Relational Calculus Chapter 4 – Part II

Formal Relational Query Languages Two mathematical Query Languages form the basis for “real” languages (e.g. SQL), and for implementation: Relational Algebra: how to compute it , operational, Representing execution plans (inside) Relational Calculus: what you want, declarative. Influent SQL design (outside)

Relational Calculus Calculus has Comes in two flavours: variables, constants, comparison ops (<,>,=,,.), logical connectives ( ,) Quantifiers (,) Comes in two flavours: Tuple relational calculus (TRC) : Variables range over tuples. (SQL) Domain relational calculus (DRC). Variables range over domain elements. (QBE) Both formula in TRC and DRC are simple subsets of first-order logic.

Tuple Relational Calculus (TRC) Tuple variable: variable that takes on tuples of a relation schema as values. Query form: T = tuple variable p(T) = formula that describes T Result = set of all tuples t for which p(t) with T=t evaluates to be true Formula P(T) specified in First Order Logic

Find all sailors with a rating above 7 Variable S bound to a tuple in relation Sailors

TRC Formulas Formula: an atomic formula Or recursively defined Terms. R  Rel, R.a op S.b, R.a op constant, costant op R.a Or recursively defined p, pq, pq, pq , where p, q are formulas R(p(R)), where R is a tuple variable R(p(R)), where R is a tuple varialbe Terms. bind --- the quantifier  and  are bind to variable R. free --- a variable is free if no quantifier binds to it in a formula

Semantics of TRC Queries F is an atomic formula R  Rel, and R is assigned a tuple in the instance of relation Rel {S|SSailors} F is a comparison R.a op S.b, R.a op constant, or constant op R.a, and the tuples assigned to R and S have field names R.a and S.b that make the comparison true. F is of the form p, and p is not true; the form p  q, and both p and q are true; the form p  q, and one of them is true; the form p  q, and q is true whenever p is true. F is of the form R(p( R )), and there is some assignment of tuples to the free variables in p ( R ), including the variable R that make the formula p( R ) true; F is of the form R (p ( R )), and there is some assignment of tuples to the free variables in p( R ) that makes the formula p( R ) true no matter what tuple is assigned to R.

TRC Example Reserves Sailors

Find the names and ages of sailors with rating > 7 { P | S  Sailors (S.rating > 7  P.name = S.sname  P.age = S.age)} P is a Tuple variable with two fields: name and age. - the only fields mentioned in P - P does not range over any relation in the query. Sailors name age Lubber 55.0 rusty 35.0

Find the names of the sailors who reserved boat 103 { P | S  Sailors R  Reserves (R.sid = S.sid  R.bid = 103  P.sname = S.sname)} Retrieve all sailor tuples for which there exists a tuple in Reserves having the same value in the sid field and with bid =103 Question: How is the answer tuple looks like? (columns? Rows?) What is difference? { S | S  Sailors  R  Reserves (R.sid = S.sid  R.bid = 103)}

Find the names of the sailors who have reserved a red boat { P | S  Sailors R  Reserves B  Boats (R.sid = S.sid  B.bid = R.bid  B.color ='red'  P.sname = S.sname)} Retrieve all sailor tuples S for which there exists tuples R in Reserves and B in Boats such that R.sid = S.sid  B.bid = R.bid  B.color ='red' { P | S  Sailors R  Reserves (R.sid = S.sid  P.sname = S.sname  B  Boats(B.bid = R.bid  B.color ='red'))}

Find the names of the sailors who have reserved all boat { P | S  Sailors B  Boats (R  Reserves (S.sid = R.sid  R.bid = B.bid  P.sname = S.sname))} Find sailors S such that for all boats B there is a Reserves tuple showing that sailor S has reserved boat B.

Find the sailors who have reserved all red boat { P | S  Sailors  B  Boats (B.color = 'red'  (R  Reserves (S.sid = R.sid  R.bid = B.bid))} Question: How to change the algebraic representation?

Find the sailors who have reserved all red boat Logically p  q is equivalent to pq (Why?) { P | S  Sailors  B  Boats (B.color = 'red'  (R  Reserves (S.sid = R.sid  R.bid = B.bid))} is rewritten as: { P | S  Sailors  B  Boats (B.color 'red'  (R  Reserves (S.sid = R.sid  R.bid = B.bid))} to restrict attention to those red boat.

Domain Relational Calculus Query has the form: Answer includes all tuples that make the formula be true.

DRC Formulas Or recursively defined Atomic formula: <x1,x2,…,xn>  Rel , where Rel is a relation with n variables X op Y X op constant op is one of comparison ops (<,>,=,,.), Or recursively defined p, pq, pq, pq , where p, q are formulas X(p(X)), where X is a domain variable X(p(X)), where X is a domain varialbe

Free and Bound Variables The use of quantifiers and in a formula is said to bind X. A variable that is not bound is free. Let us revisit the definition of a query: There is an important restriction: the variables x1, ..., xn that appear to the left of `|’ must be the only free variables in the formula p(...).

Find all sailors with a rating above 7 the domain variables I, N, T and A are bound to fields of the same Sailors tuple. that every tuple that satisfies T>7 is in the answer. Question: Modify to answer: Find sailors who are older than 18 or have a rating under 9, and are called ‘Joe’.

Find sailors rating > 7 who’ve reserved boat #103 Reserves Boats bid bname color 101 interlake red 103 marine green We have used as a shorthand for Note the use of to find a tuple in Reserves that `joins with’ the Sailors tuple under consideration. {<I,N,T,A>|<I,N,T,A> Sailors T>7   Ir,Br,D (<Ir,103,D>  Reserves  Ir =I )}

Find sailors rated > 7 who’ve reserved a red boat Reserves Boats bid bname color 101 interlake red 103 marine green Observe how the parentheses control the scope of each quantifier’s binding. Rewrite using 'red' as constant BN = red

Find sailors who’ve reserved all boats Reserves Boats bid bname color 101 interlake red 103 marine green Find all sailors I such that for each 3-tuple either it is not a tuple in Boats or there is a tuple in Reserves showing that sailor I has reserved it.

Algebra vs. Calculus Two formal query languages are equal in the power of expressivness? Algebra  Calculus? YES! Calculus  Algebra? Unsafe query {S | (S  Sailors)} Correct??? Safe TRC --- Dom(Q , I) answers for Q contains only values that are in Dom(Q, I) R(p(R)), tuple r contains only constant in Dom(Q,I) R(p(R)), tuple r contains a constant that is not in Dom(Q,I), then p(r) is true.

Summary Algebra and safe calculus have same expressive power Relationally complete --- if a query language can express all the queries that can be expressed in relational algebra. Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.)