x) Example x = y is translated to σAx=Ay AdomAx × AdomAy x > 1 is translated to σAx>1 AdomAx 21 / 24 From calculus to algebra Negation :' is translated into ×Adomη(x) − E x2free(') where E is the translation of ' Example For ' with free variables x; y and translation E :' is translated to AdomAx × AdomAy −E 22 / 24 From calculus to algebra Disjunction: '1 _ '2 is translated to E1 × ×Adomη(x) [ E2 × ×Adomη(x) x2X2−X1 x2X1−X2 where, for i 2 f1; 2g, I Ei is the translation of 'i I Xi = free('i) Conjunction: same as disjunction, but use \ instead of [ 23 / 24 Environment η = f x1 7! A; x2 7! B; x3 7! C; x4 7! D g πA;B;C E1 × AdomC × AdomD \ AdomA × AdomB × E2 \ σA=D(AdomA × AdomD) × AdomB × AdomC where I E1 = ρ CustID!A; Name!B(Customer) I E2 = ρ Number!C; CustID!D(Account) Example Customer : CustID, Name Account : Number, CustID Translate 9x4 Customer(x1; x2) ^ Account(x3; x4) ^ x1 = x4 24 / 24 πA;B;C E1 × AdomC × AdomD \ AdomA × AdomB × E2 \ σA=D(AdomA × AdomD) × AdomB × AdomC where I E1 = ρ CustID!A; Name!B(Customer) I E2 = ρ Number!C; CustID!D(Account) Example Customer : CustID, Name Account : Number, CustID Translate 9x4 Customer(x1; x2) ^ Account(x3; x4) ^ x1 = x4 Environment η = f x1 7! A; x2 7! B; x3 7! C; x4 7! D g 24 / 24 Example Customer : CustID, Name Account : Number, CustID Translate 9x4 Customer(x1; x2) ^ Account(x3; x4) ^ x1 = x4 Environment η = f x1 7! A; x2 7! B; x3 7! C; x4 7! D g πA;B;C E1 × AdomC × AdomD \ AdomA × AdomB × E2 \ σA=D(AdomA × AdomD) × AdomB × AdomC where I E1 = ρ CustID!A; Name!B(Customer) I E2 = ρ Number!C; CustID!D(Account) 24 / 24.