Relational Calculus
Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (DRC). Relational Calculus Calculus has variables, constants, comparison ops, logical connectives and quantifiers. TRC: Variables range over (i.e., get bound to) tuples. Chapter 4, Part B DRC: Variables range over domain elements (= field values). Both TRC and DRC are simple subsets of first-order logic. Expressions in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true.
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Domain Relational Calculus DRC Formulas
Query has the form: Atomic formula: x12,x ,...,xn∈ Rname , or X op Y, or X op constant xx12, ,..., xnpxx | 12 , ,..., xn op is one of <,,,,,> = ≤ ≥ ≠ Answer includes all tuples x 12 , x ,..., x n that Formula: make the formula px 12, x ,..., xn be true. an atomic formula, or ¬ pp,,∧ qp∨ q, where p and q are formulas, or Formula is recursively defined, starting with ∃X (())p X , where variable X is free in p(X), or simple atomic formulas (getting tuples from relations or making comparisons of values), ∀ X (())p X , where variable X is free in p(X) and building bigger and better formulas using The use of quantifiers ∃ X and ∀ X is said to bind X. the logical connectives. A variable that is not bound is free. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 3 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 4
Free and Bound Variables Find all sailors with a rating above 7
INTA,,, |,,, INTA∈∧> Sailors T 7 The use of quantifiers ∃ X and ∀ X in a formula is said to bind X. I,,,N T A ∈Sailors A variable that is not bound is free. The condition ensures that the domain variables I, N, T and A are bound to Let us revisit the definition of a query: fields of the same Sailors tuple. xx12, ,..., xnpxx | 12 , ,..., xn The term I ,,, N T A to the left of `|’ (which should be read as such that) says that every tuple I ,,, N T A There is an important restriction: the variables that satisfies T>7 is in the answer. x1, ..., xn that appear to the left of `|’ must be the only free variables in the formula p(...). Modify this query to answer: Find sailors who are older than 18 or have a rating under 9, and are called ‘Joe’. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 5 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 6 Find sailors rated > 7 who’ve reserved boat #103 Find sailors rated > 7 who’ve reserved a red boat
I,,,N T A |,,,I N T A ∈∧>∧Sailors T 7 I,,,N T A |,,,I N T A ∈Sailors ∧>∧T 7
∃∈∧=∧IrBrD,, IrBrD ,, Re serves Ir I Br= 103 ∃ Ir,, Br D Ir ,, Br D∈ Re serves∧=∧ Ir I ∃ BBNC,, BBNC ,,∈ Boats∧= B Br ∧ C= ' red ' We have used ∃ Ir ,, Br D () ... as a shorthand for ∃∃Ir Br() ∃ D ()... () Observe how the parentheses control the scope of each quantifier’s binding. Note the use of ∃ to find a tuple in Reserves that This may look cumbersome, but with a good user `joins with’ the Sailors tuple under consideration. interface, it is very intuitive. (MS Access, QBE)
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Find sailors who’ve reserved all boats Find sailors who’ve reserved all boats (again!)
I,,,N T A |,,,I N T A ∈∧Sailors I,,,N T A |,,,I N T A ∈Sailors ∧
∀ B,,BN C ∈ Boats ∀¬∈∨BBNC,, BBNC ,, Boats ∃ Ir,, Br D∈ Re serves I=∧ Ir Br= B ∃∈∧=∧IrBrD,, IrBrD ,, Re serves I Ir Br= B
Find all sailors I such that for each 3-tuple BB ,, N C Simpler notation, same query. (Much clearer!) either it is not a tuple in Boats or there is a tuple in To find sailors who’ve reserved all red boats: Reserves showing that sailor I has reserved it. ..... C≠'' red∨ ∃ Ir ,,Re Br D∈= serves I Ir∧ Br= B
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Unsafe Queries, Expressive Power Summary
It is possible to write syntactically correct calculus queries that have an infinite number of answers! Relational calculus is non-operational, and Such queries are called unsafe. users define queries in terms of what they
want, not in terms of how to compute it. e.g., S| ¬∈ S Sailors (Declarativeness.) It is known that every query that can be expressed Algebra and safe calculus have same in relational algebra can be expressed as a safe expressive power, leading to the notion of query in DRC / TRC; the converse is also true. relational completeness. Relational Completeness: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus. Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 11 Database Management Systems 3ed, R. Ramakrishnan and J. Gehrke 12